Approximate generalized Steiner systems and near-optimal constant weight codes
Miao Liu, Chong Shangguan
TL;DR
This work addresses the asymptotic sizes of $q$-ary constant-weight codes and constant-composition codes with fixed parameters and odd minimum distance $d$. By translating code construction into matching problems in carefully designed hypergraphs, it applies two powerful frameworks—the Kahn linear-programming variation of the Frankl–Rödl–Pippenger theorem and recent forbidden-configuration (conflict-free) results—to establish near-optimal, non-constructive existence results. Specifically, for all fixed $q,w$ and odd $d$, it proves $\displaystyle \lim_{n\to\infty}\frac{A_q(n,d,w)}{\binom{n}{t}}=\frac{(q-1)^t}{\binom{w}{t}}$ with $t=\frac{2w-d+1}{2}$, and a corresponding asymptotic for $A_q(n,d,\overline{w})$, thereby demonstrating near-optimal generalized Steiner systems in an asymptotic sense. The methods unify coding theory with modern hypergraph matching techniques and illuminate open questions on even distances and constructive realizations, with implications for extremal designs and coding bounds in the $q\ge3$ setting.
Abstract
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 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 $q\ge 3$. 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-Rödl-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-Kühn-Lichev on the existence of near-optimal matchings avoiding certain forbidden configurations. We also present several intriguing open questions for future research.