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More Basis Reduction for Linear Codes: Backward Reduction, BKZ, Slide Reduction, and More

Authors Surendra Ghentiyala, Noah Stephens-Davidowitz

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Author Details

Surendra Ghentiyala
  • Cornell University, Ithaca, NY, USA
Noah Stephens-Davidowitz
  • Cornell University, Ithaca, NY, USA

Funding

  • Ghentiyala, Surendra: This work is supported in part by the NSF under Grants Nos. CCF-2122230 and CCF-2312296, a Packard Foundation Fellowship, and a generous gift from Google.
  • Stephens-Davidowitz, Noah: This work is supported in part by the NSF under Grants Nos. CCF-2122230 and CCF-2312296, a Packard Foundation Fellowship, and a generous gift from Google. Some of this work was completed while the author was visiting the National University of Singapore and the Centre for Quantum Technologies

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Surendra Ghentiyala and Noah Stephens-Davidowitz. More Basis Reduction for Linear Codes: Backward Reduction, BKZ, Slide Reduction, and More. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 19:1-19:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.19

Abstract

We expand on recent exciting work of Debris-Alazard, Ducas, and van Woerden [Transactions on Information Theory, 2022], which introduced the notion of basis reduction for codes, in analogy with the extremely successful paradigm of basis reduction for lattices. We generalize DDvW’s LLL algorithm and size-reduction algorithm from codes over 𝔽₂ to codes over 𝔽_q, and we further develop the theory of proper bases. We then show how to instantiate for codes the BKZ and slide-reduction algorithms, which are the two most important generalizations of the LLL algorithm for lattices. 
Perhaps most importantly, we show a new and very efficient basis-reduction algorithm for codes, called full backward reduction. This algorithm is quite specific to codes and seems to have no analogue in the lattice setting. We prove that this algorithm finds vectors as short as LLL does in the worst case (i.e., within the Griesmer bound) and does so in less time. We also provide both heuristic and empirical evidence that it outperforms LLL in practice, and we give a variant of the algorithm that provably outperforms LLL (in some sense) for random codes.
Finally, we explore the promise and limitations of basis reduction for codes. In particular, we show upper and lower bounds on how "good" of a basis a code can have, and we show two additional illustrative algorithms that demonstrate some of the promise and the limitations of basis reduction for codes.

Subject Classification

ACM Subject Classification
  • Theory of computation → Error-correcting codes
Keywords
  • Linear Codes
  • Basis Reduction

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