×

Complete weight enumerators of a new class of linear codes. (English) Zbl 1439.94095

Summary: For an odd prime \(p\), let \(q = p^m\), and denote \(\operatorname{Tr}_e^m\) the trace function from \(\mathbb{F}_q\) onto \(\mathbb{F}_{p^e}\), where \(e\) is a divisor of \(m\). For a positive integer \(t\) and \(a \in \mathbb{F}_{p^e}\), let \(D_a = \{(x_1, x_2, \ldots, x_t) \in \mathbb{F}_q^t \setminus \{(0, 0, \dots, 0) \} : \operatorname{Tr}_e^m(x_1 + x_2 + \ldots + x_t) = a \}\), and define a \(p\)-ary linear code \(\mathcal{C}_{D_a}\) as
\[ \mathcal{C}_{D_a} = \{\mathbf{c}(x_1, x_2, \dots, x_t) :(x_1, x_2, \ldots, x_t) \in \mathbb{F}_q^t \},\]
where
\[ \mathbf{c}(x_1, x_2, \ldots, x_t) = (\operatorname{Tr}_1^m(x_1 d_1^2 + x_2 d_2^2 + \ldots + x_t d_t^2))_{(d_1, d_2, \ldots, d_t) \in D_a}. \]
The complete weight enumerators of linear codes \(\mathcal{C}_{D_a}\) will be presented for any divisor \(e\) of \(m\) and \(a \in \mathbb{F}_{p^e}\), and this new result generalizes that of both J. Ahn et al. [Des. Codes Cryptography 83, No. 1, 83–99 (2017; Zbl 1379.94054)] and S. Yang and Z.-A. Yao [Discrete Math. 340, No. 4, 729–739 (2017; Zbl 1393.94937)].

MSC:

94B05 Linear codes (general theory)
11T23 Exponential sums
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ahn, J.; Ka, D.; Li, C., Complete weight enumerators of a class of linear codes, Des. Codes Cryptogr., 83, 1, 83-99 (2017) · Zbl 1379.94054
[2] Blake, I. F.; Kith, K., On the complete weight enumerator of Reed-Solomon codes, SIAM J. Discrete Math., 4, 2, 164-171 (1991) · Zbl 0725.94006
[3] Ding, C., Linear codes from some 2-designs, IEEE Trans. Inform. Theory, 61, 6, 3265-3275 (2015) · Zbl 1359.94685
[4] Ding, K.; Ding, C., Bianry linear codes with three weights, IEEE Commun. Lett., 18, 11, 1879-1882 (2014)
[5] Ding, C.; Helleseth, T.; Kløve, T.; Wang, X., A generic construction of Cartesian authentication codes, IEEE Trans. Inform. Theory, 53, 6, 2229-2235 (2007) · Zbl 1310.94178
[6] Huffman, W. C.; Pless, V., Fundamentals of Error-Correcting Codes (2003), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1099.94030
[7] Lidl, R.; Niederreiter, H., Finite Fields (1997), Cambridge University Press: Cambridge University Press Cambridge
[8] Wang, Q.; Li, F.; Ding, K.; Lin, D., Complete weight enumerators of two classes of linear codes, Discrete Math., 340, 3, 467-480 (2017) · Zbl 1407.94183
[9] Yang, S.; Kong, X.; Tang, C., A construction of linear codes and their complete weight enumerators, Finite Fields Appl., 48, 196-226 (2017) · Zbl 1403.94115
[10] Yang, S.; Yao, Z., Complete weight enumerators of a class of linear codes, Discrete Math., 340, 4, 729-739 (2017) · Zbl 1393.94937
[11] Yang, S.; Yao, Z., Complete weight enumerators of a family of three-weight linear codes, Des. Codes Cryptogr., 82, 3, 663-674 (2017) · Zbl 1370.94578
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.