A Unique Representation of Cyclic Codes over <i>GR</i>(<i>pⁿ</i>,<i>r</i>)

Abstract

Let R be a Galois ring, GR(pn,r), of characteristic pn and of order pnr. In this article, we study cyclic codes of arbitrary length, N, over R. We use discrete Fourier transform (DFT) to determine a unique representation of cyclic codes of length, N, in terms of that of length, ps, where s=vp(N) and vp are the p-adic valuation. As a result, Hamming distance and dual codes are obtained. In addition, we compute the exact number of distinct cyclic codes over R when n=2.

Keywords

• Galois rings
• cyclic codes
• linear codes
• polynomials
• coding theory