Binary codes from $ m $-ary $ n $-cubes $ Q^m_n $

Jennifer D. Key; Bernardo G. Rodrigues

doi:10.3934/amc.2020079

Article Contents

2021, Volume 15, Issue 3: 507-524. Doi: 10.3934/amc.2020079

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Binary codes from $ m $-ary $ n $-cubes $ Q^m_n $

Jennifer D. Key^1, , and
Bernardo G. Rodrigues^2,

1.
Department of Mathematics and Applied Mathematics, University of the Western Cape, 7535 Bellville, South Africa
2.
Department of Mathematics and Applied Mathematics, University of Pretoria, Hatfield 0028, South Africa

^* Corresponding author: J. D. Key
^* Corresponding author: J. D. Key

Received: September 30, 2019

Revised: November 30, 2019

Early access: April 2020

Published: August 2021

The second author is supported by the National Research Foundation of South Africa (Grant Numbers 95725 and 106071)

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Abstract

We examine the binary codes from adjacency matrices of the graph with vertices the nodes of the $ m $-ary $ n $-cube $ Q^m_n $ and with adjacency defined by the Lee metric. For $ n = 2 $ and $ m $ odd, we obtain the parameters of the code and its dual, and show the codes to be $ LCD $. We also find $ s $-PD-sets of size $ s+1 $ for $ s < \frac{m-1}{2} $ for the dual codes, i.e. $ [m^2,2m-1,m]_2 $ codes, when $ n = 2 $ and $ m\ge 5 $ is odd.

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Mathematics Subject Classification: Primary: 05C50, 94B05; Secondary: 05B05.

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Table 1. Blocks in $ \mathcal{{B}} $$ _m $

$ m $
$ 5 $	$ 0,2 $
$ 7 $	$ 0,0 $	$ 0,3 $	$ 2,2 $
$ 9 $	$ 0,2 $	$ 0,3 $	$ 2,4 $
$ 11 $	$ 0,0 $	$ 0,3 $	$ 0,4 $	$ 2,2 $	$ 2,5 $	$ 4,4 $
$ 13 $	$ 0,2 $	$ 0,3 $	$ 0,6 $	$ 2,4 $	$ 2,5 $	$ 4,6 $
$ 15 $	$ 0,0 $	$ 0,3 $	$ 0,4 $	$ 0,7 $	$ 2,2 $	$ 2,5 $	$ 2,6 $	$ 4,4 $	$ 4,7 $	$ 6,6 $
$ 17 $	$ 0,2 $	$ 0,3 $	$ 0,6 $	$ 0,7 $	$ 2,4 $	$ 2,5 $	$ 2,8 $	$ 4,6 $	$ 4,7 $	$ 6,8 $
$ 19 $	$ 0,0 $	$ 0,3 $	$ 0,4 $	$ 0,7 $	0, 8	$ 2,2 $	$ 2,5 $	$ 2,6 $	2, 9	$ 4,4 $	$ 4,7 $	4, 8	$ 6,6 $	6, 9	8, 8

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