On Some Properties of the Hyper-Kloosterman Codes

Abstract

The hyper-Kloosterman code was first defined as a trace code by Chinen-Hiramatsu [1]. In this article, two basic parameters of it, the minimum distance and the dimension are estimated. Analysis of the dimension shows that it is one of few examples of trace codes, of which the dimensions do not reduce when taking the trace, and are determined explicitly. It is also shown that the hyper-Kloosterman code can be realized as a quasi-cyclic code. It implies a method of explicit construction of quasi-cyclic codes of a new type.