phanerozoic/Coq-Infotheo · Datasets at Hugging Face

fact stringlengths 9 8.91k	type stringclasses 20 values	library stringclasses 7 values	imports listlengths 1 12	filename stringclasses 87 values	symbolic_name stringlengths 1 38
location_polynomial_points := \row_i \prod_(j < n \| j != i) (a ``_ i - a ``_ j).	Definition	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]	ecc_classic/alternant.v	location_polynomial_points
b := \row_(i < n) (((location_polynomial_points a) ``_ i)^-1 * g.[a ``_ i]). Variable (r : nat).	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]	ecc_classic/alternant.v	b
GRS_PCM_polynomial := @GRS.PCM _ F a b r.	Definition	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]	ecc_classic/alternant.v	GRS_PCM_polynomial
ext_inj : {rmorphism F0 -> F1} := [the {rmorphism F0 -> F1} of @GRing.in_alg _ _].	Definition	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]	ecc_classic/alternant.v	ext_inj
ext_inj_tmp : {rmorphism F0 -> (FinFieldExtType F1)} := ext_inj. Variable n : nat.	Definition	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]	ecc_classic/alternant.v	ext_inj_tmp
ext_inj_rV : 'rV[F0]_n -> 'rV[F1]_n := @map_mx _ _ ext_inj 1 n.	Definition	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]	ecc_classic/alternant.v	ext_inj_rV
u := u'.+1. Hypothesis primep : prime p.	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]	ecc_classic/alternant.v	u
Fq : finFieldType := GF u primep.	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]	ecc_classic/alternant.v	Fq
q := p ^ u.	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]	ecc_classic/alternant.v	q
p_char : p \in [pchar Fq]. Proof. apply: char_GFqm. Qed. (** declare F_{q^m} *) Variable m' : nat.	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]	ecc_classic/alternant.v	p_char
m := m'.+1. Variable Fqm : fieldExtType Fq. Hypothesis card_Fqm : #\| FinFieldExtType Fqm \| = q ^ m. (** build GRS_k(kappa, g) *) Variable n : nat. Variable a : 'rV[Fqm]_n. Variable g : {poly Fqm}. Variable k : nat.	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]	ecc_classic/alternant.v	m
alternant_PCM : 'M_(k, n) := @GRS_PCM_polynomial n (FinFieldExtType Fqm) a g k.	Definition	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]	ecc_classic/alternant.v	alternant_PCM
alternant_code := Rcode.t (@ext_inj_tmp Fq Fqm) (kernel alternant_PCM). (** Goppa codes are a special case of alternant codes *)	Definition	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]	ecc_classic/alternant.v	alternant_code
goppa_code_condition := size g = (n - k).+1.	Definition	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]	ecc_classic/alternant.v	goppa_code_condition
u := u'.+1. Hypothesis primep : prime p.	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]	ecc_classic/alternant.v	u
Fq : finFieldType := GF u primep.	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]	ecc_classic/alternant.v	Fq
q := p ^ u.	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]	ecc_classic/alternant.v	q
p_char : p \in [pchar Fq]. Proof. apply: char_GFqm. Qed. (** declare F_{q^m} *) Variable m' : nat.	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]	ecc_classic/alternant.v	p_char
m := m'.+1. Variable Fqm : fieldExtType Fq. Hypothesis card_Fqm : #\| FinFieldExtType Fqm \| = q ^ m. (** we are talking about narrow-sense Goppa codes *)	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]	ecc_classic/alternant.v	m
n : nat := (q^m).-1. Variable e : Fqm. Hypothesis e_prim : n.-primitive_root e.	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]	ecc_classic/alternant.v	n
a : 'rV[Fqm]_n := rVexp e n. Variable t : nat. (** we have to instantiate Goppa codes with a monomial to recover BCH codes *)	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]	ecc_classic/alternant.v	a
g : {poly (FinFieldExtType Fqm)} := 'X^(n - t). (** from the Goppa code condition, we have only one choice for its degree *)	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]	ecc_classic/alternant.v	g
goppa_code_condition_check : goppa_code_condition n g t. Proof. by rewrite /goppa_code_condition size_polyXn. Qed. (* NB: we only have binary BCH codes, so we should maybe restrict q at this point ) (* wip *)	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]	ecc_classic/alternant.v	goppa_code_condition_check
narrow_sense_BCH_are_Goppa : @BCH.PCM (FinFieldExtType _) _ a t = @alternant_PCM _ u' primep Fqm _ a g t(?). Proof. rewrite /BCH.code /alternant_code. Abort.	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]	ecc_classic/alternant.v	narrow_sense_BCH_are_Goppa
PCM : 'M_(t, n) := \matrix_(i < t, j < n) (a ``_ j) ^+ i.*2.+1.	Definition	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	PCM
PCM_alt : 'M[F]_(t.2, n) := \matrix_(i < t.2, j < n) (a ``_ j) ^+ i.+1.	Definition	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	PCM_alt
PCM_alt_GRS : PCM_alt = GRS.PCM a a t.*2. Proof. apply/matrixP => i j. rewrite !mxE (bigD1 j) //= !mxE eqxx mulr1n exprS mulrC. rewrite (eq_bigr (fun=> 0)) ?big_const ?iter_addr0 ?addr0 // => k kj. by rewrite !mxE (negbTE kj) mulr0n mulr0. Qed.	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	PCM_alt_GRS
m := m'.+1.	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	m
F := GF2 m. Variable a : 'rV[F]_n. Variable t : nat.	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	F
BCH_PCM_altP2 (x : 'rV[F]_n) : (forall i : 'I_t.2, \sum_j x ``_ j a ``_ j ^+ i.+1 = 0) -> (forall i : 'I_t,\sum_j x ``_ j * a ``_ j ^+ i.2.+1 = 0). Proof. move=> H i. have @j : 'I_t.2 by refine (@Ordinal _ i.*2 _); rewrite -!muln2 ltn_pmul2r. rewrite -[RHS](H j); by apply: eq_bigr => /= k _. Qed.	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	BCH_PCM_altP2
