A New Approach to the Kasami Codes of Type 2

Revised version. The automorphism-group part essentially updated (in the previous versions, it contained incorrect results)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 $2$ are constructed. They have coset graphs isomorphic to that of the Kasami codes