MDS Self-Dual Codes and Antiorthogonal Matrices over Galois Rings

In this study, we explore maximum distance separable ( MDS) self- dual codes over Galois rings GR ( pm, r) with p similar to 1 ( mod 4) and odd r. Using the building- up construction, we construct MDS self- dual codes of length four and eight over GR ( pm, 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 UUT = I, which is called an antiorthogonal matrix. We prove that there is no 2 similar to 2 antiorthogonal matrix over GR ( 2m, r) with m similar to 2 and odd r.