A unique representation of cyclic codes over GR(pⁿ; r)

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Let R be a Galois ring GR(pⁿ, r) of characteristic pn and of order p
nr. 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 $s = v_p(N)$ and $v_p$ is 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
$n = 2$.

Brief Biography

Sami Alabiad is a researcher and PhD student in the department of mathematics at King Saud University, where he completed his master’s degree in coding theory under the supervision of  Prof. Yousef Alkhamees. His doctoral research focuses on the study of finite chain rings and their applications in coding theory. Among other published articles, he managed to correct a paper published in the Journal of Number Theory (1972).