We give a classification of four-circulant singly even self-dual [60,30,d] codes for d=10 and 12. These codes are used to construct extremal singly even self-dual [60,30,12]
Nearby in the stack
codes with weight enumerator for which no extremal singly even self-dual code was previously known to exist. From extremal singly even self-dual
[60,30,12]
codes, we also construct extremal singly even self-dual
[58,29,10]
codes with weight enumerator for which no extremal singly even self-dual code was previously known to exist. Finally, we give some restriction on the possible weight enumerators of certain singly even self-dual codes with shadow of minimum weight