1 research outputs found
Two classes of -ary linear codes and their duals
Let be the finite field of order , where is an
odd prime and is a positive integer. In this paper, we investigate a class
of subfield codes of linear codes and obtain the weight distribution of
\begin{equation*} \begin{split} \mathcal{C}_k=\left\{\left(\left( {\rm
Tr}_1^m\left(ax^{p^k+1}+bx\right)+c\right)_{x \in \mathbb{F}_{p^m}}, {\rm
Tr}_1^m(a)\right) : \, a,b \in \mathbb{F}_{p^m}, c \in \mathbb{F}_p\right\},
\end{split} \end{equation*}
where is a nonnegative integer. Our results generalize the results of the
subfield codes of the conic codes in \cite{Hengar}. Among other results, we
study the punctured code of , which is defined as
The parameters of these
linear codes are new in some cases. Some of the presented codes are optimal or
almost optimal. Moreover, let denote the 2-adic order function and
, the duals of and are
optimal with respect to the Sphere Packing bound if , and the dual of
is an optimal ternary linear code for the case
if and