On the codewords of generalized Reed-Muller codes reaching the fourth weight

Abstract

Recently, the fourth weight of generalized Reed-Muller codes $R_q\left(a\right(q-1)+b,m)$ have been characterized for $3\le b<\frac{q+4}{3}$ and $0\le a\le m-3$. In this paper, we study the codewords reaching the fourth weight of generalized Reed-Muller codes. We will determine the fourth weight codewords of generalized Reed-Muller codes of order $r=a(q-1)+b$ with $3\le b<\frac{q+4}{3}$ and $0\le a\le m-3$. Also, we characterize the number of fourth weight codewords of $R_q(b,m)$ for $3\le b<\frac{q+4}{3}$.