2024/06/11 by Mrinmoy Datta, Datta, Mrinmoy
Computer Science · Engineering · Medicine · #11T06 #14G05 #14G15 #Algebraic Geometry (math.AG) #Cancer Mechanisms and Therapy #Coding theory and cryptography #FOS: Mathematics #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2406.07339
openalex publication_date 2024/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Determining the weight distributions of the projective Reed-Muller codes is a very hard problem and has been studied extensively in the literature. In this article, we provide an alternative proof of the second weight of the projective Reed-Muller codes \PRM (d, m) where m ≥ 3 and 3 ≤ d ≤ (q+3)/(2). We show that the second weight is attained by codewords that correspond to hypersurfaces containing a hyperplane under the hypothesis on d. Furthermore, we compute the second weight of \PRM (d, 2) for 3 ≤ d ≤ q-1. Furthermore, we give an upper bound for the third weight of \PRM(d, 2).