An interesting open question is whether a(72, 36, 16)doubly even codeCexists. In  the odd prime numbers which can divide the order of the group ofCwere determined and23is the largest of these. Twenty-three is eliminated by reducing the problem to the consideration of348codes, each of which is shown to have minimum weight12or less. One of these codes, denoted byC', arises from the(a + x, b + x, a + b + x)construction whereaandbare in one quadratic residue code andxis in the other. The weight distribution ofC'is given.