BCH_PCM_altP1 (x : 'rV['F_2]_n) : (forall i : 'I_t, \sum_j GF2_of_F2 x ``_ j * a ``_ j ^+ i.2.+1 = 0) -> (forall i : 'I_t.2, \sum_j GF2_of_F2 x ``_ j * a ``_ j ^+ i.+1 = 0). Proof. move=> H [i]. elim: i {-2}i (leqnn i) => [\|i IH j ji i1]. move=> i; rewrite leqn0 => /eqP -> i0. destruct t. exfalso; by rewrite -muln2 mul0n ltnn in i0. by rewrite -[RHS](H ord0); apply: eq_bigr. case/boolP : (odd j) => [odd_j\|even_j]; last first. have j2t : j./2 < t by rewrite -divn2 ltn_divLR // muln2. rewrite -[in RHS](H (Ordinal j2t)) /=. apply/eq_bigr => k _. move: (odd_double_half j). by rewrite (negbTE even_j) add0n => ->. move: (IH j.-1./2). rewrite -{1}divn2 leq_divLR; last first. rewrite dvdn2 -subn1 oddB; last by destruct j. by rewrite /= addbT odd_j. have -> : (j.-1 <= i * 2)%N. rewrite muln2 -addnn -subn1 leq_subLR addnA add1n (leq_trans ji) //. by rewrite addSn ltnS leq_addr. move/(_ isT). have j2t : (j.-1)./2 < t.2. rewrite -divn2 ltn_divLR // -ltnS. destruct j => //. by rewrite /= (ltn_trans i1) // ltnS muln2 -[in X in _ <= X]addnn leq_addl. move/(_ j2t)/(congr1 (fun x => x ^+ 2)). rewrite expr0n /= sum_sqr ?char_GFqm // => H'. rewrite -[RHS]H'; apply: eq_bigr => k _. rewrite exprMn_comm; last by rewrite /GRing.comm mulrC. congr (_ _); last first. rewrite /= -exprM muln2; congr (_ ^+ _). move: (odd_double_half j). rewrite odd_j add1n => <-. by rewrite (half_bit_double (j./2) false) -doubleS. by rewrite (expr2 (GF2_of_F2 x ``_ k)) -rmorphM -expr2 f2.expr2_char2. Qed.	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	BCH_PCM_altP1
PCM_altP (x : 'rV_n) : (syndrome (BCH.PCM a t) (F2_to_GF2 m x) == 0) = (syndrome (BCH.PCM_alt a t) (F2_to_GF2 m x) == 0). Proof. apply/idP/idP. - rewrite /BCH.PCM /BCH.PCM_alt /syndrome => H. suff H' : forall j : 'I_t.2, \sum_j0 (GF2_of_F2 (x ``_ j0) (a ``_ j0) ^+ j.+1) = 0. apply/eqP/rowP => j; rewrite !mxE. rewrite -[RHS](H' j). apply: eq_bigr => /= i _. by rewrite !mxE mulrC. move=> j. apply: BCH_PCM_altP1 => i. move/eqP/rowP : H => /(_ i). rewrite !mxE => H; rewrite -[RHS]H. by apply: eq_bigr => /= k _; rewrite !mxE /= mulrC. - rewrite /BCH.PCM_alt /BCH.PCM /syndrome => H. apply/eqP/rowP => i. have @j : 'I_t.2. by refine (@Ordinal _ i.2 _); rewrite -!muln2 ltn_pmul2r. move/eqP : H => /matrixP/(_ ord0 j). rewrite !mxE => {2}<-. by apply: eq_bigr => k _; rewrite !mxE. Qed.	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	PCM_altP
code (a : 'rV_n) t := Rcode.t (@GF2_of_F2 m) (kernel (PCM a t)).	Definition	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	code
n := n'.+1. Variable (m : nat).	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	n
F : fieldType := GF2 m. Implicit Types a : 'rV[F]_n.	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	F
syndromep a y t := syndromep a (F2_to_GF2 m y) t.*2.	Definition	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	syndromep
F := GF2 m. Variable (n' : nat).	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	F
n := n'.+1. Variable a : F. Variable (t : nat). Hypothesis tn : t.*2 < n.	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	n
fdcoor_syndrome_coord (y : 'rV['F_2]_n) i (it : i < t.2) : fdcoor (rVexp a n) (F2_to_GF2 m y) (inord i.+1) = GRS.syndrome_coord (rVexp a n) (rVexp a n) i (F2_to_GF2 m y). Proof. rewrite /fdcoor /GRS.syndrome_coord horner_poly; apply: eq_bigr => j _. rewrite insubT // => jn. rewrite !mxE -mulrA; congr (GF2_of_F2 (y _ _) _); first by apply: val_inj. by rewrite -exprS -exprM mulnC exprM inordK // ltnS (leq_trans it). Qed.	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	fdcoor_syndrome_coord
BCH_syndromep_is_GRS_syndromep y : \BCHsynp_(rVexp a n, y, t) = GRS.syndromep (rVexp a n) (rVexp a n) t.*2 (F2_to_GF2 m y). Proof. apply/polyP => i. by rewrite !coef_poly; case: ifPn => // it; rewrite fdcoor_syndrome_coord. Qed.	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	BCH_syndromep_is_GRS_syndromep
n := n'.+1.	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	n
F := GF2 m. Variable a : F. Variable t : nat. Variable BCHcode : BCH.code (rVexp a n) t. Hypothesis tn : t <= n.-1./2.	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	F
BCH_codebook (c : 'rV['F_2]_n) : (c \in BCHcode) = [forall j : 'I_t, (map_mx (@GF2_of_F2 m) c) ^`_(rVexp a n, inord j.2.+1) == 0]. Proof. apply/idP/idP. - case: BCHcode => code condition /= /condition. rewrite mem_kernel_syndrome0 -trmx0 => /eqP/trmx_inj/matrixP Hc. apply/forallP => /= i; apply/eqP; rewrite /fdcoor horner_poly. move: (Hc i ord0); rewrite !mxE => H1; rewrite -[RHS]H1. apply/eq_bigr => /= j _; rewrite insubT // => jn. rewrite mulrC 3![in RHS]mxE; congr (_ (map_mx _ c) ord0 _); last by apply: val_inj. rewrite inordK //; last first. move: (ltn_ord i). rewrite -(@ltn_pmul2r 2) // !muln2 -(ltn_add2r 1) !addn1. move/leq_trans; apply. move: tn. rewrite -divn2 leq_divRL // muln2. move/leq_ltn_trans; by apply. by rewrite exprAC. - move/forallP => H. case: BCHcode => code condition /=; apply: (proj2 (condition _)). rewrite mem_kernel_syndrome0; apply/eqP/rowP => i; rewrite !mxE. rewrite -[RHS](eqP (H i)) /fdcoor horner_poly; apply/eq_bigr => /= j _. rewrite insubT // => jn. rewrite !mxE mulrC; congr (GF2_of_F2 (c ord0 _) * _); first by apply: val_inj. rewrite inordK //; first by rewrite exprAC. move: (ltn_ord i). rewrite -(@ltn_pmul2r 2) // !muln2 -(ltn_add2r 1) !addn1. move/leq_trans; apply. move: tn. by rewrite -divn2 leq_divRL // muln2 => /leq_ltn_trans; exact. Qed.	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	BCH_codebook
n := n'.+1.	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	n
F := GF2 m. Variable a : F. Variable t' : nat.	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	F
t := t'.+1.	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	t
BCH_syndrome_synp y : t.2 < n -> (syndrome (BCH.PCM (rVexp a n) t) (F2_to_GF2 m y) == 0) = (\BCHsynp_(rVexp a n, y, t) == 0). Proof. move=> tn. apply/idP/idP. - move/eqP/eqP. rewrite BCH.PCM_altP // /BCH.PCM_alt. move/eqP => K. rewrite /BCH.syndromep /syndromep. rewrite poly_def (eq_bigr (fun=> 0)) ?big_const ?iter_addr0 // => /= i _. apply/eqP; rewrite scaler_eq0. rewrite fdcoorE -size_poly_eq0 size_polyXn /= orbF. move/rowP : K => /(_ (inord i)) K. apply/eqP. rewrite !mxE in K. rewrite -[RHS]K. apply: eq_bigr => k _. rewrite !mxE mulrC; congr (_ _). rewrite -exprM mulnC exprM; congr (a ^+ _ ^+ _). by rewrite inord_val // inordK // (leq_ltn_trans (ltn_ord i)). - move=> K. apply/eqP; apply/eqP. rewrite BCH.PCM_altP //; apply/eqP. rewrite /BCH.PCM_alt. apply/rowP => i. rewrite !mxE. move/eqP: K. move/(congr1 (fun x : {poly F} => x`_i)). rewrite /BCH.syndromep /syndromep. rewrite coef0 poly_def coef_sum (bigD1 i) //= (eq_bigr (fun=> 0)); last first. move=> j ji. rewrite coefZ coefXn (_ : _ == _ = false) ?mulr0 //. by apply/negbTE; rewrite eq_sym. rewrite big_const iter_addr0 addr0 fdcoorE coefZ coefXn eqxx mulr1 => K. rewrite -[RHS]K. apply: eq_bigr => k _. rewrite !mxE mulrC; congr (_ * _). by rewrite -exprM mulnC exprM inordK // (leq_ltn_trans _ tn). Qed.	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	BCH_syndrome_synp
F := GF2 m. Variable a : 'rV[F]_n.	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	F
