2025/03/24 by Cheng, Kaimin, Sheng, Du · 1 citation
#94B05 #Cryptography and Security (cs.CR) #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.2503.19141
Let p be a prime, and let N be a positive integer such that p is a primitive root modulo N. Define q = pe, where e = ϕ(N), and let \mathbbFq be the finite field of order q with \mathbbFp as its prime subfield. Denote by Tr the trace function from \mathbbFq to \mathbbFp. For α∈ \mathbbFp and β∈ \mathbbFq, let D be the set of nonzero solutions in \mathbbFq to the equation Tr(x(q-1)/(N) + βx) = α. Writing D = \d1, …, dn\, we define the code Cα,β = \(Tr(d1 x), …, Tr(dn x)) : x ∈ \mathbbFq\. In this paper, we investigate the weight distribution of Cα,β for all α∈ \mathbbFp and β∈ \mathbbFq, with a focus on general odd primes p. When β= 0, we establish that Cα,0 is a two-weight code for any α∈ \mathbbFp and compute its weight distribution. For β≠ 0, we determine all possible weights of codewords in Cα,β, demonstrating that it has at most p+1 distinct nonzero weights. Additionally, we prove that the dual code C0,0⊥ is optimal with respect to the sphere packing bound. These findings extend prior results to the broader case of any odd prime p.