An optimal constant-composition or constant-weight code of weight w has
linear size if and only if its distance d is at least 2wβ1. When dβ₯2w, the determination of the exact size of such a constant-composition or
constant-weight code is trivial, but the case of d=2wβ1 has been solved
previously only for binary and ternary constant-composition and constant-weight
codes, and for some sporadic instances.
This paper provides a construction for quasicyclic optimal
constant-composition and constant-weight codes of weight w and distance
2wβ1 based on a new generalization of difference triangle sets. As a result,
the sizes of optimal constant-composition codes and optimal constant-weight
codes of weight w and distance 2wβ1 are determined for all such codes of
sufficiently large lengths. This solves an open problem of Etzion.
The sizes of optimal constant-composition codes of weight w and distance
2wβ1 are also determined for all wβ€6, except in two cases.Comment: 12 page