2015/01/06 by Li, Jiyou, Wan, Daqing, Zhang, Jun · 2 citations
#Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT)
paper · doi:10.48550/arxiv.1501.01138
Computing the minimum distance of a linear code is one of the fundamental problems in algorithmic coding theory. Vardy [14] showed that it is an \np-hard problem for general linear codes. In practice, one often uses codes with additional mathematical structure, such as AG codes. For AG codes of genus 0 (generalized Reed-Solomon codes), the minimum distance has a simple explicit formula. An interesting result of Cheng [3] says that the minimum distance problem is already \np-hard (under \rp-reduction) for general elliptic curve codes (ECAG codes, or AG codes of genus 1). In this paper, we show that the minimum distance of ECAG codes also has a simple explicit formula if the evaluation set is suitably large (at least 2/3 of the group order). Our method is purely combinatorial and based on a new sieving technique from the first two authors [8]. This method also proves a significantly stronger version of the MDS (maximum distance separable) conjecture for ECAG codes.