Three-Weight and Five-Weight Linear Codes over Finite Fields

Pavan Kumar ; Noor Mohammad Khan

Kragujevac Journal of Mathematics, Tome 48 (2024) no. 3, p. 345 .

Résumé

Recently, linear codes constructed from defining sets have been studied extensively. For an odd prime $p$, let $\mathrm{ \mathrm{Tr}}^{m}_{e}$ be the trace function from $\mathbb{F}_{p^m}$ onto $\mathbb{F}_{p^e}$, where $e$ is a divisor of $m$. In this paper, for the defining set $D=\{x\in\mathbb{F}_{p^m}^{*}: \mathrm{ \mathrm{Tr}}^{m}_{e}(x^2+x)=0\}=\{d_{1}, d_{2}, \ldots, d_{n}\}$ (say), we define a $p^e$-ary linear code $\mathcal{C}_{D}$ by $\mathcal{C}_{D}=\{\boldmath{c}_{x} =\big( \mathrm{ \mathrm{ \mathrm{Tr}}}^{m}_{e}(xd_{1}), \mathrm{ \mathrm{Tr}}^{m}_{e}(xd_{2}),..., \mathrm{ \mathrm{Tr}}^{m}_{e}(xd_{n})\big ) : xı \mathbb{F}_{p^m}\}$ and present three-weight and five-weight linear codes with their weight distributions. We show that each nonzero codeword of $\mathcal{C}_{D}$ is minimal for $\frac{m}{e}\geq5$ and, thus, such codes are applicable in secret sharing schemes.

Détail

DOI : 10.46793/KgJMat2403.345K

Classification : 11T71, 94B05
Keywords: linear code, weight distribution, Gauss sum, cyclotomic number, secret sharing