Table \[tab:comparison\] for more details). We leave it as an open question to whether coupling a good reduction with a smaller family of codes can further reduce the overall search space.\
**Organization.** This work is organized as follows. In Section \[sec:prelim\] we describe some basic definitions and notation. In Section \[sec:genRed\] we establish our main result on general reductions and several corollaries. In Section \[sec:numbFields\] we consider reductions induced by quotients of ideals in the ring of integers of a number field, proving the main corollaries. In Section \[sec:redRings\] we construct random Hurwitz and Lipschitz lattices from codes over matrix rings. In Section \[sec:conc\], we discuss an “algorithmic” version of the main theorem and draw the final conclusions.

Preliminaries and Notation {#sec:prelim}
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The Euclidean norm of ${\mathbf{x}}\in \mathbb{R}^m$ is denoted by $\left\|{\mathbf{x}}\right\| = (x_1^2+\ldots+x_m^2)^{1/2}$. The ball of radius $r$ in $\mathbb{R}^m$ is denoted by $\mathcal{B}_r = \left\{{\mathbf{x}}\in \mathbb{R}^m: \left\|{\mathbf{x}}\right\| \leq r \right\}$. A *lattice* $\Lambda$ is a discrete additive subgroup of $\mathbb{R}^m$. Denote by $\mbox{span } \Lambda$ the minimal subspace of $\mathbb{R}^m$ that contains $\Lambda$. The *rank* of $\Lambda$ is defined to be the dimension of $\mbox{span } \Lambda$. The quotient $(\mbox{span } \Lambda)/ \Lambda$ is compact, and its 