A new approach to the Kasami codes of type 2

TL;DR

This paper introduces a novel construction of Kasami codes of type 2, providing new insights into their weight distribution, automorphism groups, and related completely regular codes over finite fields of characteristic two.

Contribution

It offers a new derivation of the Kasami code's weight distribution, describes its coset graph, proves its complete regularity, and constructs generalized Kasami codes with similar properties.

Findings

Derived the weight distribution of Kasami codes.

Determined automorphism groups of Kasami and related codes.

Constructed new generalized Kasami codes with isomorphic coset graphs.

Abstract

The dual of the Kasami code of length $q^2-1$, with $q$ a power of $2$, is constructed by concatenating a cyclic MDS code of length $q+1$ over $F_q$ with a Simplex code of length $q-1$. This yields a new derivation of the weight distribution of the Kasami code, a new description of its coset graph, and a new proof that the Kasami code is completely regular. The automorphism groups of the Kasami code and the related $q$-ary MDS code are determined. New cyclic completely regular codes over finite fields a power of $2$, generalized Kasami codes, are constructed; they have coset graphs isomorphic to that of the Kasami codes. Another wide class of completely regular codes, including additive codes, as well as unrestricted codes, is obtained by combining cosets of the Kasami or generalized Kasami code.