# ALGEBRAIC COMBINATORICS

Algebraic Combinatorics, Volume 2 (2019) no. 1, pp. 119-147.

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: 2018-05-16
Accepted: 2018-05-19
Published online: 2019-02-04
DOI: https://doi.org/10.5802/alco.22
Classification: 05E30
Keywords: association schemes, symmetric designs
@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 = {alco.centre-mersenne.org/item/ALCO_2019__2_1_119_0/}
}
Kodalen, Brian G. Linked systems of symmetric designs. Algebraic Combinatorics, Volume 2 (2019) no. 1, pp. 119-147. doi : 10.5802/alco.22. https://alco.centre-mersenne.org/item/ALCO_2019__2_1_119_0/

[1] Bey, Christian; Kyureghyan, Gohar M. 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 | Article | MR 2438460 | Zbl 1196.05010

[2] Brouwer, Andries E.; Cohen, Arjeh M.; Neumaier, Arnold Distance-regular graphs, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., Volume 18, Springer, 1989, xviii+495 pages | Article | MR 1002568 | Zbl 0747.05073

[3] Cameron, Peter J. On groups with several doubly-transitive permutation representations, Math. Z., Volume 128 (1972), pp. 1-14 | Article | MR 0316538 | Zbl 0227.20001

[4] Cameron, Peter J.; Seidel, Johan J. Quadratic forms over $GF\left(2\right)$, Nederl. Akad. Wet., Proc., Ser. A, Volume 76 (1973), pp. 1-8 | MR 0327801 | Zbl 0258.05022

[5] Colbourn, Charles J.; Dinitz, Jeffrey H. Handbook of combinatorial designs, Discrete Mathematics and its Applications, Chapman & Hall/CRC, 2007, xxii+984 pages | MR 2246267 | Zbl 1101.05001

[6] Davis, James A.; Martin, William J.; Polhill, John B. Linking systems in nonelementary abelian groups, J. Comb. Theory, Ser. A, Volume 123 (2014), pp. 92-103 | Article | MR 3157802 | Zbl 1281.05131

[7] de Caen, Dominique Large equiangular sets of lines in Euclidean space, Electron. J. Comb., Volume 7 (2000), 3 pages http://www.combinatorics.org/volume_7/abstracts/v7i1r55.html | MR 1795615 | Zbl 0966.51010

[8] Goethals, Jean-Marie Nonlinear codes defined by quadratic forms over $\mathrm{GF}\left(2\right)$, Inform. and Control, Volume 31 (1976) no. 1, pp. 43-74 | Article | MR 0406682 | Zbl 0337.94007

[9] Holzmann, Wolfgang H.; Kharaghani, Hadi; Orrick, William On the real unbiased Hadamard matrices, Combinatorics and graphs (Contemporary Mathematics) Volume 531, American Mathematical Society, 2010, pp. 243-250 | Article | MR 2757803 | Zbl 1231.05043

[10] Jasper, John; Mixon, Dustin G.; Fickus, Matthew Kirkman equiangular tight frames and codes, IEEE Trans. Inf. Theory, Volume 60 (2014) no. 1, pp. 170-181 | Article | MR 3150919 | Zbl 1364.42036

[11] Jedwab, Jonathan; Li, Shuxing; Simon, Samuel Linking systems of difference sets (2017) (https://arxiv.org/abs/1708.04405 ) | Zbl 1415.05025

[12] MacNeish, Harris F. Euler squares, Ann. Math., Volume 23 (1922) no. 3, pp. 221-227 | Article | MR 1502613

[13] Martin, William J.; Muzychuk, Mikhail; Williford, Jason Imprimitive cometric association schemes: constructions and analysis, J. Algebr. Comb., Volume 25 (2007) no. 4, pp. 399-415 | Article | MR 2320370 | Zbl 1118.05100

[14] Mathon, Rudolf The systems of linked $2$-$\left(16,\phantom{\rule{0.166667em}{0ex}}6,\phantom{\rule{0.166667em}{0ex}}2\right)$ designs, Ars Comb., Volume 11 (1981), pp. 131-148 | MR 629867 | Zbl 0468.05012

[15] Muzychuk, Mikhail; Xiang, Qing 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 | Article | MR 2213691 | Zbl 1088.05020

[16] Noda, Ryuzaburo On homogeneous systems of linked symmetric designs, Math. Z., Volume 138 (1974), pp. 15-20 | Article | MR 0347636 | Zbl 0275.05016

[17] Suda, Sho Coherent configurations and triply regular association schemes obtained from spherical designs, J. Comb. Theory, Ser. A, Volume 117 (2010) no. 8, pp. 1178-1194 | Article | MR 2677683 | Zbl 1206.05105

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

[19] van Dam, Edwin R. Three-class association schemes, J. Algebr. Comb., Volume 10 (1999) no. 1, pp. 69-107 | Article | MR 1701285 | Zbl 0929.05096

[20] van Dam, Edwin R.; Martin, William J.; Muzychuk, Mikhail 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 | Article | MR 3092674 | Zbl 1314.05236

[21] van Lint, Jacobus H. Introduction to coding theory, Graduate Texts in Mathematics, Volume 86, Springer, 1999, xiv+227 pages | Article | MR 1664228

[22] Wocjan, Pawel; Beth, Thomas New construction of mutually unbiased bases in square dimensions, Quantum Inf. Comput., Volume 5 (2005) no. 2, pp. 93-101 | MR 2132048 | Zbl 1213.81108