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

Abdukhalikov, Kanat; Bannaĭ, Eiichi; Suda, Sho

doi:10.1016/j.jcta.2008.07.002

Cited by 20 publications

(31 citation statements)

References 18 publications

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“…The Cayley graphs of partial difference sets are strongly regular, so yield two-class association schemes. Therefore it is interesting to investigate whether there are translation schemes over elementary abelian 2-groups with the same parameters as those constructed in [1,6] and this paper from the various Z 4 -linear codes. When m = 3, the Gray map image of the DG code is a Z 2 -linear code, and we checked that the Gray map image of the 9-class scheme remains a scheme but with different parameters.…”

Section: Resultsmentioning

confidence: 99%

“…Henry Cohn and his collaborators in [2] conjectured that a certain 3-class association scheme on 64 points determines a universally optimal configuration in R 14 . Then in [1] Abdukhalikov, Bannai and Suda generalized this example in terms of binary and quaternary Kerdock and Preparata codes, as well as in terms of maximal set of mutually unbiased basis. To be specific, they constructed a series of 3-class association schemes using the partition of shortened Kerdock codes induced by their Lee weights, and dual schemes on the cosets of punctured Preparata codes.…”

Section: Introductionmentioning

confidence: 99%

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Association schemes related to Delsarte–Goethals codes

Hu¹,

Feng²,

Ge³

2014

J Algebr Comb

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In this paper, we construct an infinite series of 9-class association schemes from a refinement of the partition of Delsarte-Goethals codes by their Lee weights. The explicit expressions of the dual schemes are determined through direct manipulations of complicated exponential sums. As a byproduct, the other three infinite families of association schemes are also obtained as fusion schemes and quotient schemes.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Association schemes related to Delsarte–Goethals codes

Hu¹,

Feng²,

Ge³

2014

J Algebr Comb

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“…To save space, we only give the two (16, 64) codes D 16,0,i (i = 1, 2) corresponding to N 2 (RM(1, 4)). The two codes are constructed as RM (1,4), u i , where supp(u 1 ) = {1} and supp(u 2 ) = {1, 2, 3, 5, 9}. Let H 20,1 be the Paley Hadamard matrix of order 20 having the form (6), where R is the 19 × 19 circulant matrix with first row:…”

Section: Binary Codes Satisfying (16)-(18)mentioning

confidence: 99%

Quasi-unbiased Hadamard matrices and weakly unbiased Hadamard matrices: A coding-theoretic approach

2016

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This paper is concerned with quasi-unbiased Hadamard matrices and weakly unbiased Hadamard matrices, which are generalizations of unbiased Hadamard matrices, equivalently unbiased bases. These matrices are studied from the viewpoint of coding theory. As a consequence of a coding-theoretic approach, we provide upper bounds on the number of mutually quasi-unbiased Hadamard matrices. We give classifications of a certain class of self-complementary codes for modest lengths. These codes give quasi-unbiased Hadamard matrices and weakly unbiased Hadamard matrices. Some modification of the notion of weakly unbiased Hadamard matrices is also provided.

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“…Recently, four-class cometric Q-antipodal Q-bipartite association schemes were shown to be equivalent to socalled real mutually unbiased bases, and a connection to Hadamard matrices was found in [33]. We also refer to [1] for connections between real mutually unbiased bases and association schemes. Here we shall derive the connection to Hadamard matrices, and see cometric Q-antipodal Q-bipartite four-class schemes as linked systems of Hadamard symmetric nets.…”

Section: 4mentioning

confidence: 99%

Uniformity in association schemes and coherent configurations: Cometric Q-antipodal schemes and linked systems

Dam¹,

Martin²,

Muzychuk³

2013

Journal of Combinatorial Theory, Series A

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Abstract. Inspired by some intriguing examples, we study uniform association schemes and uniform coherent configurations. Perhaps the most important subclass of these objects is the class of cometric Q-antipodal association schemes. The concept of a cometric association scheme (the dual version of a distance-regular graph) is well-known; however, until recently it has not been studied well outside the area of distance-regular graphs. Uniformity is a concept introduced by Higman, but this likewise has not been well-studied. After a review of imprimitivity, we show that an imprimitive association scheme is uniform if and only if it is dismantlable, and we cast these schemes in the broader context of certain -uniform -coherent configurations. We also give a third characterization of uniform schemes in terms of the Krein parameters, and derive information on the primitive idempotents of such a scheme.In the second half of the paper, we apply these results to cometric association schemes. We show that each such scheme is uniform if and only if it is Q-antipodal, and derive results on the parameters of the subschemes and dismantled schemes of cometric Q-antipodal schemes. We revisit the correspondence between uniform indecomposable three-class schemes and linked systems of symmetric designs, and show that these are cometric Q-antipodal. We obtain a characterization of cometric Q-antipodal four-class schemes in terms of only a few parameters, and show that any strongly regular graph with a ("non-exceptional") strongly regular decomposition gives rise to such a scheme. Hemisystems in generalized quadrangles provide interesting examples of such decompositions. We finish with a short discussion of five-class schemes as well as a list of all feasible parameter sets for cometric Q-antipodal four-class schemes with at most six fibres and fibre size at most 2000, and describe the known examples.

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

Cited by 20 publications

References 18 publications

Association schemes related to Delsarte–Goethals codes

Association schemes related to Delsarte–Goethals codes

Quasi-unbiased Hadamard matrices and weakly unbiased Hadamard matrices: A coding-theoretic approach

Uniformity in association schemes and coherent configurations: Cometric Q-antipodal schemes and linked systems

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