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

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 \(\mathbb {F}_{p^{m}}\) 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 \(\mathbb {F}_{p^{m}}\) 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 \(\mathbb {F}_{p^{m}}\) exist only when p = 2 and list all self-dual cyclic (negacyclic) codes of length ℓ ^t 2 ^s over \(\mathbb {F}_{2^{m}}\). We also determine all self-orthogonal cyclic and negacyclic codes of length ℓ ^t p ^s over \(\mathbb {F}_{p^{m}}\). To illustrate our results, we determine all constacyclic codes of length 175 over \(\mathbb {F}_{5}\) and all constacyclic codes of lengths 147 and 3087 over \(\mathbb {F}_{7}\).