A linked system of symmetric designs (LSSD) is a $w$-partite graph ($w\ge 2$) where the incidence between any two parts corresponds to a symmetric design and the designs arising from three parts are related. The original construction for LSSDs by Goethals used Kerdock sets, in which $v$ is a power of two. Some four decades later, new examples were given by Davis et. al. and Jedwab et. al. using difference sets, again with $v$ a power of two. In this paper we develop a connection between LSSDs and “linked simplices”, full-dimensional regular simplices with two possible inner products between vertices of distinct simplices. We then use this geometric connection to construct sets of equiangular lines and to find an equivalence between regular unbiased Hadamard matrices and certain LSSDs with Menon parameters. We then construct examples of non-trivial LSSDs in which $w$ can be made arbitarily large for fixed even part of $v$. Finally we survey the known infinite families of symmetric designs and show, using basic number theoretic conditions, that $w=2$ in most cases.

Revised:

Accepted:

Published online:

DOI: 10.5802/alco.22

Keywords: association schemes, symmetric designs

^{1}

@article{ALCO_2019__2_1_119_0, author = {Kodalen, Brian G.}, title = {Linked systems of symmetric designs}, journal = {Algebraic Combinatorics}, pages = {119--147}, publisher = {MathOA foundation}, volume = {2}, number = {1}, year = {2019}, doi = {10.5802/alco.22}, mrnumber = {3912170}, zbl = {1405.05195}, language = {en}, url = {http://www.numdam.org/articles/10.5802/alco.22/} }

Kodalen, Brian G. Linked systems of symmetric designs. Algebraic Combinatorics, Volume 2 (2019) no. 1, pp. 119-147. doi : 10.5802/alco.22. http://www.numdam.org/articles/10.5802/alco.22/

[1] On Boolean functions with the sum of every two of them being bent, Des. Codes Cryptography, Volume 49 (2008) no. 1-3, pp. 341-346 | DOI | MR | Zbl

[2] Distance-regular graphs, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 18, Springer, 1989, xviii+495 pages | DOI | MR | Zbl

[3] On groups with several doubly-transitive permutation representations, Math. Z., Volume 128 (1972), pp. 1-14 | DOI | MR | Zbl

[4] Quadratic forms over $GF\left(2\right)$, Nederl. Akad. Wet., Proc., Ser. A, Volume 76 (1973), pp. 1-8 | MR | Zbl

[5] Handbook of combinatorial designs (Colbourn, Charles J.; Dinitz, Jeffrey H., eds.), Discrete Mathematics and its Applications, Chapman & Hall/CRC, 2007, xxii+984 pages | MR | Zbl

[6] Linking systems in nonelementary abelian groups, J. Comb. Theory, Ser. A, Volume 123 (2014), pp. 92-103 | DOI | MR | Zbl

[7] Large equiangular sets of lines in Euclidean space, Electron. J. Comb., Volume 7 (2000), R55, 3 pages http://www.combinatorics.org/volume_7/abstracts/v7i1r55.html | MR | Zbl

[8] Nonlinear codes defined by quadratic forms over $\mathrm{GF}\left(2\right)$, Inform. and Control, Volume 31 (1976) no. 1, pp. 43-74 | DOI | MR | Zbl

[9] On the real unbiased Hadamard matrices, Combinatorics and graphs (Contemporary Mathematics), Volume 531, American Mathematical Society, 2010, pp. 243-250 | DOI | MR | Zbl

[10] Kirkman equiangular tight frames and codes, IEEE Trans. Inf. Theory, Volume 60 (2014) no. 1, pp. 170-181 | DOI | MR | Zbl

[11] Linking systems of difference sets (2017) (https://arxiv.org/abs/1708.04405) | Zbl

[12] Euler squares, Ann. Math., Volume 23 (1922) no. 3, pp. 221-227 | DOI | MR

[13] Imprimitive cometric association schemes: constructions and analysis, J. Algebr. Comb., Volume 25 (2007) no. 4, pp. 399-415 | DOI | MR | Zbl

[14] The systems of linked $2$-$(16,\phantom{\rule{0.166667em}{0ex}}6,\phantom{\rule{0.166667em}{0ex}}2)$ designs, Ars Comb., Volume 11 (1981), pp. 131-148 | MR | Zbl

[15] Symmetric Bush-type Hadamard matrices of order $4{m}^{4}$ exist for all odd $m$, Proc. Am. Math. Soc., Volume 134 (2006) no. 8, pp. 2197-2204 | DOI | MR | Zbl

[16] On homogeneous systems of linked symmetric designs, Math. Z., Volume 138 (1974), pp. 15-20 | DOI | MR | Zbl

[17] Coherent configurations and triply regular association schemes obtained from spherical designs, J. Comb. Theory, Ser. A, Volume 117 (2010) no. 8, pp. 1178-1194 | DOI | MR | Zbl

[18] An Asymptotic Existence Theory on Incomplete Mutually Orthogonal Latin Squares. (2015) master thesis, University of Victoria (Canada)

[19] Three-class association schemes, J. Algebr. Comb., Volume 10 (1999) no. 1, pp. 69-107 | DOI | MR | Zbl

[20] Uniformity in association schemes and coherent configurations: cometric Q-antipodal schemes and linked systems, J. Comb. Theory, Ser. A, Volume 120 (2013) no. 7, pp. 1401-1439 | DOI | MR | Zbl

[21] Introduction to coding theory, Graduate Texts in Mathematics, 86, Springer, 1999, xiv+227 pages | DOI | MR

[22] New construction of mutually unbiased bases in square dimensions, Quantum Inf. Comput., Volume 5 (2005) no. 2, pp. 93-101 | MR | Zbl

*Cited by Sources: *