Density/length profiles and trellis complexity of lattices

Author

Forney, G. David, Jr.

Author_Institution

Motorola Inc., Mansfield, MA, USA

Volume

40

Issue

6

fYear

1994

fDate

11/1/1994 12:00:00 AM

Firstpage

1753

Lastpage

1772

Abstract

The density/length profile (DCP) of a lattice Λ is analogous to the dimension/length profile of a linear code. The DLP is a geometrical invariant of Λ that includes the coding gain of Λ. Duality results analogous to those of linear block codes are derived for lattices. Bounds on the DLP may be derived from bounds on Hermite´s constants; these hold with equality for many dense lattices. In turn, the DLP lowerbounds the state complexity profile of a minimal trellis diagram for Λ in any coordinate system. It is shown that this bound can be met for the E₈ lattice by a laminated lattice construction with a novel trellis diagram. Bounds and constructions for other important low-dimensional lattices are given. Two laminated lattice constructions of the Leech lattice yield trellis diagrams with maximum state space sizes 1024 and 972

Keywords

block codes; computational complexity; linear codes; Hermite´s constants; Leech lattice; bounds; coding gain; coordinate system; density/length profiles; dimension/length profile; laminated lattice construction; linear code; low-dimensional lattices; minimal trellis diagram; state complexity profile; state space sizes; trellis complexity; Block codes; Hamming distance; Hamming weight; Helium; Information theory; Lattices; Linear code; State-space methods;

fLanguage

English

Journal_Title

Information Theory, IEEE Transactions on

Publisher

ieee

ISSN

0018-9448

Type

jour

DOI

10.1109/18.340453

Filename

340453