Dualities over the cross product of the cyclic groups of order 2

Steven T. Dougherty; Serap Şahinkaya

doi:10.3934/amc.2023005

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2024, Volume 18, Issue 5: 1531-1546. Doi: 10.3934/amc.2023005

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Dualities over the cross product of the cyclic groups of order 2

Steven T. Dougherty^1, , and
Serap Şahinkaya^{1, 2,}

1.
Department of Mathematics, University of Scranton, Scranton, PA 18518, USA
2.
Tarsus University, Faculty of Engineering, Department of Natural and Mathematical Sciences, Mersin, Turkey

^*Corresponding author: Steven T. Dougherty
^*Corresponding author: Steven T. Dougherty

Received: July 2022

Revised: February 2023

Early access: February 2023

Published: October 2024

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Abstract

We determine the number of symmetric dualities on the $ s $-fold cross product of the cyclic group of order $ 2, $ which is the additive group of the finite field $ {\mathbb{F}}_{2^s}. $ We show that the ratio of symmetric dualities over all dualities goes to $ 0 $ as $ s $ goes to infinity.We also prove a surprising result that given any two binary codes $ C $ and $ D $ of the same length $ n $ with $ |C||D| = 2^n $, then viewing them as groups there is a symmetric duality $ M $ with $ C^M = D $, which also relates their weight enumerators as additive codes in a group via the MacWilliams relations. Using this theorem we show that any additive code in this setting can be viewed as an additive complementary dual code of length $ 1 $ with respect to some duality.

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Mathematics Subject Classification: Primary: 11T71 Secondary: 94B05.

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