Abstract
A \((v,k,\lambda )\) difference set in a group G is a subset \(\{d_1, d_2, \ldots ,d_k\}\) of G such that \(D=\sum d_i\) in the group ring \({\mathbb {Z}}[G]\) satisfies
where \(n=k-\lambda \). If \(D=\sum s_i d_i\), where the \(s_i \in \{ \pm 1\}\), satisfies the same equation, we will call it a signed difference set. This generalizes both difference sets (all \(s_i=1\)) and circulant weighing matrices (G cyclic and \(\lambda =0\)). We will show that there are other cases of interest, and give some results on their existence.
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The datasets generated during this research [8] are available free online at Zenodo.
References
Arasu K.T., Dillon J.F.: Perfect ternary arrays. In: Pott A., Kumaran V., Helleseth T., Jungnickel D. (eds.) Difference Sets, Sequences and Their Correlation Properties, pp. 1–15. Kluwer, Boston (1999).
Arasu K.T., Hollon J.R.: Group developed weighing matrices. Australas. J Comb. 55, 205–234 (2013).
Arasu K.T., Gordon D.M., Zhang Y.: New nonexistence results on circulant weighing matrices. Cryptogr. Commun. 13, 775–789 (2021).
Berndt B., Williams K., Evans R.: Gauss and Jacobi Sums. Wiley, New York (1998).
Beth T., Jungnickel D., Lenz H.: Design Theory, Encyclopedia of Mathematics and Its Applications, vol. 1, 2nd edn Cambridge University Press, New York (1999).
Golomb S.W., Gong G.: Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar. Cambridge University Press, Cambridge (2005).
Gordon, D.M.: La Jolla combinatorics repository. https://www.dmgordon.org (2022)
Gordon, D. M.: La Jolla signed difference set repository, Zenodo, 1.1. https://doi.org/10.5281/zenodo.7473882 (2023).
Helleseth T., Kumar P.V.: Sequences with low correlation. In: Pless V., Brualdi R.A., Huffman W.C. (eds.) Handbook of Coding Theory II, pp. 1765–1853. Elsevier, Amsterdam (1998).
Hu, H., Gong, G.: A new class of ternary and quaternary sequences with two-level autocorrelation. In: Proceedings of the 2010 IEEE International Symposium on Information Theory, pp. 1292–1296 (2010)
Jungnickel D., Pott A., Smith K.W.: Difference sets. In: Colbourn C.J. (ed.) CRC Handbook of Combinatorial Designs, 2nd edn, pp. 419–435. CRC Press, Boca Raton (2007).
Lehmer E.: On residue difference sets. Can. J. Math. 5, 425–432 (1953).
Tan M.M.: Group invariant weighing matrices. Des. Codes Cryptogr. 86, 2677–2702 (2018).
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Communicated by K. T. Arasu.
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Gordon, D.M. Signed difference sets. Des. Codes Cryptogr. 91, 2107–2115 (2023). https://doi.org/10.1007/s10623-022-01171-8
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DOI: https://doi.org/10.1007/s10623-022-01171-8