Patterson's decoding algorithm for Goppa codes

Question

From this Wiki page: given a Goppa code $\Gamma(g, L)$ and a binary word $v=(v_0,...,v_{n-1})$, its syndrome is defined as $$s(x)=\sum_{i=0}^{n-1}\frac{v_i}{x-L_i} \mod g(x).$$ To do error correction, Patterson's algorithm goes as follows:

• Calculate $$v(x)=\sqrt{s(x)^{-1}-x}\mod g(x)$$ (this assumes that $s(x)\ne 0$, which is always the case unless $v$ belongs to $\Gamma(g,L)$ and no correction is required).

• Use the EEA to obtain the polynomials $a(x)$ and $b(x)$ such that $$ a(x)=b(x)v(x)\mod g(x)$$

• Calculate the polynomial $p(x)=a(x)+xb(x)^2$. Assuming that the original word is decodable, this should be the same as the factorized error locator polynomial $$\sigma(x)=\prod_{i\in B}(x-L_i) $$ where $B=\{i \text{ s.t. } e_i\ne 0\}$.

• Assuming that the original word is decodable (that is, $|B|<d$ where $d$ is the minimum distance of the code), we simply calculate the roots of $\sigma(x)$: if $L_i$ is a root, then an error as occurred in the $i$-th position.

I only have one question: how do we show that the polynomial $p(x)$ obtained in the third step corresponds exactly to $\sigma(x)$ as defined above?

Answer 1

First note that $$\frac{\sigma'(x)}{\sigma(x)}=\sum_{i\in B}\frac 1{x-L_i}$$ and that if we write $C$ for the bit positions of a codeword, the Goppa code is defined by $$\sum_{i\in C}\frac 1{x-L_i}\equiv 0\pmod{g(x)}$$ so that $$\frac{\sigma'(x)}{\sigma(x)}\equiv\sum_{i\in B\ominus C}\frac 1{x-L_i}\pmod{g(x)}$$ and the right hand side is just $s(x)$.

We split $\sigma(x)$ into odd and even degree monomials so that we can find polynomials $\sigma_{odd}$ and$\sigma_{even}$ such that $$\sigma(x)=x\sigma_{odd}(x^2)+\sigma_{even}(x^2). \qquad (1)$$ Because we are in a field of a characteristic 2, polynomials with only square terms are squares of other polynomials of degree at most $(\deg g-1)/2$ and we aim to recover $a(x)$ and $b(x)$ satisfying this degree bound such that $a(x)^2=\sigma_{even}(x^2)$ and $b(x)^2=\sigma_{odd}$.

Differentiating (1) gives $$\sigma'(x)=\sigma_{odd}(x^2)$$ and so $$\frac{\sigma(x)}{\sigma'(x)}=x+\frac{\sigma_{even}(x^2)}{\sigma_{odd}(x^2)}$$ which tells us that $$\frac1{s(x)}-x\equiv \frac{\sigma_{even}(x^2)}{\sigma_{odd}(x^2)}\pmod{g(x)}.$$ Thus the polynomial $v(x)$ is equivalent to the rational function $a(x)/b(x)$ that we seek modulo $g(x)$. The extended Euclidean algorithm will return two polynomials such that $$\frac{c(x)}{d(x)}\equiv v(x)\pmod{g(x)}$$ with $\deg c$ and $\deg d$ both less than $(\deg g)/2$ and these polynomials must be equal to our sought after $a(x)$ and $b(x)$ otherwise $a(x)d(x)-b(x)c(x)$ is a polynomial of degree less than $d$ and divisible by $g(x)$. Having recovered $a(x)$ and $b(x)$ we can now construct $\sigma(x)=a(x)^2+xb(x)^2$ which is the definition of $p(x)$ that is usually given.