BCH_det_mlinear t r' (f : 'I_r'.+1 -> 'I_n) (rt : r' < t.2) : let B := \matrix_(i, j) BCH.PCM_alt a t (widen_ord rt i) (f j) in let V := Vandermonde r'.+1 (row 0 B) in \det B = \prod_(i < r'.+1) BCH.PCM_alt a t (widen_ord rt 0) (f i) \det V. Proof. move=> B V. set h := Vandermonde r'.+1 (row 0 B). set g := fun i => BCH.PCM_alt a t (widen_ord (rt) 0) (f i). transitivity (\det (\matrix_(i, j) (h i j * g j))). congr (\det _). apply/matrixP => i j. by rewrite !mxE /h /g /BCH.PCM_alt !mxE -!exprD /= -exprM mul1n addn1. rewrite det_mlinear; congr (_ * _). by congr (\det _); apply/matrixP => i j; rewrite !mxE /h -exprM. Qed. Hypothesis a_neq0 : distinct_non_zero a.	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	BCH_det_mlinear
det_B_neq0 t r f (Hr' : r.+1 < t.2) (Hinj : injective f) : let B := \matrix_(i < r.+1, j < r.+1) BCH.PCM_alt a t (widen_ord (ltnW Hr') i) (f j) in \det B != 0. Proof. rewrite /=; set B := \matrix_(_, _) _. rewrite BCH_det_mlinear det_Vandermonde mulf_neq0 //. - rewrite prodf_seq_neq0 /=; apply/allP => /= j _. by rewrite !mxE /= expr1 (proj2 a_neq0). - rewrite prodf_seq_neq0 /=; apply/allP => /= j _. rewrite prodf_seq_neq0 /=; apply/allP => /= k _. apply/implyP => jk; rewrite !mxE /= !expr1 subr_eq0. apply/negP => /eqP; move: (proj1 a_neq0) => /(_ (f k) (f j)). by rewrite 2!ffunE => akj /akj/Hinj; exact/eqP/negbT/gtn_eqF. Qed. (* parity-check matrix of a binary (n, k) code capable ot correcting all error patterns of weight <= t with dimension k >= n - m * t (see [McEliece 2002], thm 9.1) *)	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	det_B_neq0
BCH_min_dist1 t x (x0 : x != 0) (t0 : 0 < t) (C : BCH.code a t) : x \in C -> t < wH x. Proof. move/(proj1 (Rcode.P C _)). rewrite mem_kernel_syndrome0 BCH.PCM_altP // => /eqP H. (* there are r <= t columns in the PCM s.t. their linear comb. w.r.t. is 0 ) rewrite leqNgt; apply/negP => abs. have /eqP : #\| wH_supp x \| = (wH x).-1.+1. by rewrite card_wH_supp prednK // lt0n wH_eq0. rewrite cardE => supp_wH. have Hinj : injective [ffun i : 'I_(wH x).-1.+1 => tnth (Tuple supp_wH) i]. move=> /= i j. rewrite !ffunE /tnth /= => /eqP. rewrite (set_nth_default (tnth_default (Tuple supp_wH) j)) //; last first. rewrite -cardE card_wH_supp; move: i; by rewrite prednK // lt0n wH_eq0. rewrite nth_uniq ?enum_uniq //. by move/eqP/val_inj. rewrite -cardE card_wH_supp; move: i; by rewrite prednK // lt0n wH_eq0. rewrite -cardE card_wH_supp; move: j; by rewrite prednK // lt0n wH_eq0. set f := finfun (tnth (Tuple supp_wH)). move=> [:tmp]. have @f' : 'I_n -> 'I_(wH x).-1.+1. move=> i. set tmp' := fun i => if x ``_ i != 0 then index i (enum (wH_supp x)) else O. refine (@Ordinal _ (tmp' i) _). move: i. abstract: tmp. move=> i. case: ifP => // H'. by rewrite -(eqP supp_wH) index_mem mem_enum -mem_wH_supp. have f'K : forall j, x ``_ j != 0 -> f (f' j) = j. move=> i xj. by rewrite ffunE /tnth /= xj nth_index // mem_enum -mem_wH_supp. have Hf : \sum_(i < (wH x).-1.+1) (GF2_of_F2 x``_(f i)) : col (f i) (BCH.PCM_alt a t) = 0 :> 'cV__. move/(congr1 trmx) : H; rewrite trmx0 => H. rewrite -[RHS]H. apply/colP => i. rewrite ![in RHS]mxE ![in LHS]summxE [in RHS](bigID (fun j => x ``_ j == 0)). rewrite /= [X in _ = X + _](_ : _ = 0) ?add0r; last first. rewrite (eq_bigr (fun x=> 0)); last first. by move=> j /eqP xj0; rewrite !mxE xj0 rmorph0 mulr0. by rewrite big_const iter_addr0. have ? : (wH x).-1.+1 <= n by rewrite -ltnS prednK ?lt0n ?wH_eq0 // ltnS max_wH. rewrite [in RHS](reindex_onto (fun x => f x) f' f'K) /= [in RHS]big_mkcond /=. apply/eq_bigr => j _. case: ifPn => [_\|]; first by rewrite!mxE mulrC. rewrite negb_and negbK. case/orP => [/eqP ->\|abs']; first by rewrite rmorph0 scale0r mxE. have [->\|abs''] := eqVneq (x ``_ (f j)) 0. by rewrite rmorph0 scale0r mxE. exfalso. move/eqP: abs'; apply. rewrite /f'; apply/val_inj => /=. rewrite abs'' /f ffunE /tnth /= index_uniq //. by rewrite -cardE card_wH_supp -[in X in _ < X](@prednK (wH x)) // lt0n wH_eq0. by rewrite enum_uniq. set r' := (wH x).-1. have Hr' : r'.+1 < t.2. by rewrite /r' prednK // ?lt0n ?wH_eq0 // (leq_trans abs) // -addnn -{1}(add0n t) ltn_add2r. have {}Hf : \sum_(i < r'.+1) GF2_of_F2 x ``_ (f i) : \col_(j < r'.+1) (BCH.PCM_alt a t (widen_ord (ltnW Hr') j) (f i)) = 0. apply/colP => j. rewrite mxE summxE. move/colP : Hf => /(_ (widen_ord (ltnW Hr') j)). rewrite !mxE summxE => Hf. rewrite -[RHS]Hf. by apply: eq_bigr => /= i _; rewrite !mxE. have /negP := det_B_neq0 Hr' Hinj. rewrite -det_tr; apply. apply/det0P; exists (\row_i GF2_of_F2 x ``_ (f i)). suff -> : \row_i GF2_of_F2 x ``_ (f i) = const_mx 1. apply/eqP => /rowP/(_ 0). rewrite !mxE; apply/eqP; exact: oner_neq0. move=> n0; apply/rowP => k. rewrite !mxE /f ffunE /tnth /=. have -> : forall i, i \in (enum (wH_supp x)) -> x ``_ i = 1. by move=> /= i; rewrite mem_enum inE -f2.F2_eq1 => /eqP. by rewrite rmorph1. apply/(nthP (tnth_default (Tuple supp_wH) k)); exists k => //. rewrite -cardE card_wH_supp. move: (ltn_ord k); by rewrite {2}prednK // lt0n wH_eq0. apply/rowP => i; rewrite !mxE. move/colP : Hf => /(_ i). rewrite !mxE => Hf; rewrite -[RHS]Hf. by rewrite summxE; apply: eq_bigr=> k _; rewrite !mxE. Qed.	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	BCH_min_dist1
BCH_erreval := erreval (const_mx 1) a.	Definition	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	BCH_erreval
n := n'.+1. Variable a : F. (** see [McEliece 2002], thm 9.4 *)	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	n
BCH_key_equation_old (y : 'rV[F]_n) : a ^+ n = 1 -> \sigma_(rVexp a n, y) * rVpoly (dft (rVexp a n) n y) = \BCHomega_(rVexp a n, y) * (1 - 'X^n). Proof. move=> an1. rewrite dftE big_distrr /=. under eq_bigr. move=> i ie. rewrite -scalerAr. rewrite (_ : \sigma_(rVexp a n, y) = \sigma_(rVexp a n, y, i) * (1 - ((rVexp a n) ``_ i) : 'X)); last first. rewrite /errloc (bigD1 i) //= mulrC; congr (_ _). by apply: eq_bigl => ij; rewrite in_setD1 andbC. over. transitivity (\sum_(i in supp y) y ``_ i : (\sigma_(rVexp a n, y, i) (1 - a ^+ (i * n) : 'X^n))). apply: eq_bigr => /= i ie; congr (_ : _). rewrite -mulrA; congr (_ * _). rewrite mulrDl mul1r mulNr big_distrr /=. rewrite [X in _ - X](eq_bigr (fun j : 'I_n => a ^+ (i * j.+1) : 'X^(j.+1))); last first. move=> j _. rewrite !mxE -scalerAr -scalerAl scalerA -exprM -exprD. by rewrite inord_val addnC -mulSn mulnC -exprS. rewrite big_ord_recl !mxE inord_val expr0 expr1n scale1r expr0. rewrite big_ord_recr /= opprD addrA. rewrite (_ : forall a b c d, b = c -> a + b - c - d = a - d) //. move=> a0 b c d -> //; by rewrite addrK. apply: eq_bigr => j _; by rewrite mxE -exprM mulnC inordK // ltnS. rewrite /BCH_erreval [in RHS]big_distrl /=. apply: eq_bigr => i ie. rewrite mxE -scalerAl mulr1; congr (_ : (_ * _)). by rewrite mulnC exprM an1 expr1n scale1r. Qed.	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	BCH_key_equation_old
