The linearity of Carlet’s Gray image of linear codes over ${\mathbb {Z}}_{8}$

Abstract

A ${\mathbb {Z}}_{2^s}$-additive code ${\mathcal {C}}$ of length n is a subgroup of ${\mathbb {Z}}_{2^s}^n$. Carlet (IEEE Trans Inf Theory 44:1543–1547, 1998) introduced a Gray map $\Phi$ (Carlet’s Gray map) on ${\mathbb {Z}}_{2^s}$ and Tapia-Recillas and Vega (SIAM J Discret Math 17:103–113, 2003) have been shown that $\Phi ({\mathcal {C}})$ is a binary linear code if and only if for every ${\mathbf {u}}, {\mathbf {v}}\in {\mathcal {C}}$, $2({\mathbf {u}}\odot {\mathbf {v}})\in {\mathcal {C}}$, which their number is ${\mid {\mathcal {C}}\mid }^2 -\left( {\begin{array}{c}\mid {\mathcal {C}}\mid \\ 2\end{array}}\right)$. Let ${\mathcal {C}}$ be a linear code over ${\mathbb {Z}}_8$ of type $\{\delta _{0}, \delta _{1}, \delta _{2}\}$. In this paper, by using the rows of the generator matrix (in standard form) of ${\mathcal {C}}$, we introduce a set ${\mathcal {S}}$ with $\mid {\mathcal {S}} \mid \le \frac{\delta _{0}(\delta _{0}+3)}{2}+\delta _{1}$ and show that $\Phi ({\mathcal {C}})$ is linear if and only if for every ${\mathbf {u}}, {\mathbf {v}}\in {\mathcal {S}}$, $2({\mathbf {u}}\odot {\mathbf {v}})\in {\mathcal {C}}$. The results show that for a linear code $\mathcal{C}$ over ${\mathbb {Z}}_8$ calculations to check the linearity of $\Phi ({\mathcal {C}})$ are less than Tapia-Recillas and Vega’s method. In addition, for a cyclic code of odd length over ${\mathbb {Z}}_8$, a condition on the generator polynomial of this code is given which its Carlet’s Gray image is linear. Also, as a result, we show that the Carlet’s Gray image of a quadratic residue code over ${\mathbb {Z}}_8$ is not linear.