The fourth smallest Hamming weight in the code of the projective plane over ℤ/p ℤ

2017/12/20 by Bhaskar Bagchi, Bagchi, Bhaskar
Computer Science · Engineering · #Coding theory and cryptography #Combinatorics (math.CO) #Cooperative Communication and Network Coding #FOS: Mathematics #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.1712.07391

openalex publication_date 2017/12/20 · openalex created_date 2018/01/05 · openalex updated_date 2026/07/28

Abstract

Let p be a prime and let C_p denote the p-ary code of the projective plane over \mathbb Z/pℤ. It is well known that the minimum weight of non-zero words in C_p is p+1, and Chouinard proved that, for p ≥ 3, the second and third minimum weights are 2p and 2p+1. In 2007, Fack et. al. determined, for p≥ 5, all words of C_p of these three weights. In this paper we recover all these results and also prove that, for p ≥ 5, the fourth minimum weight of C_p is 3p-3. The problem of determining all words of weight 3p-3 remains open.

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