Nonexistence of certain singly even self-dual codes with minimal shadow

It is known that there is no extremal singly even self-dual [n, n/2, d] code with minimal shadow for (n, d) = (24m+2, 4m+4), (24m+4, 4m+4), (24m+6, 4m+4), (24m + 10, 4m + 4) and (24m + 22, 4m + 6). In this paper, we study singly even self-dual codes with minimal shadow having minimum weight d − 2 for these (n, d). For n = 24m + 2, 24m + 4 and 24m + 10, we show that the weight enumerator of a singly even self-dual [n, n/2, 4m + 2] code with minimal shadow is uniquely determined and we also show that there is no singly even self-dual [n, n/2, 4m + 2] code with minimal shadow for m ≥ 155, m ≥ 156 and m ≥ 160, respectively. We demonstrate that the weight enumerator of a singly even self-dual code with minimal shadow is not uniquely determined for parameters [24m + 6, 12m + 3, 4m + 2] and [24m + 22, 12m + 11, 4m + 4].