On additive MDS codes over small fields

Simeon Ball; Guillermo Gamboa; Michel Lavrauw

doi:10.3934/amc.2021024

Article Contents

2023, Volume 17, Issue 4: 828-844. Doi: 10.3934/amc.2021024

This issue Previous Article Some subfield codes from MDS codes Next Article Four by four MDS matrices with the fewest XOR gates based on words

On additive MDS codes over small fields

1.
Departament de Matemàtiques, Universitat Politècnica de Catalunya, Jordi Girona 1-3, 08034 Barcelona, Spain
2.
Faculty of Engineering and Natural Sciences, Sabancı University, Istanbul, Turkey

^* Corresponding author: Simeon Ball
^* Corresponding author: Simeon Ball

Received: December 2020

Revised: April 2021

Early access: July 2021

Published: August 2023

The first author acknowledges the support of the project MTM2017-82166-P of the Spanish Ministerio de Ciencia y Innovación

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Abstract

Let $ C $ be a $ (n,q^{2k},n-k+1)_{q^2} $ additive MDS code which is linear over $ {\mathbb F}_q $. We prove that if $ n \geq q+k $ and $ k+1 $ of the projections of $ C $ are linear over $ {\mathbb F}_{q^2} $ then $ C $ is linear over $ {\mathbb F}_{q^2} $. We use this geometrical theorem, other geometric arguments and some computations to classify all additive MDS codes over $ {\mathbb F}_q $ for $ q \in \{4,8,9\} $. We also classify the longest additive MDS codes over $ {\mathbb F}_{16} $ which are linear over $ {\mathbb F}_4 $. 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 \in \{ 2,3\} $.

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Mathematics Subject Classification: Primary: 94B27; Secondary: 51E22.

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Table 1. The classification of arcs of lines of $\mathrm{PG}(5, 2)$

size	4	5	6
number of arcs of points of $\mathrm{PG}(2, 4)$	1	1	1
number of arcs of lines of $\mathrm{PG}(5, 2)$	1	1	1

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Table 2. The classification of arcs of planes of PG(8, 2)

size	4	5	6	7	8	9	10
number of arcs of points of PG(2, 8)	1	1	3	2	2	2	1
number of arcs of planes of PG(8, 2)	1	2	4	2	2	2	1

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Table 3. The classification of arcs of lines of PG(5, 3)

size	4	5	6	7	8	9	10
# of arcs of points of PG(2, 9)	1	2	6	3	2	1	1
# of arcs of lines of PG(5, 3)	1	4	13	4	3	1	1

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Table 4. The classification of arcs of lines of PG(3, 3)

size	4	5	6	7	8	9	10
# of arcs of points of PG(1, 9)	2	2	2	1	1	1	1
# of arcs of lines of PG(3, 3)	3	4	5	4	3	2	2

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Table 5. The classification of arcs of lines of $\mathrm{PG}(5, 4)$

size	5	6	7	8	9	10	11
# of arcs of $\mathrm{PG}(2, 16)$	3	22	125	865	1534	1262	300
# of line-arcs of $\mathrm{PG}(5, 4)$	10	360	8294	15162	2869	1465	301
size	12	13	14	15	16	17	18
# of arcs of $\mathrm{PG}(2, 16)$	159	70	30	9	5	3	2
# of line-arcs of $\mathrm{PG}(5, 4)$	159	70	30	9	5	3	2

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