Association schemes on triples and a ternary algebra

Motivated by symmetric association schemes (which are known to approximate generously unitransitive group actions), we formulate combinatorial approximations to transitive extensions of generously unitransitive permutation groups. Specifically, the notions of compatible and coherent partitions are suggested and investigated in terms of the orbits of an ambient group (H, X) on the k-subsets of X, k = 2, 3, 4. We apply these ideas to investigate transitive extensions of the automorphism groups of the classical Johnson and Hamming schemes. In the latter case, we further provide algorithmic details and computer-generated data for the particular series of Hamming schemes H(m, 3), m ≥ 2. Finally, our approach is compared to the concept of a symmetric association scheme on triples in the sense of Mesner and Bhattacharya. q

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“…The following result appears as Theorem 2.13 in [15]; however, a detailed proof is not given there. Theorem 9.1.…”

Section: B Contraction Of a Compatible Partitionmentioning

confidence: 90%

“…Indeed, partially balanced block designs comprise their main focus, and appear as early as the first page of [15].…”

Section: A Genesismentioning

confidence: 99%

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A combinatorial approach to transitive extensions of generously unitransitive permutation groups

Klin

Mesner

Woldar

2010

J of Combinatorial Designs

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“…The Bhattacharya-Mesner product, or BM product for short, was first developed in [MB90,MB94]. The BM product provides a natural generalization to the matrix product.…”

Section: Introductionmentioning

confidence: 99%

On the spectra of hypermatrix direct sum and Kronecker products constructions

Gnang

Filmus

2017

Linear Algebra and its Applications

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We extend to hypermatrices definitions and theorem from matrix theory. Our main result is an elementary derivation of the spectral decomposition of hypermatrices generated by arbitrary combinations of Kronecker products and direct sums of cubic side length 2 hypermatrices. The method is based on a generalization of Parseval's identity. We use this general formulation of Parseval's identity to introduce hypermatrix Fourier transforms and discrete Fourier hypermatrices. We extend to hypermatrices a variant of the Gram-Schmidt orthogonalization process as well as Sylvester's classical Hadamard matrix construction. We conclude the paper with illustrations of spectral decompositions of adjacency hypermatrices of finite groups and a short proof of the hypermatrix formulation of the Rayleigh quotient inequality.

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“…Our initial interest in the study of multi-dimensional arrays came from our recent work on combinatorics [11]. The concept of association schemes is well-known in combinatorics with applications in coding theory and design of experiments.…”

Section: Introductionmentioning

confidence: 99%

A New Multi-Dimensional Transform for Digital Signal Processing Using Generalized Association Schemes

Bhattacharya¹

1994

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Association schemes on triples and a ternary algebra

Cited by 21 publications

References 6 publications

A combinatorial approach to transitive extensions of generously unitransitive permutation groups

A combinatorial approach to transitive extensions of generously unitransitive permutation groups

On the spectra of hypermatrix direct sum and Kronecker products constructions

A New Multi-Dimensional Transform for Digital Signal Processing Using Generalized Association Schemes

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