  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
$A(18,6)\leq 673$, $A(19,6)\leq 1237$, $A(20,6)\leq 2279$, $A(23,6)\leq 13674$,
$A(19,8)\leq 135$, $A(25,8)\leq 5421$, $A(26,8)\leq 9275$, $A(21,10)\leq 47$,
$A(22,10)\leq 84$, $A(24,10)\leq 268$, $A(25,10)\leq 466$, $A(26,10)\leq 836$,
$A(27,10)\leq 1585$, $A(25,12)\leq 55$, and $A(26,12)\leq 96$.
  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 $d$.
