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

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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 \(\mathbb {F}_{p^{m}}\) 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 t2s 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 \(\mathbb {F}_{7}\).

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  1. Department of Mathematics, Indian Institute of Technology Delhi, New Delhi, 110016, India

    Anuradha Sharma

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  1. Anuradha Sharma
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Sharma, A. Repeated-root constacyclic codes of length ℓ t p s and their dual codes. Cryptogr. Commun. 7, 229–255 (2015). https://doi.org/10.1007/s12095-014-0106-5

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  • DOI: https://doi.org/10.1007/s12095-014-0106-5

Keywords

Mathematics Subject Classification 2010