On the weights of linear codes with prescribed automorphisms

Abstract

The number of nonzero weights of a linear code is essential in coding theory as it unveils salient properties of the code, such as its covering radius. In this paper, we establish two upper bounds on the number of nonzero weights of a linear code with prescribed automorphism. Our bounds are applicable for almost all linear codes and tighter than previously known bounds. Examples confirm that our bounds are sharp on numerous occasions. In addition, we give an infinite family of linear codes that attain our bounds with equality.