On the minimal weight of some singly-even codes

It is shown that the minimal distance d of a singly-even self-dual [24t+8, 12t+4] code is at most 4t+2 if its shadow contains a weight 4 vector, t is even, and (_t^5t) is odd. It is proved particularly that there does not exist a [56, 28, 12] singly even self-dual code with 4862 words of weight 12. This answers a question raised by Conway and Sloane (see ibid., vol.36, p.1319-33, 1990)