More basis reduction for linear codes: backward reduction, BKZ, slide reduction, and more

TL;DR

This paper extends basis reduction techniques from lattices to codes over finite fields, introduces a new efficient reduction algorithm called full backward reduction, and analyzes its theoretical and practical advantages over existing methods.

Contribution

It generalizes lattice basis reduction algorithms to codes, develops new algorithms like full backward reduction, and provides theoretical and empirical analysis of their effectiveness.

Findings

Full backward reduction finds short vectors as effectively as LLL.

The new algorithm outperforms LLL in practice and in some cases provably.

Bounds on the quality of bases for codes are established.

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 $\mathbb{F}_2$ to codes over $\mathbb{F}_q$, 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.