MDS Self-Dual Codes and Antiorthogonal Matrices over Galois Rings
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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.
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2018 ◽
Vol 43
(1-4)
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pp. 13-45
2007 ◽
Vol 45
(2)
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pp. 247-258
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1984 ◽
Vol 32
(9)
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pp. 998-1006
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