n := n'.+1.	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	n
F := GF2 m. Variable a : F. Variable t' : nat.	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	F
t := t'.+1.	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	t
H := BCH.PCM (rVexp a n) t. Hypothesis a_neq0 : a != 0.	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	H
BCH_mod (y : 'rV[F]_n) : {poly F} := \sum_(i in supp y) y ``_ i : (\prod_(j in supp y :\ i) (1 - a ^+ j : 'X) * - (a ^+ (t.2.+1 i))%:P).	Definition	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	BCH_mod
BCH_mod_is_GRS_mod y : BCH_mod y = GRS_mod y (rVexp a n) (rVexp a n) t.2. Proof. rewrite /BCH_mod /GRS_mod; apply/eq_bigr => i iy. rewrite !mulrN scalerN; congr (- _). rewrite -!scalerAl; congr (_ : _). rewrite mxE -mulrA; congr (_ * _). apply/eq_big. by move=> j; rewrite in_setD1 andbC. move=> j _; by rewrite mxE. by rewrite -polyCM -exprSr -exprM mulnC. Qed. Hypothesis (tn : t.*2 < n).	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	BCH_mod_is_GRS_mod
BCH_key_equation y : \sigma_(rVexp a n, F2_to_GF2 m y) * \BCHsynp_(rVexp a n, y, t) = \BCHomega_(rVexp a n, twisted a (F2_to_GF2 m y)) + BCH_mod (F2_to_GF2 m y) * 'X^(t.2). Proof. move: (@GRS_key_equation F n (F2_to_GF2 m y) (rVexp a n) (rVexp a n) t.2) => H0. rewrite BCH_syndromep_is_GRS_syndromep // H0; congr (_ + _ * _). rewrite /Omega /BCH_erreval; apply/polyP => i. rewrite !coef_sum supp_twisted //. apply: eq_bigr => /= j jy. by rewrite !mxE mulr1. by rewrite -BCH_mod_is_GRS_mod. Qed.	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	BCH_key_equation
errloc_twisted (y : 'rV[F]_n) : errloc (rVexp a n) (supp y) = errloc (rVexp a n) (supp (twisted a y)). Proof. by rewrite /errloc supp_twisted. Qed.	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	errloc_twisted
size_erreval_twisted (e : 'rV_n) : #\|supp (F2_to_GF2 m e)\| <= t -> size (\omega_(const_mx 1, rVexp a n, twisted a (F2_to_GF2 m e))) <= t. Proof. move=> et. have et' : #\|supp (twisted a (F2_to_GF2 m e))\| <= t by rewrite supp_twisted. by apply: (size_erreval (rVexp a n) (Errvec et') (const_mx 1)). Qed. Local Notation "'v'" := (Euclid.v).	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	size_erreval_twisted
BCH_err y : {poly 'F_2} := let r0 : {poly F} := 'X^(t.2) in let r1 := \BCHsynp_(rVexp a n, y, t) in let vstop := v r0 r1 (stop t r0 r1) in let s := vstop.[0]^-1 : vstop in \poly_(i < n) (if s.[a^- i] == 0 then 1 else 0).	Definition	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	BCH_err
BCH_repair : repairT 'F_2 'F_2 n := [ffun y => if \BCHsynp_(rVexp a n, y, t) == 0 then Some y else let ret := y + poly_rV (BCH_err y) in if \BCHsynp_(rVexp a n, ret, t) == 0 then Some ret else None].	Definition	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	BCH_repair
BCH_err_is_correct (a_not_uroot : not_uroot_on a n) l (e y : 'rV_n) : let r0 := 'X^(t.2) : {poly F} in let r1 := \BCHsynp_(rVexp a n, y, t) in let vj := Euclid.v r0 r1 (stop t r0 r1) in l <> 0 -> vj = l : \sigma_(rVexp a n, F2_to_GF2 m e) -> e = poly_rV (BCH_err y). Proof. have H1 := distinct_non_zero_rVexp a_neq0 a_not_uroot. move=> r0 r1 vj /eqP l0 Hvj; apply/rowP => i. set r_ := \BCHomega_(rVexp a n, twisted a (F2_to_GF2 m e)). rewrite !mxE coef_poly ltn_ord; case: ifPn. rewrite -/r0 -/r1 -/r_ -/vj Hvj. rewrite !hornerZ !mulf_eq0 invrM; last 2 first. by rewrite unitfE. by rewrite unitfE horner_errloc_0 oner_neq0. rewrite mulf_eq0 !invr_eq0 (negbTE l0) /= orbF horner_errloc_0 oner_eq0 /=. move: (errloc_zero (supp (F2_to_GF2 m e)) i H1); rewrite mxE => ->. by rewrite supp_F2_to_GF2 inE -F2_eq1 => /eqP. rewrite -/r0 -/vj in Hvj *. rewrite Hvj !hornerZ !mulf_eq0 invrM; last 2 first. by rewrite unitfE. by rewrite unitfE horner_errloc_0 oner_neq0. rewrite mulf_eq0 !invr_eq0 (negbTE l0) /= orbF horner_errloc_0 oner_eq0 /=. move: (errloc_zero (supp (F2_to_GF2 m e)) i H1); rewrite mxE => ->. by rewrite supp_F2_to_GF2 inE negbK => /eqP ->. Qed. Local Open Scope ecc_scope.	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	BCH_err_is_correct
BCH_repair_is_correct (C : BCH.code (rVexp a n) t) : not_uroot_on a n -> t.-BDD (C, BCH_repair). Proof. move=> a_not_uroot c e. set y := c + e. set r0 : {poly F} := 'X^(t.2). set r1 := \BCHsynp_(rVexp a n, y, t). set vj := Euclid.v r0 r1 (stop t r0 r1). move=> Hc et. set r : {poly F} := \BCHomega_(rVexp a n, twisted a (F2_to_GF2 m e)). have H1 := distinct_non_zero_rVexp a_neq0 a_not_uroot. rewrite /BCH_repair. have syn_c : \BCHsynp_(rVexp a n, c, t) = 0. move/(proj1 (Rcode.P C _)) : Hc. by rewrite mem_kernel_syndrome0 BCH_syndrome_synp // => /eqP ->. have syn_y_e : \BCHsynp_(rVexp a n, y, t) = \BCHsynp_(rVexp a n, e, t). rewrite {1}/BCH.syndromep /F2_to_GF2 map_mxD syndromepD. by rewrite -!/(BCH.syndromep _ _ _) syn_c add0r. rewrite ffunE. case: ifPn => syn_y. suff e0 : e = 0 by rewrite /y e0 addr0. apply/eqP/negPn/negP => e0. move: et. apply/negP. rewrite -ltnNge (@BCH_min_dist1 _ _ _ H1 _ _ e0 _ C) //. rewrite (_ : e = y - c); last by rewrite /y addrAC subrr add0r. apply: (@Lcode0.aclosed _ _ _).2 => //; last by rewrite Lcode0.oclosed. apply: (proj2 (Rcode.P C _)). by rewrite mem_kernel_syndrome0 BCH_syndrome_synp. have size_r1 : size r1 <= size ('X^(t.2) : {poly F}). by rewrite /r1 /BCH.syndromep size_polyXn (leq_trans (size_syndromep _ _ _)). have eqn : \sigma_(rVexp a n, F2_to_GF2 m e) * r1 = r + - - BCH_mod (F2_to_GF2 m e) * 'X^(t.2). by rewrite opprK -(BCH_key_equation e) /r1 syn_y_e. have size_sigma : size \sigma_(rVexp a n, F2_to_GF2 m e) <= t.+1. by rewrite (leq_trans (size_errloc _ _)) // ltnS supp_F2_to_GF2 -wH_card_supp. have size_r : size r <= t. move: et. rewrite /r wH_card_supp -(@supp_F2_to_GF2 _ m _) -(supp_twisted a_neq0) => et'. by rewrite (leq_trans (size_erreval (rVexp a n) (Errvec et') _)). have deg : (t.+1 + t)%N = size ('X^(t.2) : {poly F}). by rewrite size_polyXn addSn addnn. have cop : coprimep \sigma_(rVexp a n, F2_to_GF2 m e) r. have tmp : (forall i, ((const_mx 1) : 'rV[F]_n) ``_ i != 0 :> F). by move=> ?; rewrite mxE ?oner_neq0. rewrite /r errloc_twisted; by apply: (coprime_errloc_erreval H1 tmp). case: (solve_key_equation_coprimep size_r1 (errloc_neq0 _ _) syn_y eqn size_sigma size_r deg cop) => [l [l0 [Hvj _]]]. by rewrite -(@BCH_err_is_correct _ l e) // -addrA F2_addmx addr0 syn_c eqxx. Qed.	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	BCH_repair_is_correct
