Abstract
In this paper, we construct MDS Euclidean self-dual codes which are extended cyclic duadic codes. And we obtain many new MDS Euclidean self-dual codes. We also construct MDS Hermitian self-dual codes from generalized Reed-Solomon codes and constacyclic codes. And we give some results on Hermitian self-dual codes, which are the extended cyclic duadic codes.
Highlights
In this paper, we construct maximum distance separable (MDS) Euclidean self-dual codes which are extended cyclic duadic codes
MDS codes are related to geometric objects called n-arcs
[11] Let D1 and D2 be a pair of odd-like duadic codes of length n over q, ( ) μ−1 Di = Di+1(mod 2)
Summary
A cyclic code C of length n over q can be considered as an ideal, g ( x) , of the ring. Lemma 1 [6] Let n | q −1 and n be an odd integer. Lemma 2 [11] Let D1 and D2 be a pair of odd-like duadic codes of length n over q , ( ) μ−1 Di = Di+1(mod 2). Lemma 3 (Law of Quadratic Reciprocity) [12] Let p and r be odd primes,. Theorem 2 Let q = rt be a prime power, n | q −1 and n be an odd integer. There exists a pair D1 , D2 of MDS odd-like duadic codes of length n and μ−1 ( Di ) = Di+1(mod2) , where even-like duadic codes are MDS self-orthogonal, and. 2t , MDS Euclidean self-dual codes by Lemma 2. We list some new MDS Euclidean self-dual codes in the Table 1
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