Near-optimal constructions of constant weight codes and constant composition codes asymptotically attaining the Johnson bound

Constant weight codes (CWCs) and constant compo-sition codes (CCCs) are two important classes of codes that have been extensively studied in the fields of combinatorics and coding theory for nearly 60 years. In this paper, we demonstrate that for all fixed odd distance, there are near-optimal CWCs and CCCs asymptotically achieving the classic Johnson-type upper bounds. Let
$A_{q}(n,\ d,\ w)$
denote the maximum size of q-ary CWCs of length
$n$
with constant weight
$w$
and minimum distance
$d$
. One of our main results shows that for all fixed
$q, w$
and odd
$d$
, we have
$\displaystyle \lim_{n\rightarrow\infty}\frac{A_{q}(n,d,w)}{(_{t}^{n})}=\frac{(q-1)^{t}}{(_{t}^{w})}$
, where
$t=\displaystyle \frac{2w-d+1}{2}$
. This implies the existence of near-optimal generalized Steiner systems originally introduced by Etzion. It can also be viewed as a counterpart of a celebrated result of Rödl on the existence of near-optimal Steiner systems. Note that prior to our work, very little is known about
$A_{q}(n,\ w,\ d)$
for
$q\geq 3$
. A similar result is proved for the maximum size of CCCs. We provide different proofs for our two main results, based on two strengthenings of the well-known Frankl-Rödl-Pippenger theorem on the existence of near-optimal matchings in hyper-graphs: the first proof follows by Kahn's linear programming variation of the above theorem, and the second follows by the recent independent work of Delcourt-Postle, and Glock-Joos-Kim-Kühn-Lichev on the existence of near-optimal matchings avoiding certain forbidden configurations. A full version of this paper can be found in [1].