Weight distribution of a class of $p$-ary codes

Let $p$ be a prime, and let $N$ be a positive integer such that $p$ is a primitive root modulo $N$. Define $q = p^e$, where $e = \phi(N)$, and let $\mathbb{F}_q$ be the finite field of order $q$ with $\mathbb{F}_p$ as its prime subfield. Denote by $\mathrm{Tr}$ the trace function from $\mathbb{F}_q$ to $\mathbb{F}_p$. For $\alpha \in \mathbb{F}_p$ and $\beta \in \mathbb{F}_q$, let $D$ be the set of nonzero solutions in $\mathbb{F}_q$ to the equation $\mathrm{Tr}(x^{\frac{q-1}{N}} + \beta x) = \alpha$. Writing $D = \{d_1, \ldots, d_n\}$, we define the code $\mathcal{C}_{\alpha,\beta} = \{(\mathrm{Tr}(d_1 x), \ldots, \mathrm{Tr}(d_n x)) : x \in \mathbb{F}_q\}$. In this paper, we investigate the weight distribution of $\mathcal{C}_{\alpha,\beta}$ for all $\alpha \in \mathbb{F}_p$ and $\beta \in \mathbb{F}_q$, with a focus on general odd primes $p$. When $\beta = 0$, we establish that $\mathcal{C}_{\alpha,0}$ is a two-weight code for any $\alpha \in \mathbb{F}_p$ and compute its weight distribution. For $\beta \neq 0$, we determine all possible weights of codewords in $\mathcal{C}_{\alpha,\beta}$, demonstrating that it has at most $p+1$ distinct nonzero weights. Additionally, we prove that the dual code $\mathcal{C}_{0,0}^{\perp}$ is optimal with respect to the sphere packing bound. These findings extend prior results to the broader case of any odd prime $p$.