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^{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 $k$ 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 $${\bar{\mathcal{C}}}_k=\{{({\mathrm{T}\mathrm{r}}_1^m(a{x}^{p^k+1}+bx)+c)}_{x\in {\mathbb{F}}_{p^m}}:a,b\in {\mathbb{F}}_{p^m},c\in {\mathbb{F}}_p\}\mathrm{.}$$ The parameters of these linear codes are new in some cases. Some of the presented codes are optimal or almost optimal. Moreover, let $v_2(\cdot )$ denote the 2-adic order function and $v_2(0)=\mathrm{\infty }$, the duals of ${\mathcal{C}}_k$ and ${\bar{\mathcal{C}}}_k$ are optimal with respect to the Sphere Packing bound if $p>3$, and the dual of ${\bar{\mathcal{C}}}_k$ is an optimal ternary linear code for the case $v_2(m)\le v_2(k)$ if $p=3$ and $m>1$.