Gleason’s Theorem on Self-Dual Codes and Its Generalizations

One of the most remarkable theorems in coding theory is Gleason’s 1970 theorem about the weight enumerators of self-dual codes. In the past 36 years there have been hundreds of papers written about generalizations and applications of this theorem to different types of codes, always on a case-by-case basis. In this talk I will state the theorem and then describe the far-reaching generalization that Gabriele Nebe, Eric Rains and I have developed which includes all the earlier generalizations at once. The full proof has just appeared in our book Self-Dual Codes and Invariant Theory (Springer, 2006). This paper is based on my talk at the conference on Algebraic Combinatorics in honor of Eiichi Bannai, held in Sendai, Japan, June 26–30, 2006. 1. Motivation Self-dual codes are important because they intersect with • communications • combinatorics • block designs, spherical designs • group theory • number theory • sphere packing • quantum codes • conformal field theory, string theory 2. Introduction In classical coding theory (as for example in MacWilliams and Sloane [13]), a code C of length N over a field is a subspace of N . The dual code is C = {u ∈ N : u · c = 0, ∀c ∈ C} . Example: C = {000, 111}, C⊥ = {000, 011, 101, 110} with = 2 . The weight enumerators of these two codes are WC(x, y) = x 3 + y, WC⊥(x, y) = x 3 + 3xy . A code is self-dual if C = C⊥ For example, the binary code i2 = {00, 11} is self-dual, with weight enumerator Wi2(x, y) = x 2 + y . (1) Example: The Hamming code h8 of length 8 is self-dual. This is the binary code with generator matrix: ∞ 0 1 2 3 4 5 6 1 1 1 1 1 1 1 1 0 1 1 1 0 1 0 0 0 0 1 1 1 0 1 0 0 0 0 1 1 1 0 1 The second row of the matrix has 1’s under the quadratic residues 0, 1, 2 and 4 mod 7. The remaining rows are obtained by fixing the infinity coordinate and cycling the other coordinates. This code has weight enumerator Wh8(x, y) = x 8 + 14xy + y . (2) As can be seen from the generator matrix, this code is closely related to the incidence matrix of the projective plane of order 2. If we replace the prime 7 in this construction by 23, we get the binary Golay self-dual code