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Erschienen in:
27.05.2021 | Original Paper
An explicit expression for all distinct self-dual cyclic codes of length \(p^k\) over Galois ring \(\mathrm{GR}(p^2,m)\)
Erschienen in: Applicable Algebra in Engineering, Communication and Computing | Ausgabe 3/2023
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Abstract
Let p be any odd prime number and let m, k be arbitrary positive integers. The construction for self-dual cyclic codes of length \(p^k\) over the Galois ring \(\mathrm{GR}(p^2,m)\) is the key to construct self-dual cyclic codes of length \(p^kn\) over the integer residue class ring \({\mathbb {Z}}_{p^2}\) for any positive integer n satisfying \(\mathrm{gcd}(p,n)=1\). So far, existing literature has only determined the number of these self-dual cyclic codes (Des Codes Cryptogr 63:105–112, 2012). In this paper, we give an efficient construction for all distinct self-dual cyclic codes of length \(p^k\) over \(\mathrm{GR}(p^2,m)\) by using column vectors of Kronecker products of matrices with specific types. On this basis, we further obtain an explicit expression for all these self-dual cyclic codes by using binomial coefficients.