Type II codes over F2 + uF2

Steven T. Dougherty; Philippe Gaborit; Masaaki Harada; Patrick Sole

doi:10.1109/18.746770

Type II codes over F₂ + uF₂

Steven T. Dougherty, Philippe Gaborit, Masaaki Harada, Patrick Sole

INF - System Information Sciences

University of Scranton

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

171 Citations (Scopus)

Abstract

The alphabet F₂ + uF₂ is viewed here as a quotient of the Gaussian integers by the ideal (2). Self-dual F₂ + uF₂ codes with Lee weights a multiple of 4 are called Type II. They give even uniniodular Gaussian lattices by Construction A, while Type I codes yield uniniodular Gaussian lattices. Construction B makes it possible to realize the Leech lattice as a Gaussian lattice. There is a Gray map which maps Type II codes into Type II binary codes with a fixed point free involution in their automorphism group. Combinatorial constructions use weighing matrices and strongly regular graphs. Gleason-type theorems for the symmetrized weight enumerators of Type II codes are derived. All self-dual codes are classified for length up to 8. The shadow of Type I codes yields bounds on the highest minimum Hamming and Lee weights.

Original language	English
Pages (from-to)	32-45
Number of pages	14
Journal	IEEE Transactions on Information Theory
Volume	45
Issue number	1
DOIs	https://doi.org/10.1109/18.746770
Publication status	Published - 1999

Keywords

Automorphism groups
Codes over rings
Gray map
Lattices and shadows

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10.1109/18.746770

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N2 - The alphabet F2 + uF2 is viewed here as a quotient of the Gaussian integers by the ideal (2). Self-dual F2 + uF2 codes with Lee weights a multiple of 4 are called Type II. They give even uniniodular Gaussian lattices by Construction A, while Type I codes yield uniniodular Gaussian lattices. Construction B makes it possible to realize the Leech lattice as a Gaussian lattice. There is a Gray map which maps Type II codes into Type II binary codes with a fixed point free involution in their automorphism group. Combinatorial constructions use weighing matrices and strongly regular graphs. Gleason-type theorems for the symmetrized weight enumerators of Type II codes are derived. All self-dual codes are classified for length up to 8. The shadow of Type I codes yields bounds on the highest minimum Hamming and Lee weights.

AB - The alphabet F2 + uF2 is viewed here as a quotient of the Gaussian integers by the ideal (2). Self-dual F2 + uF2 codes with Lee weights a multiple of 4 are called Type II. They give even uniniodular Gaussian lattices by Construction A, while Type I codes yield uniniodular Gaussian lattices. Construction B makes it possible to realize the Leech lattice as a Gaussian lattice. There is a Gray map which maps Type II codes into Type II binary codes with a fixed point free involution in their automorphism group. Combinatorial constructions use weighing matrices and strongly regular graphs. Gleason-type theorems for the symmetrized weight enumerators of Type II codes are derived. All self-dual codes are classified for length up to 8. The shadow of Type I codes yields bounds on the highest minimum Hamming and Lee weights.

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Type II codes over F₂ + uF₂

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Keywords

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