Linear size optimal q-ary constant-weight codes and constant-composition codes

• Main Authors: Ling, Alan C. H., Chee, Yeow Meng, Dau, Son Hoang, Ling, San
• Other Authors: School of Physical and Mathematical Sciences
• Format: Article
• Language: English
• Published: 2012
• Subjects: DRNTU::Science::Mathematics
• Online Access: https://hdl.handle.net/10356/92390; http://hdl.handle.net/10220/7637
• Institution: Nanyang Technological University

An optimal constant-composition or constant-weight code of weight w has linear size if and only if its distance d is at least 2w-1. When d ≥ 2w, the determination of the exact size of such a constant-composition or constant-weight code is trivial, but the case of d=2w-1 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 2w-1 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 2w-1 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 2w-1 are also determined for all w ≤ 6, except in two cases.