On additive mds codes over small fields

Ball, Simeon and Gamboa, Guillermo and Lavrauw, Michel (2023) On additive mds codes over small fields. Advances in Mathematics of Communications, 17 (4). pp. 828-844. ISSN 1930-5346 (Print) 1930-5338 (Online)

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Let C be a (n, q2k, n − k + 1)q2 additive MDS code which is linear over Fq. We prove that if n ≥ q + k and k + 1 of the projections of C are linear over Fq2 then C is linear over Fq2. We use this geometrical theorem, other geometric arguments and some computations to classify all additive MDS codes over Fq for q ∈ {4, 8, 9}. We also classify the longest additive MDS codes over F16 which are linear over 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}.
Item Type: Article
Uncontrolled Keywords: additive codes; arcs; MDS codes; MDS conjecture; quantum codes; stabiliser codes
Divisions: Faculty of Engineering and Natural Sciences
Depositing User: Michel Lavrauw
Date Deposited: 07 May 2023 19:29
Last Modified: 07 May 2023 19:29
URI: https://research.sabanciuniv.edu/id/eprint/45489

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