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Several classes of optimal $p$-ary cyclic codes with minimal distance four (2208.14404v1)

Published 30 Aug 2022 in cs.IT and math.IT

Abstract: Cyclic codes are a subclass of linear codes and have wide applications in data storage systems, communication systems and consumer electronics due to their efficient encoding and decoding algorithms. Let $p\ge 5$ be an odd prime and $m$ be a positive integer. Let $\mathcal{C}{(1,e,s)}$ denote the $p$-ary cyclic code with three nonzeros $\alpha$, $\alphae$, and $\alphas$, where $\alpha $ is a generator of ${\mathbb F}{pm}*$, $s=\frac{pm-1}{2}$, and $2\le e\le pm-2$. In this paper, we present four classes of optimal $p$-ary cyclic codes $\mathcal{C}{(1,e,s)}$ with parameters $[pm-1,pm-2m-2,4]$ by analyzing the solutions of certain polynomials over finite fields. Some previous results about optimal quinary cyclic codes with parameters $[5m-1,5m-2m-2,4]$ are special cases of our constructions. In addition, by analyzing the irreducible factors of certain polynomials over ${\mathbb F}{5m}$, we present two classes of optimal quinary cyclic codes $\mathcal{C}_{(1,e,s)}$.

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