Punctured Low-Bias Codes Behave Like Random Linear Codes

Venkatesan Guruswami, Jonathan Mosheiff

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

5 Scopus citations

Abstract

Random linear codes are a workhorse in coding theory, and are used to show the existence of codes with the best known or even near-optimal trade-offs in many noise models. However, they have little structure besides linearity, and are not amenable to tractable error-correction algorithms. In this work, we prove a general derandomization result applicable to random linear codes. Namely, in settings where the coding-theoretic property of interest is 'local' (in the sense of forbidding certain bad configurations involving few vectors-code distance and list-decodability being notable examples), one can replace random linear codes (RLCs) with a significantly derandomized variant with essentially no loss in parameters. Specifically, instead of randomly sampling coordinates of the (long) Hadamard code (which is an equivalent way to describe RLCs), one can randomly sample coordinates of any code with low bias. Over large alphabets, the low bias requirement can be weakened to just large distance. Furthermore, large distance suffices even with a small alphabet in order to match the current best known bounds for RLC list-decodability. In particular, by virtue of our result, all current (and future) achievability bounds for list-decodability of random linear codes extend automatically to random puncturings of any low-bias (or large alphabet) 'mother' code. We also show that our punctured codes emulate the behavior of RLCs on stochastic channels, thus giving a derandomization of RLCs in the context of achieving Shannon capacity as well. Thus, we have a randomness-efficient way to sample codes achieving capacity in both worst-case and stochastic settings that can further inherit algebraic or other algorithmically useful structural properties of the mother code. This is an extended abstract. The full version is available at https://arxiv.org/abs/2109.11725.

Original languageEnglish
Title of host publicationProceedings - 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science, FOCS 2022
PublisherInstitute of Electrical and Electronics Engineers
Pages36-45
Number of pages10
ISBN (Electronic)9781665455190
DOIs
StatePublished - 1 Jan 2022
Event63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022 - Denver, United States
Duration: 31 Oct 20223 Nov 2022

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
Volume2022-October
ISSN (Print)0272-5428

Conference

Conference63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022
Country/TerritoryUnited States
CityDenver
Period31/10/223/11/22

Keywords

  • Code Derandomization
  • Local Properties
  • Random Linear Codes
  • Random Puncturing

ASJC Scopus subject areas

  • General Computer Science

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