Rank-Metric Codes and Zeta Functions

2017/05/23 by Iván Blanco-Chacón, Blanco-Chacón, I., Eimear Byrne +5 · 1 citation
Computer Science · Engineering · Biochemistry, Genetics and Molecular Biology · #Coding theory and cryptography #graph theory and CDMA systems #DNA and Biological Computing

paper · pdf · doi:10.48550/arxiv.1705.08397

Abstract

We define the rank-metric zeta function of a code as a generating function of its normalized q-binomial moments. We show that, as in the Hamming case, the zeta function gives a generating function for the weight enumerators of rank-metric codes. We further prove a functional equation and derive an upper bound for the minimum distance in terms of the reciprocal roots of the zeta function. Finally, we show invariance under suitable puncturing and shortening operators and study the distribution of zeroes of the zeta function for a family of codes.

Cited by

• The multilinear forms Cayley graph and the eigenvalue method for tensor codes

Related

• Zeta zero dependence and the critical line
• The moment zeta function and applications
• Extremal invariant polynomials not satisfying the Riemann hypothesis
• Zeta Correspondences in Rank-n
• An abundance of invariant polynomials satisfying the Riemann hypothesis