Upper bounds for ternary constant weight codes from semidefinite programming and representation theory

In this thesis we use a semidefinite programming approach to find explicit upper bounds on the size of ternary constant weight codes with prescribed minimum distance d. By constructing a graph Γ = (X,E), on the set, X, of all possible ternary words of weight w letting {x, y} ∈ E ⇔ 0 < dH(x, y) < d, we can view this problem as a special case of the stable set problem. Using symmetry, the constraint matrices of the semidefinite program can be seen to belong to the algebra, A, of G0-invariant matrices. Where G0 is the subgroup of the automorphism group of Γ, stabilizing a special point of X. Using representation theory we give a general method for explicitly finding the block diagonalization of the algebra of G-invariant matrices for a finite group G. This is applied to the Terwilliger algebra of the binary Hamming scheme and makes it possible to interpret the entries of the block diagonalized matrices as Hahn polynomials. Moreover, we find a block diagonalization of the algebra A. This is used to reduce the complexity of the semidefinite program and makes it possible to explicitly calculate upper bounds on the size of ternary constant weight codes. One new bound has been obtained.