Association schemes related to universally optimal configurations, Kerdock codes and extremal Euclidean line-sets

Kanat Abdukhalikov; Eiichi Bannai; Sho Suda

doi:10.1016/j.jcta.2008.07.002

Association schemes related to universally optimal configurations, Kerdock codes and extremal Euclidean line-sets

Kanat Abdukhalikov, Eiichi Bannai, Sho Suda

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

17 Citations (Scopus)

Abstract

H. Cohn et al. proposed an association scheme of 64 points in R¹⁴ which is conjectured to be a universally optimal code. We show that this scheme has a generalization in terms of Kerdock codes, as well as in terms of maximal collections of real mutually unbiased bases. These schemes are also related to extremal line-sets in Euclidean spaces and Barnes-Wall lattices. D. de Caen and E.R. van Dam constructed two infinite series of formally dual 3-class association schemes. We explain this formal duality by constructing two dual abelian schemes related to quaternary linear Kerdock and Preparata codes.

Original language	English
Pages (from-to)	434-448
Number of pages	15
Journal	Journal of Combinatorial Theory. Series A
Volume	116
Issue number	2
DOIs	https://doi.org/10.1016/j.jcta.2008.07.002
Publication status	Published - Feb 2009
Externally published	Yes

Keywords

Association schemes
Barnes-Wall lattices
Dual schemes
Kerdock codes
Mutually unbiased bases
Preparata codes
Universally optimal configurations

ASJC Scopus subject areas

Theoretical Computer Science
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics

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10.1016/j.jcta.2008.07.002

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title = "Association schemes related to universally optimal configurations, Kerdock codes and extremal Euclidean line-sets",

abstract = "H. Cohn et al. proposed an association scheme of 64 points in R14 which is conjectured to be a universally optimal code. We show that this scheme has a generalization in terms of Kerdock codes, as well as in terms of maximal collections of real mutually unbiased bases. These schemes are also related to extremal line-sets in Euclidean spaces and Barnes-Wall lattices. D. de Caen and E.R. van Dam constructed two infinite series of formally dual 3-class association schemes. We explain this formal duality by constructing two dual abelian schemes related to quaternary linear Kerdock and Preparata codes.",

keywords = "Association schemes, Barnes-Wall lattices, Dual schemes, Kerdock codes, Mutually unbiased bases, Preparata codes, Universally optimal configurations",

author = "Kanat Abdukhalikov and Eiichi Bannai and Sho Suda",

note = "Funding Information: E-mail addresses: abdukhalikov@math.kz (K. Abdukhalikov), bannai@math.kyushu-u.ac.jp (E. Bannai), suda@ims.is.tohoku.ac.jp (S. Suda). 1 Supported by Japan Society for the Promotion of Science.",

year = "2009",

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doi = "10.1016/j.jcta.2008.07.002",

language = "English",

volume = "116",

pages = "434--448",

journal = "Journal of Combinatorial Theory - Series A",

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T1 - Association schemes related to universally optimal configurations, Kerdock codes and extremal Euclidean line-sets

AU - Abdukhalikov, Kanat

AU - Bannai, Eiichi

AU - Suda, Sho

N1 - Funding Information: E-mail addresses: abdukhalikov@math.kz (K. Abdukhalikov), bannai@math.kyushu-u.ac.jp (E. Bannai), suda@ims.is.tohoku.ac.jp (S. Suda). 1 Supported by Japan Society for the Promotion of Science.

PY - 2009/2

Y1 - 2009/2

N2 - H. Cohn et al. proposed an association scheme of 64 points in R14 which is conjectured to be a universally optimal code. We show that this scheme has a generalization in terms of Kerdock codes, as well as in terms of maximal collections of real mutually unbiased bases. These schemes are also related to extremal line-sets in Euclidean spaces and Barnes-Wall lattices. D. de Caen and E.R. van Dam constructed two infinite series of formally dual 3-class association schemes. We explain this formal duality by constructing two dual abelian schemes related to quaternary linear Kerdock and Preparata codes.

AB - H. Cohn et al. proposed an association scheme of 64 points in R14 which is conjectured to be a universally optimal code. We show that this scheme has a generalization in terms of Kerdock codes, as well as in terms of maximal collections of real mutually unbiased bases. These schemes are also related to extremal line-sets in Euclidean spaces and Barnes-Wall lattices. D. de Caen and E.R. van Dam constructed two infinite series of formally dual 3-class association schemes. We explain this formal duality by constructing two dual abelian schemes related to quaternary linear Kerdock and Preparata codes.

KW - Association schemes

KW - Barnes-Wall lattices

KW - Dual schemes

KW - Kerdock codes

KW - Mutually unbiased bases

KW - Preparata codes

KW - Universally optimal configurations

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DO - 10.1016/j.jcta.2008.07.002

M3 - Article

AN - SCOPUS:56749170736

SN - 0097-3165

VL - 116

SP - 434

EP - 448

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

IS - 2

ER -

Association schemes related to universally optimal configurations, Kerdock codes and extremal Euclidean line-sets

Abstract

Keywords

ASJC Scopus subject areas

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