Weight distributions of a class of cyclic codes from $\Bbb F_l$-conjugates

Chengju  Li; Qin  Yue; Ziling  Heng

doi:10.3934/amc.2015.9.341

Article Contents

2015, Volume 9, Issue 3: 341-352. Doi: 10.3934/amc.2015.9.341

This issue Previous Article Ideal forms of Coppersmith's theorem and Guruswami-Sudan list decoding Next Article An improved certificateless strong key-insulated signature scheme in the standard model

Weight distributions of a class of cyclic codes from $\Bbb F_l$-conjugates

1.
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211100

Received: July 31, 2014

Published: October 2015

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Abstract

Let $\Bbb F_{q^k}$ be a finite field and $\alpha$ a primitive element of $\Bbb F_{q^k}$, where $q=l^f$, $l$ is a prime power, and $f$ is a positive integer. Suppose that $N$ is a positive integer and $m_{g^{l^u}}(x)$ is the minimal polynomial of $g^{l^u}$ over $\Bbb F_q$ for $u=0, 1, \ldots, f-1$, where $g=\alpha^{-N}$. Let $\mathcal C$ be a cyclic code over $\Bbb F_q$ with check polynomial $$m_g(x)m_{g^l}(x) \cdots m_{g^{l^{f-1}}}(x).$$ In this paper, we shall present a method to determine the weight distribution of the cyclic code $\mathcal C$ in two cases: (1) $\gcd(\frac {q^k-1} {l-1}, N)=1$; (2) $l=2$ and $f=2$. Moreover, we will obtain a class of two-weight cyclic codes and a class of new three-weight cyclic codes.

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Mathematics Subject Classification: Primary: 11T71, 11T24; Secondary: 94B15.

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