Satins, lattices, and extended Euclid's algorithm
Motivated by the design of satins with draft of period m and step a, we draw our attention to the lattices L(m,a)=¿(1,a),(0,m)¿ where 1=a<m are integers with gcd(m,a)=1. We show that the extended Euclid's algorithm applied to m and a produces a shortest no null vector of L(m, a) and that the...
| Authors: | , |
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| Format: | article |
| Publication Date: | 2021 |
| Country: | España |
| Institution: | Universitat Politècnica de Catalunya (UPC) |
| Repository: | UPCommons. Portal del coneixement obert de la UPC |
| Language: | English |
| OAI Identifier: | oai:upcommons.upc.edu:2117/365868 |
| Online Access: | https://hdl.handle.net/2117/365868 https://dx.doi.org/10.1007/s13160-021-00477-9 |
| Access Level: | Open access |
| Keyword: | Lattice theory Algorithms Satins Square satins Symmetric satins Extended Euclid’s algorithm Lagrange–Gauss lattice basis reduction Shortest vector Optimal basis Reticles, Teoria de Algorismes Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra::Ordres, reticles, estructures algebraiques ordenades |
| Summary: | Motivated by the design of satins with draft of period m and step a, we draw our attention to the lattices L(m,a)=¿(1,a),(0,m)¿ where 1=a<m are integers with gcd(m,a)=1. We show that the extended Euclid's algorithm applied to m and a produces a shortest no null vector of L(m, a) and that the algorithm can be used to find an optimal basis of L(m, a). We also analyze square and symmetric satins. For square satins, the extended Euclid's algorithm produces directly the two vectors of an optimal basis. It is known that symmetric satins have either a rectangular or a rombal basis; rectangular basis are optimal, but rombal basis are not always optimal. In both cases, we give the optimal basis directly in terms of m and a. |
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