A class of few-Lee weight ℤ₂[u]-linear codes using simplicial complexes and minimal codes via Gray map

2022/05/13 by Kewat, Pramod Kumar, Mondal, Nilay Kumar
#94A62 #94B05 #FOS: Computer and information sciences #Information Theory (cs.IT)

paper · doi:10.48550/arxiv.2205.06470

Abstract

Recently some mixed alphabet rings are involved in constructing few-Lee weight additive codes with optimal or minimal Gray images using suitable defining sets or down-sets. Inspired by these works, we choose the mixed alphabet ring ℤ₂ℤ₂[u] to construct a special class of linear code C_L over ℤ₂[u] with u²=0 by employing simplicial complexes generated by a single maximal element. We show that C_L has few-Lee weights by determining the Lee weight distribution of C_L. Theoretically, this shows that we may employ simplicial complexes to obatin few-weight codes even in the case of mixed alphabet rings. We show that the Gray image of C_L is self-orthogonal and we have an infinite family of minimal codes over ℤ₂ via Gray map, which can be used to secret sharing schemes.

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