Ultra-Resilient Superimposed Codes: Near-Optimal Construction and Applications

TL;DR

This paper introduces Ultra-Resilient Superimposed Codes (URSCs), a new class of codes that are robust against asynchronous shifts and partial corruption, with near-optimal construction and broad applications in distributed systems.

Contribution

The paper presents the first polynomial-time construction of URSCs that are resilient to adversarial shifts and corruption, without prior knowledge of the number of codewords, advancing superimposed code capabilities.

Findings

URSCs outperform previous codes in asynchronous and fault-prone environments.

URSCs enable nearly two orders of magnitude faster local broadcast in networks.

Applications include improved contention resolution and communication efficiency.

Abstract

A superimposed code is a collection of binary vectors (codewords) with the property that no vector is contained in the Boolean sum of any $k$ others, enabling unique identification of codewords within any group of $k$. Superimposed codes are foundational combinatorial tools with applications in areas ranging from distributed computing and data retrieval to fault-tolerant communication. However, classical superimposed codes rely on strict alignment assumptions, limiting their effectiveness in asynchronous and fault-prone environments, which are common in modern systems and applications. We introduce Ultra-Resilient Superimposed Codes (URSCs), a new class of codes that extends the classic superimposed framework by ensuring a stronger codewords' isolation property and resilience to two types of adversarial perturbations: arbitrary cyclic shifts and partial bitwise corruption (flips). Additionally, URSCs exhibit universality, adapting seamlessly to any number $k$ of concurrent codewords without prior knowledge. This is a combination of properties not achieved in any previous construction. We provide the first polynomial-time construction of URSCs with near-optimal length, significantly outperforming previous constructions with less general features, all without requiring prior knowledge of the number of concurrent codewords, $k$. % We demonstrate that our URSCs significantly advance the state of the art in multiple applications, including uncoordinated beeping networks, where our codes reduce time complexity for local broadcast by nearly two orders of magnitude, and generalized contention resolution in multi-access channel communication.