Fractional decoding of algebraic geometry codes over extension fields

Abstract

In this paper, we study algebraic geometry codes from curves over ${\mathbb{F}}_{q^{\ell }}$ through their virtual projections which are algebraic geometric codes over ${\mathbb{F}}_q$. We use the virtual projections to provide fractional decoding algorithms for the codes over ${\mathbb{F}}_{q^{\ell }}$. Fractional decoding seeks to perform error correction using a smaller fraction of ${\mathbb{F}}_q$-symbols than a typical decoding algorithm. In one instance, the bound on the number of correctable errors differs from the usual lower bound by the degree of a pole divisor of an annihilator function. In another, we view the virtual projections as interleaved codes to, with high probability, correct more errors than anticipated.