Crnković, Dean; Danilović, Doris Dumičić; Egan, Ronan; Švob, Andrea Periodic Golay pairs and pairwise balanced designs. (English) Zbl 1493.05055 J. Algebr. Comb. 55, No. 1, 245-257 (2022). In this paper, a classification of periodic Golay pairs of length less than \(40\) is provided, making use of a relation between certain pairwise balanced designs with \(v\) points and periodic Golay pairs of length \(v\).A pair \((a,b)\) of \(\{\pm 1\}\) sequences is a periodic Golay pair (PGP) if \[\sum_{i=0}^{v-1}a_i a_{i+s} + \sum_{i=0}^{v-1}b_i b_{i+s}=0\]for every shift \(s\), with \(0\leq s\leq v-1\).Periodic Golay pairs generalize Golay pairs, which are known to have applications in multi-slit spectroscopy, signal processing, digital communications, and other areas. PGPs exist at lengths where no Golay pairs may exist, and they also possess the properties required for the applications.In this paper, the authors construct pairwise balanced designs (PBDs) using orbit matrices, with suitable parameters and given cyclic automorphism group, and they use them to construct PGPs. They classify PBDs with these conditions which correspond to PGPs of length \(v\) for all \(v \leq 34\), and they observe that there are no periodic Golay pairs of length \(36\) and \(38\). Moreover, to complete the classification of PBDs and PGPs up to isomorphism, a suitable notion of isomorphism is introduced. Finally, the authors show how the PGPs constructed and their related orbit matrices can be used to construct quasi-cyclic self-orthogonal linear codes over suitable finite fields. Reviewer: Gianira N. Alfarano (Zürich) Cited in 1 Review MSC: 05B30 Other designs, configurations 05E18 Group actions on combinatorial structures 11B83 Special sequences and polynomials 94B05 Linear codes (general theory) Keywords:periodic Golay pair; pairwise balanced design; self-orthogonal codes Software:Magma; GAP PDFBibTeX XMLCite \textit{D. Crnković} et al., J. Algebr. Comb. 55, No. 1, 245--257 (2022; Zbl 1493.05055) Full Text: DOI References: [1] Arasu, KT; Xiang, Q., On the existence of periodic complementary binary sequences, Des. Codes Cryptogr., 2, 257-262 (1992) · Zbl 0763.94017 [2] Balonin, NA; Djoković, D. Ž., Symmetry of two-circulant Hadamard matrices and periodic Golay pairs, (in Russian) Inf, Control Syst., 3, 2-16 (2015) [3] Bosma, W., Cannon, J.: Handbook of Magma Functions. University of Sydney, Department of Mathematics (1994). http://magma.maths.usyd.edu.au/magma · Zbl 0964.68595 [4] Crnković, D., Symmetric (70,24,8) designs having \({\rm Frob}_{21} \times Z_2\) as an automorphism group, Glas. Mat. Ser. III, 34, 54, 109-121 (1999) · Zbl 0948.05010 [5] Crnković, D.; Danilović, D. Dumičić; Rukavina, S., On symmetric (78,22,6) designs and related self-orthogonal codes, Util. Math., 109, 227-253 (2018) · Zbl 1415.94478 [6] Crnković, D.; Rukavina, S., Construction of block designs admitting an Abelian automorphism group, Metrika, 62, 175-183 (2005) · Zbl 1080.05011 [7] Djoković, D. Ž., Equivalence classes and representatives of Golay sequences, Discrete Math., 189, 79-93 (1998) · Zbl 0949.94001 [8] Djoković, D. Ž.; Kotsireas, IS; Colbourn, C. J., Periodic Golay Pairs of Length 72, Algebraic Design Theory and Hadamard Matrices, Springer Proceedings in Mathematics and Statistics, 83-92 (2015), Cham: Springer, Cham · Zbl 1329.05035 [9] Djoković, D. Ž.; Kotsireas, IS, Some new periodic Golay pairs, Numer. Algor., 69, 523-530 (2015) · Zbl 1317.05023 [10] Djoković, D. Ž, Kotsireas, I.S.: Compression of periodic complementary sequences and applications. Des. Codes Cryptogr. 74, 365-377 (2015) · Zbl 1307.05033 [11] Egan, R., On equivalence of negaperiodic Golay pairs, Des. Codes Cryptogr., 85, 523-532 (2017) · Zbl 1372.05025 [12] Eliahou, S., Kervaire, M., Saffari, B.: A new restriction on the lengths of Golay complementary sequences. J. Combin. Theory Ser. A 55(1), 49-59 (1990) · Zbl 0705.94012 [13] The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.8.4; 2016. http://www.gap-system.org [14] Golay, MJE, Multislit spectrometry, J. Opt. Soc. Am., 39, 437-444 (1949) [15] Harada, M.; Tonchev, VD, Self-orthogonal codes from symmetric designs with fixed-point-free automorphisms, Discrete Math., 264, 81-90 (2003) · Zbl 1019.94019 [16] Janko, Z.: Coset enumeration in groups and constructions of symmetric designs, Combinatorics ’90 (Gaeta, 1990). Ann. Discrete Math 52, 275-277 (1992) · Zbl 0773.05010 [17] Tonchev, VD; Colbourn, CJ; Dinitz, JH, Codes, Handbook of combinatorial designs, 677-702 (2007), CRC, Boca Raton: Chapman and Hall, CRC, Boca Raton · Zbl 1101.05001 [18] Weathers, G.; Holiday, E., Group-complementary array coding for radar clutter rejection, IEEE Trans. Aerosp. Electron. Syst., AES-29, 369-379 (1983) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.