Abstract
Generalized perfect binary arrays (GPBAs) were used by Jedwab to construct perfect binary arrays. A non-trivial GPBA can exist only if its energy is 2 or a multiple of 4. This paper introduces generalized optimal binary arrays (GOBAs) with even energy not divisible by 4, as analogs of GPBAs. We give a procedure to construct GOBAs based on a characterization of the arrays in terms of 2-cocycles. As a further application, we determine negaperiodic Golay pairs arising from generalized optimal binary sequences of small length.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Álvarez V., Armario J.A., Frau M.D., Real P.: A system of equations for describing cocyclic Hadamard matrices. J. Comb. Des. 16, 276–290 (2008).
Arasu K.J., Chen Y.Q., Pott A.: Hadamard and conference matrices. J. Algebr. Comb. 14, 103–117 (2001).
Armario J.A., Flannery D.L.: On quasi-orthogonal cocycles. J. Comb. Des. 26, 401–411 (2018).
Balonin N., Djokovic D.: Negaperiodic Golay pairs and Hadamard matrices. Inf. Control Syst. 5, 2–17 (2015).
Carlet C., Ding C.: Highly nonlinear mappings. J. Complexity 20, 205–244 (2004).
de Launey W., Flannery D.L., Horadam K.J.: Cocyclic Hadamard matrices and difference sets. Discret. Appl. Math. 102, 47–62 (2000).
de Launey W., Flannery D.L.: Algebraic Design Theory. Mathematical Surveys and Monographs, vol. 175. American Mathematical Society, Providence, RI (2011).
Egan R.: On equivalence of negaperiodic Golay pairs. Des. Codes Cryptogr. 85, 523–532 (2017).
Flannery D.L.: Cocyclic Hadamard matrices and Hadamard groups are equivalent. J. Algebra 192, 749–779 (1997).
Hiramine Y.: On \((2n,2,2n, n)\)-relative difference sets. J. Comb. Theory Ser. A 101, 281–284 (2003).
Hughes G.: Non-splitting abelian \((4t,2,4t, 2t)\) relative difference sets and Hadamard cocycles. Eur. J. Combin. 21, 323–331 (2000).
Ito N.: On Hadamard groups. J. Algebra 168, 981–987 (1994).
Ito N.: On Hadamard groups III. Kyushu J. Math. 51, 369–379 (1997).
Jedwab J.: Generalized perfect arrays and Menon difference sets. Des. Codes Cryptogr. 2, 19–68 (1992).
Pott A.: Nonlinear functions in abelian groups and relative difference sets. Discret. Appl. Math. 138, 177–193 (2004).
Schmidt B.: Williamson matrices and a conjecture of Ito’s. Des. Codes Cryptogr. 17, 61–68 (1999).
Acknowledgements
The authors thank Kristeen Cheng for reading the manuscript, and Víctor Álvarez for his assistance with computations. We are also grateful to Ronan Egan, who shared his insights on NGPs with us. Remark 5 and reference [2] were kindly provided by one of the referees. This research has been supported by project FQM-016 funded by JJAA (Spain).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by J. Jedwab.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Armario, J.A., Flannery, D.L. Generalized binary arrays from quasi-orthogonal cocycles. Des. Codes Cryptogr. 87, 2405–2417 (2019). https://doi.org/10.1007/s10623-019-00626-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-019-00626-9
Keywords
- Generalized optimal binary arrays
- Autocorrelation
- Cocycle
- (Quasi)-orthogonal
- Difference set
- Negaperiodic Golay pair