The triply shortened binary Hamming code is optimal

Abstract By explicit evaluation of the linear programming bound for the case q =2, d =3 (after adding one inequality when n =0 (mod4)), we prove that A[n, 3]⩽2 n−2 [ 1 4 n+1] . In particular the binary Hamming code is shown to remain optimal when it is shortened one, two or three times. Furthermore some general relations between solutions of the LP problem are derived.