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

Authors Gianluca De Marco , Dariusz R. Kowalski

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ACM Subject Classification
  • Mathematics of computing → Information theory
  • Theory of computation → Distributed algorithms
Keywords
  • superimposed codes
  • ultra-resiliency
  • deterministic algorithms
  • uncoordinated beeping networks
  • contention resolution

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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.

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Gianluca De Marco and Dariusz R. Kowalski. Ultra-Resilient Superimposed Codes: Near-Optimal Construction and Applications. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 65:1-65:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.65

Author Details

Gianluca De Marco
  • University of Salerno, Italy
Dariusz R. Kowalski
  • Augusta University, GA, USA

Funding

Partly supported by the Fondo di Ateneo per la Ricerca di Base (FARB) of University of Salerno and by the National Science Center, Poland (NCN), grant 2020/39/B/ST6/03288.

Acknowledgements

The authors would like to thank Ugo Vaccaro for many inspiring and valuable discussions, and are also grateful to the anonymous reviewers for the constructive comments and suggestions that helped to improve the overall quality of the paper.

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