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Open AccessArticle

MDS Self-Dual Codes and Antiorthogonal Matrices over Galois Rings

by Sunghyu Han School of Liberal Arts, KoreaTech, Cheonan 31253, Korea
Information 2019, 10(4), 153; https://doi.org/10.3390/info10040153
Received: 21 March 2019 / Revised: 15 April 2019 / Accepted: 24 April 2019 / Published: 25 April 2019
(This article belongs to the Section Information Theory and Methodology)
In this study, we explore maximum distance separable (MDS) self-dual codes over Galois rings $G R ( p m , r )$ with $p ≡ − 1 ( mod 4 )$ and odd r. Using the building-up construction, we construct MDS self-dual codes of length four and eight over $G R ( p m , 3 )$ with ( $p = 3$ and $m = 2 , 3 , 4 , 5 , 6$ ), ( $p = 7$ and $m = 2 , 3$ ), ( $p = 11$ and $m = 2$ ), ( $p = 19$ and $m = 2$ ), ( $p = 23$ and $m = 2$ ), and ( $p = 31$ and $m = 2$ ). In the building-up construction, it is important to determine the existence of a square matrix U such that $U U T = − I$ , which is called an antiorthogonal matrix. We prove that there is no $2 × 2$ antiorthogonal matrix over $G R ( 2 m , r )$ with $m ≥ 2$ and odd r. View Full-Text
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MDPI and ACS Style

Han, S. MDS Self-Dual Codes and Antiorthogonal Matrices over Galois Rings. Information 2019, 10, 153.

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