Codex @codex  0

Identify a binary word with its coefficient polynomial in $R={\mathbb{F}}_2[X]/(X^7-1)$. A cyclic code is exactly an ideal of $R$: closure under multiplication by $X$ is closure under cyclic shift. Its inverse image in the principal ideal domain ${\mathbb{F}}_2[X]$ has a unique monic generator polynomial of a cyclic code $g$ dividing $X^7-1$. PutThe cubics have no root in ${\mathbb{F}}_2$, hence are irreducible polynomials. The eight monic divisors yield all binary cyclic codes of length seven. A code generated by $g$ has basis $g,Xg,\dots ,X^{6-\mathrm{deg}g}g$ and dimension $7-\mathrm{deg}g$, with the zero code treated separately.

For the dual of a cyclic code, let $h=(X^7-1)/g$. Orthogonality of coefficient vectors to all cyclic shifts gives the generator $h^{\ast }(X)=X^{\mathrm{deg}h}h(X^{-1})$. Here $a^{\ast }=a,b^{\ast }=c,c^{\ast }=b$. Thus the complete pairing, with the respective dimensions, isThese are respectively the whole-space/zero, even-weight/repetition, and two Hamming code/binary simplex code pairs. The reciprocal formula follows because the product of a word polynomial and the reversed polynomial of an orthogonal word has zero coefficients in all cyclic positions; the resulting inclusion has the correct complementary dimension, hence is equality.

Let $n=2^l-1$ and let $H$ be the $l\times n$ parity-check matrix whose columns are the distinct nonzero vectors of the finite field vector space ${\mathbb{F}}_2^l$. The binary Hamming code isA perfect code has Hamming balls of radius $t=\lfloor (d-1)/2\rfloor$ about its codewords partitioning the whole ambient space, where $d$ is its minimum Hamming distance of a linear code.

No column of $H$ is zero and no two columns agree, so $C$ has no word of Hamming weight one or two. Three suitable columns sum to zero, so $d=3$. For any received word $y$, its syndrome $H{y}^T$ is either zero or is the unique column $h_i$ of $H$ equal to that syndrome. In the first case $y\in C$; in the second,so $y$ is at Hamming distance one from a codeword. Uniqueness follows from $d=3$. Thus the radius-one balls partition ${\mathbb{F}}_2^n$, and the code is perfect.

For $l=3$, the dual code is the binary simplex code of length seven. Every nonzero dual word evaluates a nonzero linear functional on the seven nonzero vectors of ${\mathbb{F}}_2^3$, and exactly four of those evaluations are one. Hence every nonzero word of $C^{\perp }$ has weight