Some two-weight and three-weight linear codes
Chengju Li, Sunghan Bae, Shudi Yang · Advances in Mathematics of Communications · 2018
Let $\Bbb F_q$ be the finite field with $q = p^m$ elements, where $p$ is an odd prime and $m$ is a positive integer. For a positive integer $t$, let $D \subset \Bbb F_q^t$ and let $\mbox{Tr}_m$ be the trace function from $\Bbb F_q$ onto $\Bbb F_p$. We define a $p$-ary linear code $\mathcal C_D$ by $ \mathcal C_D = \{\textbf{c}(a_1,a_2, ..., a_t): a_1, a_2, ..., a_t ∈ \Bbb F_{p^m}\}, $ where $\textbf{c}(a_1,a_2, ..., a_t) = \big(\mbox{Tr}_m(a_1x_1+a_2x_2+···+a_tx_t)\big)_{(x_1,x_2, ..., x_t)∈ D}.$ In this paper, we will present the weight enumerators of the linear codes $\mathcal C_D$ in the following two cases:1. $D = \{(x_1,x_2, ..., x_t) ∈ \Bbb F_q^t \setminus \{(0,0, ..., 0)\}: \mbox{Tr}_m(x_1^2+x_2^2+···+x_t^2) = 0\}$;2. $D = \{(x_1,x_2, ..., x_t) ∈ \Bbb F_q^t: \mbox{Tr}_m(x_1^2+x_2^2+···+x_t^2) = 1\}$.It is shown that $\mathcal C_D$ is a two-weight code if $tm$ is even and three-weight code if $tm$ is odd in both cases. The weight enumerators of $\mathcal C_D$ in the first case generalize the results in [17] and [18]. The complete weight enumerators of $\mathcal C_D$ are also investigated.