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Erschienen in:
10.06.2021
A class of linear codes with their complete weight enumerators over finite fields
Erschienen in: Cryptography and Communications | Ausgabe 5/2021
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Abstract
For any positive integer m > 2 and an odd prime p, let \(\mathbb {F}_{p^{m}}\) be the finite field with pm elements and let \( \text {Tr}^{m}_{e}\) be the trace function from \(\mathbb {F}_{p^{m}}\) onto \(\mathbb {F}_{p^{e}}\) for a divisor e of m. In this paper, for the defining set \(D=\{x\in \mathbb {F}_{p^{m}}:\text {Tr}^{m}_{e}(x)=1\text { and } \text {Tr}^{m}_{e}(x^{2})=0\}=\{d_{1}, d_{2}, \ldots , d_{n}\}\)(say), we define a pe-ary linear code \(\mathcal {C}_{D}\) by
Then we determine the complete weight enumerator and weight distribution of the linear code \(\mathcal {C}_{D}\). The presented code is optimal with respect to the Griesmer bound provided that \(\frac {m}{e}=3\). In fact, it is MDS when \(\frac {m}{e}=3\). This paper gives the results of S. Yang, X. Kong and C. Tang (Finite Fields Appl. 48 (2017)) if we take e = 1. In addition to the generalization of the results of Yang et al., we study the dual code \(\mathcal {C}_{D}^{\perp }\) of the code \(\mathcal {C}_{D}\) as well as find some optimal constant composition codes.
$$ \mathcal{C}_{D}=\{\textbf{c}_{a} =\left( \text{Tr}^{m}_{e}(ad_{1}), \text{Tr}^{m}_{e}(ad_{2}),\ldots,\text{Tr}^{m}_{e}(ad_{n})\right) : a\in \mathbb{F}_{p^{m}}\}. $$