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Designs, Codes and Cryptography
Published in: Designs, Codes and Cryptography 2/2016

01-08-2016

Complete weight enumerators of some cyclic codes

Authors: Chengju Li, Qin Yue, Fang-Wei Fu

Published in: Designs, Codes and Cryptography | Issue 2/2016

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Abstract

Let \({\mathbb {F}}_r\) be a finite field with \(r=q^m\) elements, \(\alpha \) a primitive element of \({\mathbb {F}}_r\), \(\hbox {Tr}_{r/q}\) the trace function from \({\mathbb {F}}_r\) onto \({\mathbb {F}}_q\), \(r-1=nN\) for two integers \(n, N \ge 1\), and \(\theta =\alpha ^N\). In this paper, we use Gauss sums to investigate the complete weight enumerators of irreducible cyclic codes
$$\begin{aligned} {{\mathcal {C}}}=\big \{{\mathbf{c }}(a)=(\hbox {Tr}_{r/q}(a), \hbox {Tr}_{r/q}(a\theta ), \ldots , \hbox {Tr}_{r/q}\big (a\theta ^{n-1}\big ): a \in {\mathbb {F}}_r\big \} \end{aligned}$$
and explicitly present the complete weight enumerators of some irreducible cyclic codes when \(\gcd (n, q-1)=q-1 \text{ or } \frac{q-1}{2}\). Moreover, we determine the complete weight enumerators of a class of cyclic codes with the check polynomials \(h_1(x)h_2(x)\) by using Gauss sums, where \(h_i(x)\) are the minimal polynomials of \(\alpha _i^{-1}\) over \({\mathbb {F}}_q\) and \({\mathbb {F}}_{q^{m_i}}^*=\langle \alpha _i \rangle \) for \(i=1,2\). We shall obtain their explicit complete weight enumerators if \(\gcd (m_1,m_2)=1\) and \(q=3\) or \(4\).
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Metadata
Title
Complete weight enumerators of some cyclic codes
Authors
Chengju Li
Qin Yue
Fang-Wei Fu
Publication date
01-08-2016
Publisher
Springer US
Published in
Designs, Codes and Cryptography / Issue 2/2016
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI
https://doi.org/10.1007/s10623-015-0091-5

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