Repeated-root constacyclic codes of length ℓ t p s and their dual codes

Abstract

Constacyclic codes form an interesting family of error-correcting codes due to their rich algebraic structure, and are generalizations of cyclic and negacyclic codes. In this paper, we classify repeated-root constacyclic codes of length l t p s over the finite field F p m $\mathbb {F}_{p^{m}}$ containing p m elements, where l ? 1(mod 2), p are distinct primes and t, s, m are positive integers. Based upon this classification, we explicitly determine the algebraic structure of all repeated-root constacyclic codes of length l t p s over F p m $\mathbb {F}_{p^{m}}$ and their dual codes in terms of generator polynomials. We also observe that self-dual cyclic (negacyclic) codes of length l t p s over F p m $\mathbb {F}_{p^{m}}$ exist only when p = 2 and list all self-dual cyclic (negacyclic) codes of length l t 2 s over F 2 m $\mathbb {F}_{2^{m}}$ . We also determine all self-orthogonal cyclic and negacyclic codes of length l t p s over F p m $\mathbb {F}_{p^{m}}$ . To illustrate our results, we determine all constacyclic codes of length 175 over F 5 $\mathbb {F}_{5}$ and all constacyclic codes of lengths 147 and 3087 over F 7 $\mathbb {F}_{7}$ .