Two classes of $p$-ary linear codes and their duals

Let $\mathbb{F}{p^m}$ be the finite field of order $p^m$, where $p$ is an odd prime and $m$ 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{pk+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 $k$ 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 $\mathcal{C}k$, which is defined as $$\mathcal{\bar{C}}k=\left{\left( {\rm Tr}1^m\left(a x^{{{pk}+1}+bx\right)+c\right)}{x \in \mathbb{F}{p^m}} : \, a,b \in \mathbb{F}{p^m}, \,\,c \in \mathbb{F}p\right}.$$ The parameters of these linear codes are new in some cases. Some of the presented codes are optimal or almost optimal. Moreover, let $v2(\cdot)$ denote the 2-adic order function and $v2(0)=\infty$, the duals of $\mathcal{C}k$ and $\mathcal{\bar{C}}k$ are optimal with respect to the Sphere Packing bound if $p>3$, and the dual of $\mathcal{\bar{C}}k$ is an optimal ternary linear code for the case $v2(m)\leq v2(k)$ if $p=3$ and $m>1$.