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Advances in Mathematics of Communications (AMC)
 

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

Pages: 341 - 352, Volume 9, Issue 3, August 2015      doi:10.3934/amc.2015.9.341

 
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Chengju Li - Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211100, China (email)
Qin Yue - Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211100, China (email)
Ziling Heng - Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211100, China (email)

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.

Keywords:  Weight distribution, cyclic codes, Gauss periods, character sums.
Mathematics Subject Classification:  Primary: 11T71, 11T24; Secondary: 94B15.

Received: August 2014;      Available Online: July 2015.

 References