n := n'.+1.	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	n
F := GF2 m. Variable a : F. Variable t : nat. Hypothesis tn : t <= n.-1./2. Local Open Scope cyclic_code_scope.	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	F
rcsP_BCH_cyclic (C : BCH.code (rVexp a n) t) : a ^+ n = 1 -> rcsP [set cw in C]. Proof. move=> a1 x. rewrite inE (BCH_codebook C tn) => /forallP /= Hx. rewrite inE (BCH_codebook C tn); apply/forallP => /= i. rewrite map_mx_rcs rcs_rcs_poly /rcs_poly. set x' := map_mx _ _. have [M HM] : exists M, `[ 'X * rVpoly x' ]_ n = 'X * rVpoly x' + M * ('X^n - 1). exists (- ('X * rVpoly x') %/ ('X^n - 1)). move/eqP: (divp_eq ('X * rVpoly x') ('X^n - 1)); rewrite addrC -subr_eq => /eqP <-. by rewrite divpN mulNr. rewrite HM /fdcoor poly_rV_K //; last first. rewrite -HM. move: (ltn_modp ('X * rVpoly x') ('X^n - 1)). rewrite size_XnsubC // ltnS => ->. by rewrite monic_neq0 // monicXnsubC. rewrite !(hornerE,hornerXn). move: (Hx i); rewrite /fdcoor => /eqP ->; rewrite mulr0 add0r. by rewrite mxE exprAC a1 expr1n subrr mulr0. Qed.	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv finalg zmodp.", "From mathcomp Require Import matrix mxalgebra mxpoly vector fieldext finfield.", "Require Import ssralg_ext hamming linearcode decoding cyclic_code poly_decoding.", "Require Import dft euclid grs f2." ]	ecc_classic/bch.v	rcsP_BCH_cyclic
rcs_perm_ffun n : {ffun 'I_n.+1 -> 'I_n.+1} := [ffun x : 'I_n.+1 => if x == ord0 then ord_max else inord x.-1].	Definition	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv fingroup perm.", "From mathcomp Require Import finalg zmodp matrix mxalgebra mxpoly polydiv.", "From mathcomp Require Import vector.", "Require Import ssralg_ext poly_ext f2 hamming linearcode dft." ]	ecc_classic/cyclic_code.v	rcs_perm_ffun
rcs_perm_ffun_injectiveb n : injectiveb (@rcs_perm_ffun n). Proof. apply/injectiveP => x y. rewrite /rcs_perm_ffun 2!ffunE. case: ifPn => [/eqP ->\|x0]. case: ifPn => [/eqP -> //\|y0]. move/eqP; rewrite -val_eqE /= => /eqP n0y. exfalso; move: (ltn_ord y). rewrite ltnNge => /negP; apply. rewrite [in X in X < _]n0y /= inordK //; first by rewrite prednK // lt0n. by apply: ltn_trans (ltn_ord y); rewrite prednK // lt0n. case: ifPn => [/eqP -> //\|y0 /eqP]. move/eqP; rewrite -val_eqE /= => /eqP n0x. exfalso; move: (ltn_ord x). rewrite ltnNge => /negP; apply. rewrite -[in X in X < _]n0x /= inordK //; first by rewrite prednK // lt0n. apply: ltn_trans (ltn_ord x); by rewrite prednK // lt0n. rewrite -val_eqE /= inordK; last first. by rewrite (ltn_trans _ (ltn_ord x))// prednK// lt0n. rewrite inordK; last first. by rewrite (ltn_trans _ (ltn_ord y))// prednK // lt0n. move/eqP/(congr1 S). by rewrite prednK ?lt0n // prednK ?lt0n // => ?; exact: val_inj. Defined.	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv fingroup perm.", "From mathcomp Require Import finalg zmodp matrix mxalgebra mxpoly polydiv.", "From mathcomp Require Import vector.", "Require Import ssralg_ext poly_ext f2 hamming linearcode dft." ]	ecc_classic/cyclic_code.v	rcs_perm_ffun_injectiveb
rcs_perm n : {perm 'I_n} := if n is n.+1 then Perm (rcs_perm_ffun_injectiveb n) else 1.	Definition	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv fingroup perm.", "From mathcomp Require Import finalg zmodp matrix mxalgebra mxpoly polydiv.", "From mathcomp Require Import vector.", "Require Import ssralg_ext poly_ext f2 hamming linearcode dft." ]	ecc_classic/cyclic_code.v	rcs_perm
rcs (R : idomainType) n (x : 'rV[R]_n) := col_perm (rcs_perm n) x.	Definition	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv fingroup perm.", "From mathcomp Require Import finalg zmodp matrix mxalgebra mxpoly polydiv.", "From mathcomp Require Import vector.", "Require Import ssralg_ext poly_ext f2 hamming linearcode dft." ]	ecc_classic/cyclic_code.v	rcs
size_rcs (R : idomainType) n (x : 'rV[R]_n.+1) : size (rVpoly (rcs x)) < size ('X^(n.+1) - 1%:P : {poly R}). Proof. by rewrite size_XnsubC // ltnS /rVpoly size_poly. Qed.	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv fingroup perm.", "From mathcomp Require Import finalg zmodp matrix mxalgebra mxpoly polydiv.", "From mathcomp Require Import vector.", "Require Import ssralg_ext poly_ext f2 hamming linearcode dft." ]	ecc_classic/cyclic_code.v	size_rcs
map_mx_rcs (R0 R1 : idomainType) n (x : 'rV[R0]_n.+1) (f : R0 -> R1) : map_mx f (rcs x) = rcs (map_mx f x). Proof. by rewrite /rcs map_col_perm. Qed.	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv fingroup perm.", "From mathcomp Require Import finalg zmodp matrix mxalgebra mxpoly polydiv.", "From mathcomp Require Import vector.", "Require Import ssralg_ext poly_ext f2 hamming linearcode dft." ]	ecc_classic/cyclic_code.v	map_mx_rcs
rcs' (R : idomainType) n (x : 'rV[R]_n.+1) := \row_(i < n.+1) (if i == ord0 then x ord0 (inord n) else x ord0 (inord i.-1)).	Definition	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv fingroup perm.", "From mathcomp Require Import finalg zmodp matrix mxalgebra mxpoly polydiv.", "From mathcomp Require Import vector.", "Require Import ssralg_ext poly_ext f2 hamming linearcode dft." ]	ecc_classic/cyclic_code.v	rcs'
rcs'0 (R : idomainType) n (x : 'rV[R]_n.+1) : (rcs' x == 0) = (x == 0). Proof. apply/idP/idP => [\|/eqP ->]. move/eqP/rowP => H; apply/eqP/rowP => i; rewrite !mxE. have ni : i.+1 %% n.+1 < n.+1 by apply: ltn_pmod. move: (H (Ordinal ni)); rewrite !mxE -val_eqE /=. case: ifPn => [in' Hx\|in' Hx]. have ? : n = i. case/dvdnP : in' => -[//\|[\|k abs]] /=; last first. exfalso; move: (ltn_ord i); rewrite ltnNge => /negP; apply. by rewrite -ltnS abs ltn_Pmull. by rewrite mul1n => /eqP; rewrite eqSS => /eqP. rewrite -[RHS]Hx; congr (x _ _); apply/val_inj => /=; by rewrite inordK. rewrite -[RHS]Hx modn_small /=; last first. rewrite ltnS ltn_neqAle -ltnS (ltn_ord i) andbT; apply/eqP => in''. by rewrite in'' modnn eqxx in in'. congr (x _ _); apply: val_inj => /=; by rewrite inordK. apply/eqP/rowP => i; rewrite !mxE; by case: ifPn. Qed.	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv fingroup perm.", "From mathcomp Require Import finalg zmodp matrix mxalgebra mxpoly polydiv.", "From mathcomp Require Import vector.", "Require Import ssralg_ext poly_ext f2 hamming linearcode dft." ]	ecc_classic/cyclic_code.v	rcs'0
