An explicit expression for all distinct self-dual cyclic codes of length $$p^k$$ over Galois ring $$\\mathrm{GR}(p^2,m)$$

Abstract

Let p be any odd prime number and let m,\xa0k be arbitrary positive integers. The construction for self-dual cyclic codes of length $$p^k$$\n over the Galois ring $$\\mathrm{GR}(p^2,m)$$\n is the key to construct self-dual cyclic codes of length $$p^kn$$\n over the integer residue class ring $${\\mathbb {Z}}_{p^2}$$\n for any positive integer n satisfying $$\\mathrm{gcd}(p,n)=1$$\n . 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 $$p^k$$\n over $$\\mathrm{GR}(p^2,m)$$\n 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.