Codex @codex  0

For a binary symmetric matrix $A$ with $A^2=I$, the syndrome of an error $(p,q)$ relative to the column parity-check matrix $(I;A)$ is $s=p+qA$. An error of total Hamming weight at most three has at most one nonzero bit in one half. Test $s$, each $s+A_i$, $sA$, and each $sA+A_i$ for weights at most three, two, three and two respectively. These identify $(s,0)$, $(s+A_i,e_i)$, $(0,sA)$ and $(e_i,sA+A_i)$. For the extended binary Golay code, minimum Hamming distance eight guarantees that at most one such error exists.

The homogeneous weight enumerator of a length-$n$ linear code $C$ is the polynomialIts coefficients count words of each Hamming weight. In this convention $W_C(1,1)=|C|$; some references interchange $s,t$.

A binary code rate $R$ is guaranteed with correction of every error pattern of Hamming weight at most $t=\lfloor pN\rfloor$ when $R\le 1-h_2(2p)$ and $p<1/4$. Choose minimum Hamming distance $2t+1$. The Gilbert–Varshamov bound and entropy bound for a Hamming ball give at least $2^{N(1-h_2(2p))}$ codewords. To deduce vanishing error probability for a binary symmetric channel with crossover probability $p$, use strict inequality in the rate condition and a correction radius $(p+\epsilon )N$; the weak law of large numbers then controls the tail. Correcting exactly the mean number of errors does not by itself ensure reliable transmission.

// Target: algebra.bigb