Shadow lattices and shadow codes

Steven T. Dougherty; Masaaki Harada; Patrick Solé

doi:10.1016/S0012-365X(99)00368-4

Shadow lattices and shadow codes

Steven T. Dougherty, Masaaki Harada, Patrick Solé

Research output: Contribution to journal › Article › peer-review

11 Citations (Scopus)

Abstract

Recently shadows of unimodular lattices have been defined to derive upper bounds on the minimum norm of Type I lattices. In this paper we study shadows of lattices mostly to construct unimodular lattices in higher dimensions. We introduce shadow-optimal lattices which are a class of optimal Type I lattices by investigating theta series of optimal Type I lattices. These lattices are used to construct unimodular lattices by the above construction. An analogy between self-dual codes over the ring ℤ_2k of integers and unimodular lattices is found. We also study shadow codes for Type I codes over ℤ_2k by considering shadow lattices.

Original language	English
Pages (from-to)	49-64
Number of pages	16
Journal	Discrete Mathematics
Volume	219
Issue number	1-3
DOIs	https://doi.org/10.1016/S0012-365X(99)00368-4
Publication status	Published - 2000 May 28
Externally published	Yes

Keywords

Self-dual codes
Shadows
Unimodular lattices

ASJC Scopus subject areas

Theoretical Computer Science
Discrete Mathematics and Combinatorics

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10.1016/S0012-365X(99)00368-4

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title = "Shadow lattices and shadow codes",

abstract = "Recently shadows of unimodular lattices have been defined to derive upper bounds on the minimum norm of Type I lattices. In this paper we study shadows of lattices mostly to construct unimodular lattices in higher dimensions. We introduce shadow-optimal lattices which are a class of optimal Type I lattices by investigating theta series of optimal Type I lattices. These lattices are used to construct unimodular lattices by the above construction. An analogy between self-dual codes over the ring ℤ2k of integers and unimodular lattices is found. We also study shadow codes for Type I codes over ℤ2k by considering shadow lattices.",

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author = "Dougherty, {Steven T.} and Masaaki Harada and Patrick Sol{\'e}",

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T1 - Shadow lattices and shadow codes

AU - Dougherty, Steven T.

AU - Harada, Masaaki

AU - Solé, Patrick

PY - 2000/5/28

Y1 - 2000/5/28

N2 - Recently shadows of unimodular lattices have been defined to derive upper bounds on the minimum norm of Type I lattices. In this paper we study shadows of lattices mostly to construct unimodular lattices in higher dimensions. We introduce shadow-optimal lattices which are a class of optimal Type I lattices by investigating theta series of optimal Type I lattices. These lattices are used to construct unimodular lattices by the above construction. An analogy between self-dual codes over the ring ℤ2k of integers and unimodular lattices is found. We also study shadow codes for Type I codes over ℤ2k by considering shadow lattices.

AB - Recently shadows of unimodular lattices have been defined to derive upper bounds on the minimum norm of Type I lattices. In this paper we study shadows of lattices mostly to construct unimodular lattices in higher dimensions. We introduce shadow-optimal lattices which are a class of optimal Type I lattices by investigating theta series of optimal Type I lattices. These lattices are used to construct unimodular lattices by the above construction. An analogy between self-dual codes over the ring ℤ2k of integers and unimodular lattices is found. We also study shadow codes for Type I codes over ℤ2k by considering shadow lattices.

KW - Self-dual codes

KW - Shadows

KW - Unimodular lattices

UR - http://www.scopus.com/inward/record.url?scp=0037711148&partnerID=8YFLogxK

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DO - 10.1016/S0012-365X(99)00368-4

M3 - Article

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VL - 219

SP - 49

EP - 64

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 1-3

ER -

Shadow lattices and shadow codes

Abstract

Keywords

ASJC Scopus subject areas

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