The fourth smallest Hamming weight in the code of the projective plane over $\mathbb{Z}/p \mathbb{Z}$

Abstract

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