Design Theory from the Viewpoint of Algebraic Combinatorics

The notion of T-design in a symmetric association scheme was introduced by Delsarte. A harmonic index t-design is the particular case in which the index set T consists of a single index ftg. Zhu et al. studied a harmonic index t-design in the binary Hamming scheme and gave a Fisher type lower bound on the cardinality. Also they defined the notion of a tight harmonic index design using the bound, and considered the classification problem. In this paper, we extend their results to the general Hamming scheme and improve their Fisher type lower bound slightly. Also using the improved Fisher type lower bound, we redefine the notion of a tight harmonic index design and consider the classification problem. Furthermore, we give a natural characterization for a harmonic index t-design and an analogue in the Hamming scheme for the construction of spherical harmonic index t-designs which was given by Bannai-Okuda-Tagami.

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“…We set B n;t to be the right-hand side in (2). The values C n;t 's have been studied in Brouwer et al [6].…”

Section: Now Let Us Give a Fisher Type Lower Bound For An Himentioning

confidence: 99%

Harmonic Index t-Designs in the Hamming Scheme for Arbitrary q

Tagami

Hori

2021

Graphs and Combinatorics

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“…The aim of design theory is to approximate a space by its finite subset. There have been numerous study of designs on intervals, spheres, and X k , namely the ksubsets of a v-element set X [1,7]. These designs are useful in areas such as numerical computation and experiment design.…”

Section: Motivation and Backgroundmentioning

confidence: 99%

The complex conjugate invariants of Clifford groups

Bannaĭ

Oura

Zhao

2020

Preprint

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Nebe, Rains and Sloane studied the polynomial invariants for real and complex Clifford groups and they relate the invariants to the space of complete weight enumerators of certain self-dual codes. The purpose of this paper is to show that very similar results can be obtained for the invariants of the complex Clifford group X m acting on the space of conjugate polynomials in 2 m variables of degree N 1 in x f and of degree N 2 in their complex conjugates x f . In particular, we show that the dimension of this space is 2, for (N 1 , N 2 ) = (5, 5). This solves the Conjecture 2 given in Zhu, Kueng, Grassl and Gross affirmatively. In other words if an orbit of the complex Clifford group is a projective 4-design, then it is automatically a projective 5-design.

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“…This point of view was discussed in detail by Bannai and Bannai [3]. See also [7,8]. The success and the depth of the theory of Euclidean t-designs (cf.…”

Section: Introductionmentioning

confidence: 99%

“…The success and the depth of the theory of Euclidean t-designs (cf. [38]) has been one driving force for the recent research activity on relative t-designs in Q-polynomial association schemes; see, e.g., [3,5,6,7,8,9,11,32,51,53,54].…”

Section: Introductionmentioning

confidence: 99%

Tight relative t-designs on two shells in hypercubes, and Hahn and Hermite polynomials

Bannaĭ

Tanaka²,

Zhu³

2022

Ars Math. Contemp.

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Relative t-designs in the n-dimensional hypercube Q n are equivalent to weighted regular t-wise balanced designs, which generalize combinatorial t-(n, k, λ) designs by allowing multiple block sizes as well as weights. Partly motivated by the recent study on tight Euclidean t-designs on two concentric spheres, in this paper we discuss tight relative t-designs in Q n supported on two shells. We show under a mild condition that such a relative t-design induces the structure of a coherent configuration with two fibers. Moreover, from this structure we deduce that a polynomial from the family of the Hahn hypergeometric orthogonal polynomials must have only integral simple zeros. The Terwilliger algebra is the main tool to establish these results. By explicitly evaluating the behavior of the zeros of the Hahn polynomials when they degenerate to the Hermite polynomials under an appropriate limit process, we prove a theorem which gives a partial evidence that the non-trivial tight relative t-designs in Q n supported on two shells are rare for large t.

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Design Theory from the Viewpoint of Algebraic Combinatorics

Cited by 19 publications

References 84 publications

Harmonic Index t-Designs in the Hamming Scheme for Arbitrary q

Harmonic Index t-Designs in the Hamming Scheme for Arbitrary q

The complex conjugate invariants of Clifford groups

Tight relative t-designs on two shells in hypercubes, and Hahn and Hermite polynomials

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