phanerozoic / threshold-hamming1511-encoder

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

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

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)

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