### Abstract

We present theoretical and computational results concerning an optimization problem for lattices, related to a generalization of the concept of dual lattices. Let Λ be a k-dimensional lattice in ℝ^{n} (with 0 < k ≤ n), and p, q ε ℝ^{+} ∪. We define the p, qnorm Np,q() of the lattice Λ and show that this norm always exists. In fact, our results yield an algorithm for the calculation of Np,q(). Further, since this general algorithm is not efficient, we discuss more closely two particular choices for p, q that arise naturally. Namely, we consider the case (p, q) = (2,), and also the choice (p, q) = (1,). In both cases,we showthat in general, an optimal basis of Λ as well as N_{p,q} (λ) can be calculated. Finally, we illustrate our methods by several numerical examples.

Original language | English |
---|---|

Pages (from-to) | 443-455 |

Number of pages | 13 |

Journal | Experimental Mathematics |

Volume | 22 |

Issue number | 4 |

DOIs | |

Publication status | Published - Oct 2 2013 |

### Fingerprint

### Keywords

- LLL-reduction
- bases of lattices
- basis reduction
- dual lattices
- lattices

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Experimental Mathematics*,

*22*(4), 443-455. https://doi.org/10.1080/10586458.2013.833489