The extended and generalized rank weight enumerator of a code

This paper investigates the rank weight enumerator of a code over L, where L is a nite extension of a eld K. This is a generalization of the case where K = Fq and L = Fqm of Gabidulin codes to arbitrary characteristic. We use the notion of counting polynomials, to dene the (extended) rank weight enumerator, since in this generality the set of codewords of a given rank weight is no longer nite. Also the extended and generalized rank weight enumerator are studied in analogy with previous work on codes with respect to the Hamming metric.