On the Optimum of Delsarteʹs Linear Program

We are interested in the maximal size A(n, d) of a binary error-correcting code of length n and distance d, or, alternatively, in the best packing of balls of radius (d−1)/2 in the n-dimensional Hamming space. The best known lower bound on A(n, d) is due to Gilbert and Varshamov and is obtained by a covering argument. The best know upper bound is due to McEliece, Rodemich, Rumsey, and Welch, and is obtained using Delsarteʹs linear programming approach. It is not known whether this is the best possible bound one can obtain from Delsarteʹs linear program. We show that the optimal upper bound obtainable from Delsarteʹs linear program will strictly exceed the Gilbert–Varshamov lower bound. In fact, it will be at least as big as the average of the Gilbert–Varshamov bound and the McEliece, Rodemich, Rumsey, and Welch upper bound. Similar results hold for constant weight binary codes. The average of the Gilbert–Varshamov bound and the McEliece, Rodemich, Rumsey, and Welch upper bound might be the true value of Delsarteʹs bound. We provide some evidence for this conjecture.