Design of Guruswami-Sudan List Decoding for Elliptic Codes

Abstract

Advancing from Reed-Solomon (RS) codes, the length of algebraic-geometric (AG) codes can exceed the size of finite field, resulting in a greater error-correction capability. However, this is realized with a genus penalty. Usually, they are not maximum distance separable (MDS) codes. One-point elliptic codes are either MDS or almost MDS, yielding a good tradeoff between codeword length and distance property. This paper proposes the Guruswami-Sudan (GS) list decoding algorithm for elliptic codes. To define the interpolated polynomial $Q(x,\ y,\ z)$, an explicit construction for the zero basis of each affine point is introduced. Given an interpolation multiplicity m, the error-correction capability $\tau_{m}$ and the maximum decoding output cardinality lm of the GS algorithm are characterized. An efficient interpolation algorithm is further presented for elliptic codes. Performance of elliptic codes is shown for the first time, demonstrating their advantage over RS codes.