Let A(n,d) be the maximum number of 0,1 words of length n, any two
having Hamming distance at least d. We prove A(20,8)=256, which implies
that the quadruply shortened Golay code is optimal. Moreover, we show
A(18,6)≤673, A(19,6)≤1237, A(20,6)≤2279, A(23,6)≤13674,
A(19,8)≤135, A(25,8)≤5421, A(26,8)≤9275, A(21,10)≤47,
A(22,10)≤84, A(24,10)≤268, A(25,10)≤466, A(26,10)≤836,
A(27,10)≤1585, A(25,12)≤55, and A(26,12)≤96.
The method is based on the positive semidefiniteness of matrices derived from
quadruples of words. This can be put as constraint in a semidefinite program,
whose optimum value is an upper bound for A(n,d). The order of the matrices
involved is huge. However, the semidefinite program is highly symmetric, by
which its feasible region can be restricted to the algebra of matrices
invariant under this symmetry. By block diagonalizing this algebra, the order
of the matrices will be reduced so as to make the program solvable with
semidefinite programming software in the above range of values of n and d.Comment: 15 page