Classification of optimal $(v,4,1)$ binary cyclically permutable constant-weight codes and cyclic $2$-$(v,4,1)$ designs with $v\le76$

We classify up to isomorphism optimal $(v,4,1)$ binary cyclically permutable constant-weight (CPCW) codes with $v\le76$ and cyclic $2$-$(73,4,1)$ and $2$-$(76,4,1)$ designs. There is a one-to-one correspondence between optimal $(v,4,1)$ CPCW codes, optimal cyclic binary constant-weight codes with weight $4$ and minimum distance $6$, $(v,4;\lfloor(v-1)/12\rfloor)$ difference packings, and optimal $(v,4,1)$ optical orthogonal codes. Therefore, the classification of CPCW codes holds for them too. Perfect $(v,4,1)$ CPCWcodes are equivalent to $(v,4,1)$ cyclic difference families, and thus $(73,4,1)$ cyclic difference families are classified too.