Abstract
Let p be an odd prime, and m, k and d be positive integers such that \(2 \le k\le \frac{m+1}{2}\) and \(\hbox {gcd}(m,d)=1. \pi \) is a primitive element of the finite field \({\mathbb {F}}_{p^{m}}\). The weight enumerator of cyclic codes over \({\mathbb {F}}_{p}\) whose duals have 2k zeros \(\pi ^{-(p^{jd}+1)/2}\) and \(-\pi ^{-(p^{jd}+1)/2} (j=0,1,\ldots ,k-1)\) is determined in the present paper. The weight enumerator of cyclic codes over \({\mathbb {F}}_{p}\) whose duals have \(2k-1\) zeros \(\pi ^{-(p^{(k-1)d}+1)/2}, \pi ^{-(p^{jd}+1)/2}\) and \(-\pi ^{-(p^{jd}+1)/2} (j=0,1,\ldots ,k-2)\) is also determined when \(2\not \mid \frac{m}{gcd(m,k-1)}\) holds.
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The authors are grateful to the referees for their careful reading of the original version of this paper, their detailed comments and suggestions, which have much improved the quality of this paper.
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Communicated by T. Helleseth.
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Yan, H., Liu, C. Two classes of cyclic codes and their weight enumerator. Des. Codes Cryptogr. 81, 1–9 (2016). https://doi.org/10.1007/s10623-015-0125-z
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DOI: https://doi.org/10.1007/s10623-015-0125-z