Constant weight codes (CWCs) and constant composition codes (CCCs) are two important classes of codes that have been studied extensively in both combinatorics and coding theory for nearly sixty years. In this paper we show that for {\it all} fixed odd distances, there exist near-optimal CWCs and CCCs asymptotically achieving the classic Johnson-type upper bounds. Let $A_q(n,w,d)$ 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 {\it all} fixed $q,w$ and odd $d$, one has $\lim_{n\rightarrow\infty}\frac{A_q(n,d,w)}{\binom{n}{t}}=\frac{(q-1)^t}{\binom{w}{t}}$, where $t=\frac{2w-d+1}{2}$. This implies the existence of near-optimal generalized Steiner systems originally introduced by Etzion, and can be viewed as a counterpart of a celebrated result of R\"odl 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 $q\ge 3$. 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-R\"odl-Pippenger theorem on the existence of near-optimal matchings in hypergraphs: the first proof follows by Kahn's linear programming variation of the above theorem, and the second follows by the recent independent work of Delcour-Postle, and Glock-Joos-Kim-K\"uhn-Lichev on the existence of near-optimal matchings avoiding certain forbidden configurations. We also present several intriguing open questions for future research.

翻译：常重码（CWCs）和常成分码（CCCs）是两类重要的码，在近六十年来的组合学与编码理论中得到了广泛研究。本文证明：对于所有固定的奇距离，存在渐近达到经典Johnson型上界的近优常重码与常成分码。设$A_q(n,w,d)$表示长度为$n$、常重$w$、最小距离$d$的$q$元常重码的最大尺寸。我们的主要结果之一表明：对于所有固定的$q,w$及奇数$d$，有$\lim_{n\rightarrow\infty}\frac{A_q(n,d,w)}{\binom{n}{t}}=\frac{(q-1)^t}{\binom{w}{t}}$，其中$t=\frac{2w-d+1}{2}$。这意味着由Etzion最初引入的近似广义斯坦纳系统的存在性，并可视为Rödl关于近优斯坦纳系统存在性著名结果的对应版本。值得注意的是，此前对于$q\ge 3$的$A_q(n,w,d)$所知甚少。对于常成分码的最大尺寸，我们同样证明了类似结果。我们为两个主要结果提供了不同证明：第一种基于Frankl-Rödl-Pippenger定理关于超图中近优匹配存在性的线性规划变体（由Kahn提出）；第二种基于Delcour-Postle以及Glock-Joos-Kim-Kühn-Lichev近期独立提出的关于避开特定禁止配置的近优匹配存在性工作。最后，我们提出若干值得未来研究的开放性问题。