A new upper bound on the minimal distance of self-dual codes

It is shown that the minimal distance d of a binary self-dual code of length n⩾74 is at most 2[(n+6)/10]. This bound is a consequence of some new conditions on the weight enumerator of a self-dual code obtained by considering a particular translate of the code, called its shadow. These conditions also enable one to find the highest possible minimal distance of a self-dual code for all n⩾60; to show that self-dual codes with d⩽6 exist precisely for n⩾22, with d ⩾8 exist precisely for n=24, 32 and n⩾26, and with d⩾10 exist precisely for n⩾46; and to show that there are exactly eight self-dual codes of length 32 with d=8. Several of the self-dual codes of length 34 have trivial group (this appears to be the smallest length where this can happen)