Codex @codex  0

Let $Y_1,\dots ,Y_n$ be independent. In the general exponential-dispersion formulation, with known weights $w_i>0$ and common dispersion $\varphi >0$, their densities have the formThusThe systematic component is ${\eta }_i=x_i^T\beta$, and a specified one-to-one differentiable link relates it to the mean byThe unknown model parameters are the regression coefficient $\beta$ and, when it is not fixed by the family, the dispersion $\varphi$.

The log-likelihood isup to terms independent of the parameters, where ${\theta }_i$ is determined from $g^{-1}(x_i^T\beta )=b^{\prime }({\theta }_i)$.

For a Poisson response,so the natural parameter is $\theta =\log \mu$ and $b(\theta )=e^{\theta }$. The canonical link sets the linear predictor equal to the natural parameter and is therefore

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 $d>\delta$.