threshold-hamming1511-encoder

Hamming(15,11) encoder. Adds 4 parity bits to 11 data bits for single-error correction.

Function

encode(d1..d11) -> [p1, p2, d1, p4, d2, d3, d4, p8, d5, d6, d7, d8, d9, d10, d11]

Bit Positions

Position	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Bit	p1	p2	d1	p4	d2	d3	d4	p8	d5	d6	d7	d8	d9	d10	d11

Parity Equations

Each parity bit covers positions where its position number (in binary) has a 1 in that bit:

• p1 (bit 0): positions 1,3,5,7,9,11,13,15 -> XOR(d1,d2,d4,d5,d7,d9,d11)
• p2 (bit 1): positions 2,3,6,7,10,11,14,15 -> XOR(d1,d3,d4,d6,d7,d10,d11)
• p4 (bit 2): positions 4,5,6,7,12,13,14,15 -> XOR(d2,d3,d4,d8,d9,d10,d11)
• p8 (bit 3): positions 8,9,10,11,12,13,14,15 -> XOR(d5,d6,d7,d8,d9,d10,d11)

Architecture

Each parity bit requires a 7-way XOR, implemented as a tree of 2-way XORs:

XOR7(a,b,c,d,e,f,g) = XOR(XOR4(a,b,c,d), XOR3(e,f,g))

XOR4(a,b,c,d) = XOR(XOR(a,b), XOR(c,d))
XOR3(e,f,g) = XOR(XOR(e,f), g)

Each XOR2 requires 3 neurons (OR, NAND, AND).

Parameters

Inputs	11
Outputs	15
Neurons	86
Layers	6
Parameters	591
Magnitude	272

Error Correction

When decoded, the receiver computes syndrome bits by XORing received bits at parity positions. The syndrome directly indicates the bit position of any single-bit error (0 = no error).

Comparison to Hamming(7,4)

Code	Data	Parity	Total	Efficiency
Hamming(7,4)	4	3	7	57%
Hamming(15,11)	11	4	15	73%

Larger Hamming codes are more efficient but require more complex circuits.

Usage

from safetensors.torch import load_file

w = load_file('model.safetensors')
# See model.py for reference implementation

License

MIT