A Syndrome Formulation of the Interpolation Step in the Guruswami-Sudan Algorithm

• Authors: Peter Beelen;Tom Høholdt
• Venue: ICMCTA '08 Proceedings of the 2nd international Castle meeting on Coding Theory and Applications
• Year: 2008

Let $\mathbb{F}_q$ be a finite field with qelements and $C \subset \mathbb{F}_q^n$ a code of length n. If c驴 Cand h= (h1h2驴 hn) is a row of a parity check matrix of C, then it is clear that h·c= 0. Such parity checks can therefore be used to test if a given word $w \in \mathbb{F}_q^n$ is an element of C, but it turns out that the expressions h·w(usually called syndromes of w) can be useful in decoding algorithms as well. The link between decoding and syndromes is an old one. Indeed the first known algorithm for the decoding of Reed-Solomon codes (Peterson's algorithm) uses syndromes. Now let ${\mathcal P}=\{x_1,\dots,x_n\}$ be a subset of $\mathbb{F}_q$ consisting of ndistinct elements. We can see an RS-code of dimension k≤ nas the set of all n-tuples that arise by evaluating all polynomial f(x) of degree less than or equal to k驴 1 in the points x1,...,xn.