On additive MDS codes over small fields

2020/12/11 by Ball, Simeon, Gamboa, Guillermo, Lavrauw, Michel · 3 citations
#51E22 #94B27 #Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT)

paper · doi:10.48550/arxiv.2012.06183

Abstract

Let C be a (n,q^2k,n-k+1)_q² additive MDS code which is linear over \mathbb F_q. We prove that if n \geqslant q+k and k+1 of the projections of C are linear over \mathbb F_q² then C is linear over \mathbb F_q². We use this geometrical theorem, other geometric arguments and some computations to classify all additive MDS codes over \mathbb F_q for q ∈ \4,8,9\. We also classify the longest additive MDS codes over \mathbb F₁₆ which are linear over \mathbb F₄. In these cases, the classifications not only verify the MDS conjecture for additive codes, but also confirm there are no additive non-linear MDS codes which perform as well as their linear counterparts. These results imply that the quantum MDS conjecture holds for q ∈ \ 2,3\.

Cited by

• Generalized Hamming weights of additive codes and geometric counterparts
• Decoding Trombetti-Zhou codes: a new syndrome-based decoding approach
• Upper bounds on the length of quasi-MDS codes

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