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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 F_2 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. 

We prove a conjecture due to Dadush, showing that if L is an n-dimensional lattice whose sublattices all have determinant at least one, then the sum over all lattice vectors y of e^{-\pi t^2 ||y||^2} is at most 3/2, where t := 10(log n + 2). From this we derive bounds on the number of short lattice vectors, which can be viewed as a partial converse to Minkowski’s celebrated first theorem. We also derive a bound on the covering radius.