rcs'_rcs (R : idomainType) n (x : 'rV[R]_n.+1) : rcs' x = rcs x. Proof. rewrite /rcs' /rcs; apply/rowP => i; rewrite !mxE. case: ifPn => [/eqP -> \|i0]. congr (x _ _). rewrite /rcs_perm/= unlock/= ffunE eqxx. apply/val_inj => /=. by rewrite inordK. congr (x _ _); apply: val_inj => /=. rewrite unlock ffunE /= inordK. by rewrite (negPf i0) inordK // (ltn_trans _ (ltn_ord i)) // prednK // lt0n. by rewrite (ltn_trans _ (ltn_ord i)) // prednK // lt0n. Qed.	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv fingroup perm.", "From mathcomp Require Import finalg zmodp matrix mxalgebra mxpoly polydiv.", "From mathcomp Require Import vector.", "Require Import ssralg_ext poly_ext f2 hamming linearcode dft." ]	ecc_classic/cyclic_code.v	rcs'_rcs
sub_XrVpoly_rcs (R : idomainType) n (x : 'rV[R]_n.+1) : 'X * rVpoly x - rVpoly (rcs x) = (x 0 (inord n))%:P * ('X^(n.+1) - 1). Proof. apply/polyP => i. rewrite coef_add_poly coefXM coefCM coef_add_poly coefXn 2!coef_opp_poly coefC. case: ifPn => [/eqP ->\|i0]; last rewrite subr0. rewrite 2!add0r mulrN1 coef_rVpoly /=. case: insubP => //= j _ j0. rewrite mxE unlock ffunE /= -val_eqE j0 /= j0 eqxx; congr (- x _ _). by apply: val_inj => /=; rewrite inordK. have [->\|in0] := eqVneq i n.+1. rewrite mulr1 2!coef_rVpoly /=. case: insubP => /= [j _ j0\|]; last by rewrite ltnS leqnn. case: insubP => /= [?\|_]; first by rewrite ltnn. rewrite subr0; congr (x _ _); apply: val_inj => /=; by rewrite j0 inordK. rewrite mulr0 2!coef_rVpoly; case: insubP => /= [j\|]. rewrite ltnS => in0' ji; case: insubP => /= [k _ ki\|]. apply/eqP; rewrite subr_eq0; apply/eqP. rewrite mxE unlock ffunE -val_eqE /= ki (negPf i0) -ji; congr (x _ _). by apply/val_inj => /=; rewrite inordK. rewrite ltnS -ltnNge => n0i; exfalso. rewrite -ltnS prednK // in in0'; last by rewrite lt0n. by move/negP : in0; apply; rewrite eq_sym eqn_leq {}n0i. rewrite -leqNgt => n0i; case: insubP => /= [j\|]; last by rewrite subrr. rewrite ltnS => in0' ji; exfalso; move/negP : in0; apply. by rewrite eqn_leq ltnW //= (leq_trans n0i (leq_pred i)). Qed.	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv fingroup perm.", "From mathcomp Require Import finalg zmodp matrix mxalgebra mxpoly polydiv.", "From mathcomp Require Import vector.", "Require Import ssralg_ext poly_ext f2 hamming linearcode dft." ]	ecc_classic/cyclic_code.v	sub_XrVpoly_rcs
rcs_poly (R : idomainType) (p : {poly R}) n := ('X * p) %% ('X^n - 1).	Definition	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv fingroup perm.", "From mathcomp Require Import finalg zmodp matrix mxalgebra mxpoly polydiv.", "From mathcomp Require Import vector.", "Require Import ssralg_ext poly_ext f2 hamming linearcode dft." ]	ecc_classic/cyclic_code.v	rcs_poly
size_rcs_poly (R : idomainType) n (x : {poly R}) (xn : size x <= n) : 0 < n -> size (rcs_poly x n) <= n. Proof. move=> n0. rewrite /rcs_poly. set xn1 := _ - _. apply: (@leq_trans (size xn1).-1); last by rewrite /xn1 size_XnsubC. rewrite -ltnS prednK; last by rewrite size_XnsubC. have : xn1 != 0 by apply/monic_neq0/monicXnsubC. by move/ltn_modpN0; exact. Qed. (* TODO: not used? *)	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv fingroup perm.", "From mathcomp Require Import finalg zmodp matrix mxalgebra mxpoly polydiv.", "From mathcomp Require Import vector.", "Require Import ssralg_ext poly_ext f2 hamming linearcode dft." ]	ecc_classic/cyclic_code.v	size_rcs_poly
size_rcs_poly_old (R : idomainType) n (x : 'rV[R]_n) : size (rcs_poly (rVpoly x) n) <= n. Proof. destruct n as [\|n']. rewrite /rcs_poly subrr modp0. have -> : x = 0 by apply/matrixP => i []. by rewrite linear0 mulr0 size_poly0. by rewrite size_rcs_poly // size_poly. Qed.	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv fingroup perm.", "From mathcomp Require Import finalg zmodp matrix mxalgebra mxpoly polydiv.", "From mathcomp Require Import vector.", "Require Import ssralg_ext poly_ext f2 hamming linearcode dft." ]	ecc_classic/cyclic_code.v	size_rcs_poly_old
rcs_rcs_poly (F : fieldType) n0 (x : 'rV[F]_n0) : rcs x = poly_rV (rcs_poly (rVpoly x) n0). Proof. destruct n0 as [\|n0]. rewrite /rcs_poly (_ : 'X^0 = 1); last first. by apply/polyP => i; rewrite coefXn. rewrite subrr modp0 (_ : rVpoly x = 0); last first. apply/polyP => i. rewrite coef_rVpoly. by case: insubP => //; rewrite coef0. rewrite mulr0 /rcs /= col_perm1. apply/rowP. case; by case. rewrite -rcs'_rcs /rcs' /rcs_poly. move/eqP: (sub_XrVpoly_rcs x); rewrite subr_eq => /eqP. by case/divp_modpP/(_ (size_rcs _)) => _ <-; rewrite rVpolyK -rcs'_rcs. Qed.	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv fingroup perm.", "From mathcomp Require Import finalg zmodp matrix mxalgebra mxpoly polydiv.", "From mathcomp Require Import vector.", "Require Import ssralg_ext poly_ext f2 hamming linearcode dft." ]	ecc_classic/cyclic_code.v	rcs_rcs_poly
rcs_poly_rcs (F : fieldType) n0 (x : {poly F}) (xn0 : size x <= n0.+1) : rcs_poly x n0.+1 = rVpoly (@rcs _ n0.+1 (poly_rV x)). Proof. by rewrite rcs_rcs_poly poly_rV_K // poly_rV_K // size_rcs_poly. Qed.	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv fingroup perm.", "From mathcomp Require Import finalg zmodp matrix mxalgebra mxpoly polydiv.", "From mathcomp Require Import vector.", "Require Import ssralg_ext poly_ext f2 hamming linearcode dft." ]	ecc_classic/cyclic_code.v	rcs_poly_rcs
rcsP (F: finFieldType) n (C : {set 'rV[F]_n}) := forall x, x \in C -> rcs x \in C.	Definition	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv fingroup perm.", "From mathcomp Require Import finalg zmodp matrix mxalgebra mxpoly polydiv.", "From mathcomp Require Import vector.", "Require Import ssralg_ext poly_ext f2 hamming linearcode dft." ]	ecc_classic/cyclic_code.v	rcsP
n := n'.+1. Variable (a : F). Local Notation "v ^`_ i" := (fdcoor (rVexp a n) v i) (at level 9).	Let	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv fingroup perm.", "From mathcomp Require Import finalg zmodp matrix mxalgebra mxpoly polydiv.", "From mathcomp Require Import vector.", "Require Import ssralg_ext poly_ext f2 hamming linearcode dft." ]	ecc_classic/cyclic_code.v	n
