On well-rounded sublattices of the hexagonal lattice

被引：7

作者：

Fukshansky, Lenny ^{[1
]}

Moore, Daniel ^{[2
]}

Ohana, R. Andrew ^{[3
]}

Zeldow, Whitney ^{[4
]}

机构：

[1] Claremont Mckenna Coll, Dept Math, Claremont, CA 91711 USA

[2] Loyola Marymount Univ, Dept Math, Los Angeles, CA 90045 USA

[3] Univ Washington, Dept Math, Seattle, WA 98195 USA

[4] San Francisco State Univ, Dept Math, San Francisco, CA 94132 USA

来源：

DISCRETE MATHEMATICS | 2010年 / 310卷 / 23期

关键词：

Hexagonal lattice; Well-rounded lattices; Binary and ternary quadratic forms; Epstein zeta function;

D O I：

10.1016/j.disc.2010.07.014

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

We produce an explicit parameterization of well-rounded sublattices of the hexagonal lattice in the plane, splitting them into similarity classes. We use this parameterization to study the number, the greatest minimal norm, and the highest signal-to-noise ratio of well-rounded sublattices of the hexagonal lattice of a fixed index. This investigation parallels earlier work by Bernstein, Sloane, and Wright where similar questions were addressed on the space of all sublattices of the hexagonal lattice. Our restriction is motivated by the importance of well-rounded lattices for discrete optimization problems. Finally, we also discuss the existence of a natural combinatorial structure on the set of similarity classes of well-rounded sublattices of the hexagonal lattice, induced by the action of a certain matrix monoid. (C) 2010 Elsevier B.V. All rights reserved.

引用

页码：3287 / 3302

页数：16

共 11 条

[1] On the complexity of decoding lattices using the Korkin-Zolotarev reduced basis [J].

Banihashemi, AH ;

Khandani, AK .

IEEE TRANSACTIONS ON INFORMATION THEORY, 1998, 44 (01) :162-171

[2]

BARNES F, PYTHAGOREAN TRIPLES

[3] On sublattices of the hexagonal lattice [J].

Bernstein, M ;

Sloane, NJA ;

Wright, PE .

DISCRETE MATHEMATICS, 1997, 170 (1-3) :29-39

[4]

BUTCHER JC, 1999, NZ MATH SOC NEWSLETT, V76, P18

[5]

Conway J H, 1999, Grundlehren der Mathematischen Wissenschaften, V3rd, DOI DOI 10.1007/978-1-4757-6568-7

[6]

FUKSHANSKY L, ELEM MATH IN PRESS

[7] On distribution of well-rounded sublattices of Z2 [J].

Fukshansky, Lenny .

JOURNAL OF NUMBER THEORY, 2008, 128 (08) :2359-2393

[8] Frobenius problem and the covering radius of a lattice [J].

Fukshansky, Lenny ;

Robins, Sinai .

DISCRETE & COMPUTATIONAL GEOMETRY, 2007, 37 (03) :471-483

[9] on similarity classes of well-rounded sublattices of Z2 [J].

Fukshansky, Lenny .

JOURNAL OF NUMBER THEORY, 2009, 129 (10) :2530-2556

[10]

Martinet J., 2003, PERFECT LATTICES EUC

← 1 2 →