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On Some Classes of Repeated-root Constacyclic Codes of Length a Power of 2 over Galois Rings

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Advances in Ring Theory

Part of the book series: Trends in Mathematics ((TM))

Abstract

Negacyclic codes of length 2s over the Galois ring GR(2a, m) are ideals of the chain ring \( \frac{{GR \left( {2^a ,m} \right)\left[ x \right]}} {{\left\langle {x^{2^s } + 1} \right\rangle }} \). This structure is used to provide the Hamming and homogeneous distances of all such negacyclic codes. The technique is then generalized to obtain the structure and Hamming and homogeneous distances of all γ-constacyclic codes of length 2s over GR(2a, m), where γ is any unit of the ring GR(2a, m) that has the form γ = (4k 0 −1)+4k 1ξ +...+ 4k m−1ξm−1, for integers k 0, k 1, ..., k m−1. Among other results, duals of such γ-constacyclic codes are studied, and necessary and sufficient conditions for the existence of a self-dual γ-constacyclic code are established.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Kent State University, 4314 Mahoning Avenue, Warren, OH, 44483, USA

    Hai Q. Dinh

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  1. Hai Q. Dinh
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  1. Department of Mathematics, Ohio University, 321 Morton Hall, Athens, OH, 45701, USA

    Dinh Van Huynh  & Sergio R. López-Permouth  & 

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Dedicated to Professor S.K. Jain on the occasion of his seventieth birthday.

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Dinh, H.Q. (2010). On Some Classes of Repeated-root Constacyclic Codes of Length a Power of 2 over Galois Rings. In: Van Huynh, D., López-Permouth, S.R. (eds) Advances in Ring Theory. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0286-0_10

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