Existence of an analogue at other residue classes

Construct an analogue of Theorem R.2 for lengths L≡5 (mod 8) or L≡1 (mod 8), if such a half-flip 32-modular Hadamard family exists.

Background

Theorem R.2 gives an explicit half-flip family for L≡3 (mod 8), based on a specific run-length comparator and a periodic sign sequence.

The paper leaves unresolved whether comparable explicit constructions can be obtained in the other residue classes L≡5 (mod 8) or L≡1 (mod 8).