Constructions of Goethals–Seidel Sequences by Using k-Partition
Abstract
:1. Introduction
2. Preliminaries
2.1. Parseval Relation
2.2. GS Sequences
2.3. T-Matrices and T-Matrix Sequences
3. Main Results
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
- Álvarez, V.; Armario, J.; Falcón, R.; Frau, M.; Gudiel, F.; Güemes, M.; Osuna, A. On Cocyclic Hadamard Matrices over Goethals–Seidel Loops. Mathematics 2020, 8, 24. [Google Scholar] [CrossRef] [Green Version]
- Álvarez, V.; Armario, J.; Frau, M.; Gudiel, F.; Güemes, M.; Osuna, A. Hadamard Matrices with Cocyclic Core. Mathematics 2021, 9, 857. [Google Scholar] [CrossRef]
- Armario, J. Boolean Functions and Permanents of Sylvester Hadamard Matrices. Mathematics 2021, 9, 177. [Google Scholar] [CrossRef]
- Barrera Acevedo, S.; Ó Catháin, P.; Dietrich, H. Cocyclic two-circulant core Hadamard matrices. J. Algebraic Combin. 2022, 55, 201–215. [Google Scholar] [CrossRef]
- Horadam, K.J. Hadamard Matrices and Their Applications; Princeton University Press: Princeton, NJ, USA, 2007. [Google Scholar]
- Seberry, J. Orthogonal Designs; Hadamard Matrices, Quadratic forms and Algebras, Revised and Updated Edition of the 1979 Original [MR0534614]; Springer: Cham, Switzerland, 2017; pp. 1–5. [Google Scholar]
- Wallis, W.D.; Street, A.P.; Wallis, J.S. Combinatorics: Room Squares, Sum-Free Sets, Hadamard Matrices; Lecture Notes in Mathematics; Springer: Berlin, Germany; New York, NY, USA, 1972. [Google Scholar]
- Xia, T.; Xia, M.; Seberry, J. The construction of regular Hadamard matrices by cyclotomic classes. Bull. Iran. Math. Soc. 2021, 47, 601–625. [Google Scholar] [CrossRef]
- Kharaghani, H. Arrays for orthogonal designs. J. Combin. Des. 2000, 8, 166–173. [Google Scholar] [CrossRef]
- Goethals, J.; Seidel, J. A skew Hadamard matrix of order 36. J. Austral. Math. Soc. 1970, 11, 343–344. [Google Scholar] [CrossRef] [Green Version]
- Whiteman, A. An infinite family of Hadamard matrices of Williamson type. J. Comb. Theory A 1973, 14, 334–340. [Google Scholar] [CrossRef] [Green Version]
- Doković, D.Ž. Construction of some new Hadamard matrices. Bull. Austral. Math. Soc. 1992, 45, 327–332. [Google Scholar] [CrossRef] [Green Version]
- Doković, D.Ž. Skew Hadamard matrices of order 4 × 37 and 4 × 43. J. Combin. Theory Ser. A 1992, 61, 319–321. [Google Scholar] [CrossRef]
- Doković, D.Ž. Ten Hadamard matrices of order 1852 of Goethals-Seidel type. Eur. J. Combin. 1992, 13, 245–248. [Google Scholar] [CrossRef] [Green Version]
- Doković, D.Ž. Ten new orders for Hadamard matrices of skew type. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 1992, 3, 47–59. [Google Scholar]
- Doković, D.Ž. Two Hadamard matrices of order 956 of Goethals-Seidel type. Combinatorica 1994, 14, 375–377. [Google Scholar] [CrossRef]
- Doković, D.Ž. Skew-Hadamard matrices of orders 436, 580, and 988 exist. J. Combin. Des. 2008, 16, 493–498. [Google Scholar] [CrossRef] [Green Version]
- Doković, D.Ž.; Golubitsky, O.; Kotsireas, I. Some new orders of Hadamard and skew-Hadamard matrices. J. Combin. Des. 2014, 22, 270–277. [Google Scholar] [CrossRef] [Green Version]
- Doković, D.Ž.; Kotsireas, I. Goethals-Seidel difference families with symmetric or skew base blocks. Math. Comput. Sci. 2018, 12, 373–388. [Google Scholar] [CrossRef] [Green Version]
- Fletcher, R.; Koukouvinos, C.; Seberry, J. New skew-Hadamard matrices of order 4 × 59 and new D-optimal designs of order 2 × 59. Discrete Math. 2004, 286, 251–253. [Google Scholar] [CrossRef]
- Whiteman, A. Skew Hadamard matrices of Goethals—Seidel type. Discrete Math. 1972, 2, 397–405. [Google Scholar] [CrossRef] [Green Version]
- Xia, M.; Xia, T.; Seberry, J.; Wu, J. An infinite family of Goethals-Seidel arrays. Discrete Appl. Math. 2005, 145, 498–504. [Google Scholar] [CrossRef] [Green Version]
- Yang, C.H. Hadamard matrices, finite sequences, and polynomials defined on the unit circle. Math. Comp. 1979, 33, 688–693. [Google Scholar] [CrossRef]
- Yang, C.H. Hadamard matrices and δ-codes of length 3n. Proc. Amer. Math. Soc. 1982, 85, 480–482. [Google Scholar] [CrossRef]
- Yang, C.H. Lagrange identity for polynomials and δ-codes of lengths 7t and 13t. Proc. Amer. Math. Soc. 1983, 88, 746–750. [Google Scholar] [CrossRef]
- Yang, C. A composition theorem for δ-codes. Proc. Amer. Math. Soc. 1983, 89, 375–378. [Google Scholar] [CrossRef]
- Yang, C. On composition of four-symbol δ-codes and Hadamard matrices. Proc. Amer. Math. Soc. 1989, 107, 763–776. [Google Scholar]
- Doković, D.Ž.; Kotsireas, I. Compression of periodic complementary sequences and applications. Des. Codes Cryptogr. 2015, 74, 365–377. [Google Scholar] [CrossRef] [Green Version]
- Fletcher, R.; Gysin, M.; Seberry, J. Application of the discrete Fourier transform to the search for generalised Legendre pairs and Hadamard matrices. Australas. J. Combin. 2001, 23, 75–86. [Google Scholar]
- Zuo, G.; Xia, M.; Xia, T. Constructions of composite T-matrices. Linear Algebra Appl. 2013, 438, 1223–1228. [Google Scholar] [CrossRef]
- Cooper, J.; Wallis, J. A construction for Hadamard arrays. Bull. Austral. Math. Soc. 1972, 7, 269–277. [Google Scholar] [CrossRef] [Green Version]
- Xia, M.; Xia, T. A family of C-partitions and T-matrices. J. Combin. Des. 1999, 7, 269–281. [Google Scholar] [CrossRef]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Shen, S.; Zhang, X. Constructions of Goethals–Seidel Sequences by Using k-Partition. Mathematics 2023, 11, 294. https://doi.org/10.3390/math11020294
Shen S, Zhang X. Constructions of Goethals–Seidel Sequences by Using k-Partition. Mathematics. 2023; 11(2):294. https://doi.org/10.3390/math11020294
Chicago/Turabian StyleShen, Shuhui, and Xiaojun Zhang. 2023. "Constructions of Goethals–Seidel Sequences by Using k-Partition" Mathematics 11, no. 2: 294. https://doi.org/10.3390/math11020294
APA StyleShen, S., & Zhang, X. (2023). Constructions of Goethals–Seidel Sequences by Using k-Partition. Mathematics, 11(2), 294. https://doi.org/10.3390/math11020294