Optimal Ternary Constant-Weight Codes With Weight 4 and Distance 5

Constant-weight codes (CWCs) play an important role in coding theory. The problem of determining the sizes for optimal ternary CWCs with length
$n$
, weight 4, and minimum Hamming distance 5 (
$(n,5,4)_{3}$
code) has been settled for all positive integers
$n\leq 10$
or
$n > 10$
and
$n\equiv 1\pmod {3}$
with
$n\in \{13,52,58\}$
undetermined. In this paper, we investigate the problem of constructing optimal
$(n,5,4)_{3}$
codes for all lengths
$n$
with the tool of group divisible codes. We determine the size of an optimal
$(n,5,4)_{3}$
code for each integer
$n\geq 4$
leaving the lengths
$n\in \{12,13,21,27,33,39,45,52\}$
unsolved.