Singly even self-dual codes of length $24k+10$ and minimum weight $4k+2$

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.