Cyclic code

A length-$n$ cyclic code over ${\mathbb{F}}_q$ is an ideal of ${\mathbb{F}}_q[X]/(X^n-1)$. It is generated by a monic divisor of $X^n-1$.

The unique monic divisor $g(X)$ of $X^n-1$ generating a cyclic code is its generator polynomial. The code has dimension $n-\mathrm{deg}g$.

Let $\alpha$ be a root of the generator polynomial of a cyclic code $g(X)$. Every codeword $c(X)$ satisfies $c(\alpha )=0$, so a received word $r=c+e$ satisfies $r(\alpha )=e(\alpha )$. For a single-symbol error $e(X)=a{X}^j$, the evaluation $r(\alpha )=a{\alpha }^j$ identifies its value and position when the relevant powers of $\alpha$ are distinct.

The dual of a cyclic code is cyclic. If $g(X)h(X)=X^n-1$, then the dual generator is the monic reciprocal polynomial $h^{\ast }(X)$.

Over ${\mathbb{F}}_2$,Its eight monic divisors generate the eight cyclic codes of length seven: the whole-space, even-weight, two Hamming, two simplex, repetition, and zero codes.

Given a primitive $n$th root $\alpha$, a BCH code of design distance $\delta$ has a generator polynomial vanishing at $\delta -1$ consecutive powers ${\alpha }^b,\dots ,{\alpha }^{b+\delta -2}$.

For error locations $i_1,\dots ,i_{\nu }$ in a cyclic code with primitive root $\alpha$, the error locator polynomial is $\sigma (Z)={\prod }_h(1-{\alpha }^{i_h}Z)$. Its roots encode the error positions.

The minimum distance of a BCH code is at least its design distance. A hypothetical word of smaller weight yields a square Vandermonde system from the consecutive-root equations; its support elements are distinct, so the determinant is nonzero and every word coefficient must vanish.