Abstract
Let m ≥ 2 be any natural number and let be a finite non-chain ring, where and q is a prime power congruent to 1 modulo (m-1). In this paper we study duadic codes over the ring and their extensions. A Gray map from to is defined which preserves self duality of linear codes. As a consequence self-dual, formally self-dual and self-orthogonal codes over are constructed. Some examples are also given to illustrate this.
Highlights
Duadic codes form a class of cyclic codes that generalizes quadratic residue codes from prime to composite lengths
In this paper we extend our results of [9] to duadic codes over the ring = q + u q + u2 q + + um−1 q, where um = u, q is a prime power congruent to 1 modulo (m −1)
The following is a well known result : Lemma 4: (i) Let C be a cyclic code of length n over a finite ring S generated by the idempotent E in S [ x] / xn −1 C⊥ is generated by the idempotent
Summary
Duadic codes form a class of cyclic codes that generalizes quadratic residue codes from prime to composite lengths. Kaya et al [4] and Zhang et al [5] studied quadratic residue codes over p + u p where p is an odd prime. Kaya et al [6] studied quadratic residue codes over 2 + u 2 + u2 2 whereas Liu et al [7] studied them over non-local ring p + u p + u2 p where u3 = u and p is an odd prime. In this paper we extend our results of [9] to duadic codes over the ring = q + u q + u2 q + + um−1 q , where um = u , q is a prime power congruent to 1 modulo (m −1).
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