fdcoor_rcs' (i : 'I_n) x : a ^+ n = 1 -> a ^+ i * x ^`_ i = 0 -> (rcs x) ^`_ i = 0. Proof. move=> an1. rewrite /fdcoor (@horner_coef_wide _ n); last by rewrite /rVpoly size_poly. rewrite big_distrr /=. rewrite (eq_bigr (fun i1 : 'I_ _ => (rVpoly x)`_i1 * a ^+ i ^+ i1.+1)); last first. move=> i1 _; rewrite mulrC -mulrA; congr (_ * _). by rewrite mxE -!exprM -exprD mulnS addnC. move=> x_RS. rewrite (@horner_coef_wide _ n); last by move: (size_rcs x); rewrite size_XnsubC. rewrite -{}[RHS]x_RS rcs_rcs_poly; apply/esym. rewrite (reindex_onto (@rcs_perm n) (perm_inv (@rcs_perm n))) /=; last first. move=> i1 _; by rewrite permKV. rewrite (eq_bigl xpredT); last by move=> i1; rewrite permK eqxx. apply: eq_bigr => i1 _. rewrite coef_rVpoly unlock ffunE. case: ifPn => [/eqP ->\|]. case: insubP => [j _ jn0\|]; last by rewrite ltnS leqnn. rewrite /= in jn0. rewrite coef_rVpoly; case: insubP => // k _ k0. rewrite mxE rcs_poly_rcs ?rVpolyK; last by rewrite size_poly. rewrite coef_rVpoly_ord mxE unlock ffunE -val_eqE k0 /= expr0 mulr1. by rewrite exprAC an1 expr1n mulr1; congr (x _ _); apply: val_inj => /=. case: insubP => /= [j\|]. rewrite ltnS => i1n0 ji1 i10; rewrite coef_rVpoly. case: insubP => /= [k\|]. rewrite ltnS => i1n0' ki1. rewrite mxE rcs_poly_rcs ?rVpolyK; last by rewrite size_poly. rewrite coef_rVpoly_ord mxE unlock ffunE -val_eqE [val k]/= ki1 val_eqE (negPf i10). rewrite inordK; last by rewrite ltnW // ltnS prednK // lt0n. rewrite prednK // ?lt0n //; congr (x _ _ * _). exact/val_inj. by rewrite mxE. by rewrite ltnS => /negP abs; exfalso; apply: abs; rewrite -ltnS. by move=> /negP abs i10; exfalso; apply: (abs); rewrite ltnS inordK. Qed.	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv fingroup perm.", "From mathcomp Require Import finalg zmodp matrix mxalgebra mxpoly polydiv.", "From mathcomp Require Import vector.", "Require Import ssralg_ext poly_ext f2 hamming linearcode dft." ]	ecc_classic/cyclic_code.v	fdcoor_rcs'
fdcoor_rcs (i : 'I_n) x : a ^+ n = 1 -> x ^`_ i = 0 -> (rcs x) ^`_ i = 0. Proof. move=> H1 H2; by rewrite (fdcoor_rcs' H1) // H2 mulr0. Qed.	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv fingroup perm.", "From mathcomp Require Import finalg zmodp matrix mxalgebra mxpoly polydiv.", "From mathcomp Require Import vector.", "Require Import ssralg_ext poly_ext f2 hamming linearcode dft." ]	ecc_classic/cyclic_code.v	fdcoor_rcs
loop (T : Type) (f : T -> T) (n : nat) (x : T) := if n is m.+1 then loop f m (f x) else x.	Fixpoint	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv fingroup perm.", "From mathcomp Require Import finalg zmodp matrix mxalgebra mxpoly polydiv.", "From mathcomp Require Import vector.", "Require Import ssralg_ext poly_ext f2 hamming linearcode dft." ]	ecc_classic/cyclic_code.v	loop
fdcoor_loop_rcs k (i : 'I_n) (x : 'rV_n) : a ^+ n = 1 -> x ^`_ i = 0 -> (loop (@rcs F n) k x) ^`_ i = 0. Proof. move: i x. elim: k => // k IH i x an1 H /=. by rewrite IH // fdcoor_rcs. Qed.	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv fingroup perm.", "From mathcomp Require Import finalg zmodp matrix mxalgebra mxpoly polydiv.", "From mathcomp Require Import vector.", "Require Import ssralg_ext poly_ext f2 hamming linearcode dft." ]	ecc_classic/cyclic_code.v	fdcoor_loop_rcs
is_pgen := [pred g \| [forall x, (x \in C) == (g %\| rVpoly x)]].	Definition	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv fingroup perm.", "From mathcomp Require Import finalg zmodp matrix mxalgebra mxpoly polydiv.", "From mathcomp Require Import vector.", "Require Import ssralg_ext poly_ext f2 hamming linearcode dft." ]	ecc_classic/cyclic_code.v	is_pgen
t := mk { lcode0 :> Lcode0.t F n ; gen : {poly F} ; size_gen : size gen < n ; P : gen \in 'pgen[[set cw in lcode0]] }.	Record	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv fingroup perm.", "From mathcomp Require Import finalg zmodp matrix mxalgebra mxpoly polydiv.", "From mathcomp Require Import vector.", "Require Import ssralg_ext poly_ext f2 hamming linearcode dft." ]	ecc_classic/cyclic_code.v	t
pcode_coercion (F : finFieldType) (n : nat) (c : Pcode.t F n) : {vspace 'rV[F]_n} := let: Pcode.mk v _ _ _ := c in v.	Coercion	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv fingroup perm.", "From mathcomp Require Import finalg zmodp matrix mxalgebra mxpoly polydiv.", "From mathcomp Require Import vector.", "Require Import ssralg_ext poly_ext f2 hamming linearcode dft." ]	ecc_classic/cyclic_code.v	pcode_coercion
t := mk { lcode0 :> Lcode0.t F n ; P : rcsP [set cw in lcode0] }.	Record	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv fingroup perm.", "From mathcomp Require Import finalg zmodp matrix mxalgebra mxpoly polydiv.", "From mathcomp Require Import vector.", "Require Import ssralg_ext poly_ext f2 hamming linearcode dft." ]	ecc_classic/cyclic_code.v	t
ccode_coercion (F : finFieldType) (n : nat) (c : Ccode.t F n) : {vspace 'rV[F]_n} := let: Ccode.mk v _ := c in v.	Coercion	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv fingroup perm.", "From mathcomp Require Import finalg zmodp matrix mxalgebra mxpoly polydiv.", "From mathcomp Require Import vector.", "Require Import ssralg_ext poly_ext f2 hamming linearcode dft." ]	ecc_classic/cyclic_code.v	ccode_coercion
is_cgen := [pred x \| non0_codeword_lowest_deg C x]. Local Notation "''cgen'" := (is_cgen).	Definition	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv fingroup perm.", "From mathcomp Require Import finalg zmodp matrix mxalgebra mxpoly polydiv.", "From mathcomp Require Import vector.", "Require Import ssralg_ext poly_ext f2 hamming linearcode dft." ]	ecc_classic/cyclic_code.v	is_cgen
is_cgenE g : g \in 'cgen = non0_codeword_lowest_deg C g. Proof. by []. Qed.	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv fingroup perm.", "From mathcomp Require Import finalg zmodp matrix mxalgebra mxpoly polydiv.", "From mathcomp Require Import vector.", "Require Import ssralg_ext poly_ext f2 hamming linearcode dft." ]	ecc_classic/cyclic_code.v	is_cgenE
size_is_cgen g : g \in 'cgen -> size (rVpoly g) <= n. Proof. by rewrite size_poly. Qed.	Lemma	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv fingroup perm.", "From mathcomp Require Import finalg zmodp matrix mxalgebra mxpoly polydiv.", "From mathcomp Require Import vector.", "Require Import ssralg_ext poly_ext f2 hamming linearcode dft." ]	ecc_classic/cyclic_code.v	size_is_cgen
canonical_cgen : 'rV[F]_n := val (exists_non0_codeword_lowest_deg C_not_trivial).	Definition	ecc_classic	[ "From mathcomp Require Import all_ssreflect ssralg poly polydiv fingroup perm.", "From mathcomp Require Import finalg zmodp matrix mxalgebra mxpoly polydiv.", "From mathcomp Require Import vector.", "Require Import ssralg_ext poly_ext f2 hamming linearcode dft." ]	ecc_classic/cyclic_code.v	canonical_cgen

