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Optimal Constant-Weight and Mixed-Weight Conflict-Avoiding Codes

Yuan-Hsun Lo, Tsai-Lien Wong, Kangkang Xu, Yijin Zhang

TL;DR

The paper addresses deterministic, grant-free CAC design for asynchronous multi-user access with no feedback, aiming to maximize the number of users supported at fixed length $L$ and weight $w$. It advances the field by (i) introducing direct $p$-ary representations and SDR-based constructions to yield optimal constant-weight CACs for lengths $L=\frac{w-1}{d}p^r$, (ii) extending optimal CACs to lengths $L=p^r$, $wp^r$, and $(2w-1)p^r$ via CRT-layer lifting and additive-combinatorics arguments (including Kneser’s theorem), and (iii) proposing mixed-weight CACs to enable priority-based throughput, with several classes of optimal two- and multi-weight codes and explicit corollaries. The methods combine group-theoretic constructions, CRT decompositions, and additive combinatorics to derive tight bounds and constructive optimal codes, offering a richer design space for URLLC-like scenarios. These results provide practical frameworks for achieving higher throughput and lower access delays in heterogeneous asynchronous networks through structured, analyzable CACs.

Abstract

A conflict-avoiding code (CAC) is a deterministic transmission scheme for asynchronous multiple access without feedback. When the number of simultaneously active users is less than or equal to $w$, a CAC of length $L$ with weight $w$ can provide a hard guarantee that each active user has at least one successful transmission within every consecutive $L$ slots. In this paper, we generalize some previously known constructions of constant-weight CACs, and then derive several classes of optimal CACs by the help of Kneser's Theorem and some techniques in Additive Combinatorics. Another spotlight of this paper is to relax the identical-weight constraint in prior studies to study mixed-weight CACs for the first time, for the purpose of increasing the throughput and reducing the access delay of some potential users with higher priority. As applications of those obtained optimal CACs, we derive some classes of optimal mixed-weight CACs.

Optimal Constant-Weight and Mixed-Weight Conflict-Avoiding Codes

TL;DR

The paper addresses deterministic, grant-free CAC design for asynchronous multi-user access with no feedback, aiming to maximize the number of users supported at fixed length and weight . It advances the field by (i) introducing direct -ary representations and SDR-based constructions to yield optimal constant-weight CACs for lengths , (ii) extending optimal CACs to lengths , , and via CRT-layer lifting and additive-combinatorics arguments (including Kneser’s theorem), and (iii) proposing mixed-weight CACs to enable priority-based throughput, with several classes of optimal two- and multi-weight codes and explicit corollaries. The methods combine group-theoretic constructions, CRT decompositions, and additive combinatorics to derive tight bounds and constructive optimal codes, offering a richer design space for URLLC-like scenarios. These results provide practical frameworks for achieving higher throughput and lower access delays in heterogeneous asynchronous networks through structured, analyzable CACs.

Abstract

A conflict-avoiding code (CAC) is a deterministic transmission scheme for asynchronous multiple access without feedback. When the number of simultaneously active users is less than or equal to , a CAC of length with weight can provide a hard guarantee that each active user has at least one successful transmission within every consecutive slots. In this paper, we generalize some previously known constructions of constant-weight CACs, and then derive several classes of optimal CACs by the help of Kneser's Theorem and some techniques in Additive Combinatorics. Another spotlight of this paper is to relax the identical-weight constraint in prior studies to study mixed-weight CACs for the first time, for the purpose of increasing the throughput and reducing the access delay of some potential users with higher priority. As applications of those obtained optimal CACs, we derive some classes of optimal mixed-weight CACs.
Paper Structure (16 sections, 30 theorems, 95 equations, 2 tables)

This paper contains 16 sections, 30 theorems, 95 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Conflict-Avoiding Codes
  3. Known Optimal Constant-Weight CACs
  4. Mixed-Weight Conflict-Avoiding Codes
  5. Main Contributions
  6. Additive Combinatorics and Kneser's Theorem
  7. New Optimal CACs based on Direct Constructions
  8. $p$-ary representation
  9. A direct construction
  10. Optimal CACs of length $\frac{w-1}{d}p^r$ and weight $w$
  11. New Optimal CACs based on Extending Smaller-length CACs
  12. Optimal CACs of length $p^r$
  13. Optimal CACs of length $wp^r$
  14. Optimal CACs of length $(2w-1)p^r$
  15. Mixed-Weight CACs
  16. ...and 1 more sections

Key Result

Theorem 1

Let $p$ be an odd prime and $w$ be an integer such that $2\leq w\leq p$. If and then there exists a code in $\mathrm{CAC^e}((w-1)p,w)$ with $(p-1)/2$ codewords. In particular, if $w-1$ is an odd prime such that $p\geq 2w-1$, then

Theorems & Definitions (59)

  • Definition 1
  • Theorem 1: SW10
  • Theorem 2: MMSJ07
  • Theorem 3: SWC10
  • Definition 2
  • Proposition 1
  • Theorem 4: Kneser53TV06
  • Corollary 1
  • proof
  • Lemma 1
  • ...and 49 more