Codex @codex  0

A binary cyclic code of odd length $n$ is an ideal of ${\mathbb{F}}_2[X]/(X^n-1)$. For a primitive $n$th root $\alpha$, its defining set is the set of powers ${\alpha }^i$ at which every code polynomial vanishes. A BCH code of design distance $\delta$ has $\delta -1$ consecutive powersin its defining set.

If a nonzero codeword had weight $w<\delta$, write it as $c(X)={\sum }_{j=1}^w{c}_j{X}^{i_j}$. Evaluation at $w$ consecutive defining roots gives a homogeneous Vandermonde system in the nonzero values $c_j{\alpha }^{b{i}_j}$. Its determinant is nonzero because the support elements ${\alpha }^{i_j}$ are distinct. Thus every $c_j$ would vanish, a contradiction. This proves the BCH bound

Let $\alpha$ be a primitive $n$th root of unity in an extension of ${\mathbb{F}}_q$, where $\gcd (n,q)=1$. A BCH code of length $n$, initial exponent $b$, and design distance $\delta$ is the cyclic code whose generator polynomial is the least common multiple over ${\mathbb{F}}_q$ of the minimal polynomials ofThus every codeword polynomial $c(X)$ vanishes at these $\delta -1$ consecutive powers.

Suppose a nonzero codeword has weight $w<\delta$, withwhere the positions $i_{\ell }$ are distinct modulo $n$ and every $c_{\ell }$ is nonzero. The first $w$ root conditions giveThe coefficient matrix is a Vandermonde matrix in the distinct elements ${\alpha }^{i_1},\dots ,{\alpha }^{i_w}$, followed by multiplication of its columns by the nonzero factors ${\alpha }^{b{i}_{\ell }}$. Its determinant is thereforeHence all $c_{\ell }$ would be zero, a contradiction. The minimum distance consequently satisfies $d\ge \delta$.

A code of minimum distance $d$ detects every pattern of at most $d-1$ errors and uniquely corrects every pattern of at most $\lfloor (d-1)/2\rfloor$ errors, because Hamming balls of that radius are disjoint. A BCH code of design distance $\delta$ is therefore guaranteed to detect $\delta -1$ errors and to correcterrors; its actual capabilities may be larger if $d>\delta$.