location_polynomial_points := \row_i \prod_(j < n | j != i) (a ``_ i - a ``_ j).

Definition

ecc_classic

[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]

ecc_classic/alternant.v

location_polynomial_points

b := \row_(i < n) (((location_polynomial_points a) ``_ i)^-1 * g.[a ``_ i]). Variable (r : nat).

Let

ecc_classic

[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]

ecc_classic/alternant.v

b

GRS_PCM_polynomial := @GRS.PCM _ F a b r.

Definition

ecc_classic

[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]

ecc_classic/alternant.v

GRS_PCM_polynomial

ext_inj : {rmorphism F0 -> F1} := [the {rmorphism F0 -> F1} of @GRing.in_alg _ _].

Definition

ecc_classic

[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]

ecc_classic/alternant.v

ext_inj

ext_inj_tmp : {rmorphism F0 -> (FinFieldExtType F1)} := ext_inj. Variable n : nat.

Definition

ecc_classic

[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]

ecc_classic/alternant.v

ext_inj_tmp

ext_inj_rV : 'rV[F0]_n -> 'rV[F1]_n := @map_mx _ _ ext_inj 1 n.

Definition

ecc_classic

[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]

ecc_classic/alternant.v

ext_inj_rV

u := u'.+1. Hypothesis primep : prime p.

Let

ecc_classic

[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]

ecc_classic/alternant.v

u

Fq : finFieldType := GF u primep.

Let

ecc_classic

[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]

ecc_classic/alternant.v

Fq

q := p ^ u.

Let

ecc_classic

[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]

ecc_classic/alternant.v

q

p_char : p \in [pchar Fq]. Proof. apply: char_GFqm. Qed. (** declare F_{q^m} *) Variable m' : nat.

Let

ecc_classic

[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]

ecc_classic/alternant.v

p_char

m := m'.+1. Variable Fqm : fieldExtType Fq. Hypothesis card_Fqm : #| FinFieldExtType Fqm | = q ^ m. (** build GRS_k(kappa, g) *) Variable n : nat. Variable a : 'rV[Fqm]_n. Variable g : {poly Fqm}. Variable k : nat.

Let

ecc_classic

[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]

ecc_classic/alternant.v

m

alternant_PCM : 'M_(k, n) := @GRS_PCM_polynomial n (FinFieldExtType Fqm) a g k.

Definition

ecc_classic

[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]

ecc_classic/alternant.v

alternant_PCM

alternant_code := Rcode.t (@ext_inj_tmp Fq Fqm) (kernel alternant_PCM). (** Goppa codes are a special case of alternant codes *)

Definition

ecc_classic

[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]

ecc_classic/alternant.v

alternant_code

goppa_code_condition := size g = (n - k).+1.

Definition

ecc_classic

[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]

ecc_classic/alternant.v

goppa_code_condition

u := u'.+1. Hypothesis primep : prime p.

Let

ecc_classic

[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]

ecc_classic/alternant.v

u

Fq : finFieldType := GF u primep.

Let

ecc_classic

[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]

ecc_classic/alternant.v

Fq

q := p ^ u.

Let

ecc_classic

[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]

ecc_classic/alternant.v

q

p_char : p \in [pchar Fq]. Proof. apply: char_GFqm. Qed. (** declare F_{q^m} *) Variable m' : nat.

Let

ecc_classic

[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]

ecc_classic/alternant.v

p_char

m := m'.+1. Variable Fqm : fieldExtType Fq. Hypothesis card_Fqm : #| FinFieldExtType Fqm | = q ^ m. (** we are talking about narrow-sense Goppa codes *)

Let

ecc_classic

[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]

ecc_classic/alternant.v

m

n : nat := (q^m).-1. Variable e : Fqm. Hypothesis e_prim : n.-primitive_root e.

Let

ecc_classic

[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]

ecc_classic/alternant.v

n

a : 'rV[Fqm]_n := rVexp e n. Variable t : nat. (** we have to instantiate Goppa codes with a monomial to recover BCH codes *)

Let

ecc_classic

[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]

ecc_classic/alternant.v

a

g : {poly (FinFieldExtType Fqm)} := 'X^(n - t). (** from the Goppa code condition, we have only one choice for its degree *)

Let

ecc_classic

[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]

ecc_classic/alternant.v

g

goppa_code_condition_check : goppa_code_condition n g t. Proof. by rewrite /goppa_code_condition size_polyXn. Qed. (* NB: we only have binary BCH codes, so we should maybe restrict q at this point *) (** wip *)

Let

ecc_classic

[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]

ecc_classic/alternant.v

goppa_code_condition_check

narrow_sense_BCH_are_Goppa : @BCH.PCM (FinFieldExtType _) _ a t = @alternant_PCM _ u' primep Fqm _ a g t(*?*). Proof. rewrite /BCH.code /alternant_code. Abort.

Lemma

ecc_classic

[ "From mathcomp Require Import all_ssreflect ssralg finalg poly polydiv.", "From mathcomp Require Import cyclic perm matrix mxpoly vector mxalgebra zmodp.", "From mathcomp Require Import finfield falgebra fieldext.", "Require Import ssr_ext ssralg_ext linearcode.", "Require Import dft poly_decoding grs bch." ]

ecc_classic/alternant.v

narrow_sense_BCH_are_Goppa

PCM : 'M_(t, n) := \matrix_(i < t, j < n) (a ``_ j) ^+ i.*2.+1.

Definition

ecc_classic