Singly even self-dual codes of length 24k+10 and minimum weight 4k+2
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Currently, the existence of an extremal singly even self-dual code of length 24k+10 is unknown for all nonnegative integers k. In this note, we study singly even self-dual [24k+10,12k+5,4k+2] codes. We give some restrictions on the possible weight enumerators of singly even self-dual [24k+10,12k+5,4k+2] codes with shadows of minimum weight at least 5 for k=2,3,4,5. We discuss a method for constructing singly even self-dual codes with minimal shadow. As an example, a singly even self-dual [82,41,14] code with minimal shadow is constructed for the first time. In addition, as neighbors of the code, we construct singly even self-dual [82,41,14] codes with weight enumerator for which no singly even self-dual code was previously known to exist.
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