Gleason´s theorem on self-dual codes

The weight enumerator of a code is the polynomial begin{equation} W(x,y)= sum_{r=0}^n A_r x^{n-r} y^r, end{equation} where denotes the block length and , denotes the number of codewords of weight . Let be a self-dual code over in which every weight is divisible by . Then Gleason\´s theorem states that 1) if = 2 and = 2, the weight enumerator of is a sum of products of the polynomials and if = 2 and = 4, the weight enumerator is a sum of products of and ; and 3) if = 3 and = 3, the weight enumerator is a sum of products of and . In this paper we give several proofs of Gleason\´s theorem.