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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 singlyeven self-dual $[60,30,12]$ codes with weight enumerator for which no extremalsingly even self-dual code was previously known to exist. From extremal singlyeven self-dual $[60,30,12]$ codes, we also construct extremal singly evenself-dual $[58,29,10]$ codes with weight enumerator for which no extremalsingly even self-dual code was previously known to exist. Finally, we give somerestriction on the possible weight enumerators of certain singly even self-dualcodes with shadow of minimum weight $1$.

Binary extremal self-dual codes of length $60$ and related codes