Abstract
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.
Highlights
In coding theory, the minimum distance is very important because it indicates the ability to perform error correction on the codes
In this study, we explore maximum distance separable (MDS) self-dual codes over Galois rings
MDS self-dual codes have been studied over finite fields and over finite rings Z pm
Summary
The minimum distance is very important because it indicates the ability to perform error correction on the codes. Codes that contain both structures, which are called MDS self-dual codes, have been investigated. We used the building-up construction method which was described in [9] to construct MDS self-dual codes. For this method, it is very important to determine the existence of a square matrix U such that UU T = − I, which is called an antiorthogonal matrix [10,11]. All of the computations in this paper were performed using the computer algebra system Magma [13]
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