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Improving Uniquely Decodable Codes in Binary Adder Channels

József Balogh, The Nguyen, Patric R. J. Ostergard, Ethan Patrick White, Michael Wigal

TL;DR

The paper addresses improving the sum rate of uniquely decodable codes for the $T$-user binary adder channel by modifying constituent codes when at least one seed code does not have average weight $d/2$, producing higher-dimensional UD codes with strictly larger rates. It provides a general existence proof using a two-block gluing construction and a Berry–Esseen–based concentration argument to ensure disjoint unions remain UD. It then offers a constructive three-step scheme to generate practical high-rate UD codes and presents explicit optimal constructions for $T=2$ to $8$, achieving new lower bounds on the zero-error capacity region. While the numerical gains are modest and the resulting codes are large, the work advances the theoretical understanding of UD codes in additive channels and hints at broader applicability to co-Sidon and multi-set union-free problems.

Abstract

We present a general method to modify existing uniquely decodable codes in the $T$-user binary adder channel. If at least one of the original constituent codes does not have average weight exactly half of the dimension, then our method produces a new set of constituent codes in a higher dimension, with a strictly higher rate. Using our method we improve the highest known rate for the $T$-user binary adder channel for all $T \geq 2$. This information theory problem is equivalent to co-Sidon problems initiated by Lindstr{ö}m in the 1960s, and also the multi-set union-free problem. Our results improve the known lower bounds in these settings as well.

Improving Uniquely Decodable Codes in Binary Adder Channels

TL;DR

The paper addresses improving the sum rate of uniquely decodable codes for the -user binary adder channel by modifying constituent codes when at least one seed code does not have average weight , producing higher-dimensional UD codes with strictly larger rates. It provides a general existence proof using a two-block gluing construction and a Berry–Esseen–based concentration argument to ensure disjoint unions remain UD. It then offers a constructive three-step scheme to generate practical high-rate UD codes and presents explicit optimal constructions for to , achieving new lower bounds on the zero-error capacity region. While the numerical gains are modest and the resulting codes are large, the work advances the theoretical understanding of UD codes in additive channels and hints at broader applicability to co-Sidon and multi-set union-free problems.

Abstract

We present a general method to modify existing uniquely decodable codes in the -user binary adder channel. If at least one of the original constituent codes does not have average weight exactly half of the dimension, then our method produces a new set of constituent codes in a higher dimension, with a strictly higher rate. Using our method we improve the highest known rate for the -user binary adder channel for all . This information theory problem is equivalent to co-Sidon problems initiated by Lindstr{ö}m in the 1960s, and also the multi-set union-free problem. Our results improve the known lower bounds in these settings as well.
Paper Structure (11 sections, 2 theorems, 37 equations, 1 table)

This paper contains 11 sections, 2 theorems, 37 equations, 1 table.

Table of Contents

  1. Introduction
  2. A General Existence Proof
  3. New Codes
  4. Two users
  5. Three users
  6. Four users
  7. Five users
  8. Six users
  9. Seven users
  10. Eight users
  11. Conclusion

Key Result

Lemma 1

Let $n \ge 1$ and let $C \subset \{0,1\}^d$ be a code with variance $\sigma^2 > 0$ and third absolute standardized central moment $\rho^3 \ge 0$. For any $t > 0$, let Then, Furthermore, let Then for any $t > 0$,

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Theorem 2
  • proof