Abstract
Two-level Cretan matrices are orthogonal matrices with two elements, x and y. At least one element per row and column is 1 and the other element has modulus ≤ 1. These have been studied in the Russian literature for applications in image processing and compression. Cretan matrices have been found by both mathematical and computational methods but this paper concentrates on mathematical solutions for the first time.
We give, for the first time, families of Cretan matrices constructed using the incidence matrix of a symmetric balanced incomplete block design and Hadamard related difference sets.
References
[1] N. A. Balonin. Existence of Mersenne Matrices of 11th and 19th Orders. Informatsionno-upravliaiushchie sistemy, 2013. 2, pp. 89 – 90 (In Russian). Search in Google Scholar
[2] N. A. Balonin and L. A. Mironovski. Hadamard matrices of odd order, Informatsionno-upravliaiushchie sistemy, 2006.3, pp. 46–50 (In Russian). Search in Google Scholar
[3] N. A. Balonin and Jennifer Seberry. Remarks on extremal and maximum determinant matrices with real entries ≤ 1. Informatsionno-upravliaiushchie sistemy, 5, (71) (2014), p2–4. (In English). 10.15217/issn1684-8853.2016.4.2Search in Google Scholar
[4] N. A. Balonin and M. B. Sergeev. On the issue of existence of Hadamard and Mersenne matrices. Informatsionnoupravliaiushchie sistemy, 2013. 5 (66), pp. 2–8 (In Russian). Search in Google Scholar
[5] J. Hadamard, Résolution d’une question relative aux déterminants. Bulletin des Sciences Mathematiques. 1893. Vol. 17. pp. 240-246. Search in Google Scholar
[6] La Jolla Difference Set Repository. URL www.ccrwest.org/ds.html. Viewed 2014:10:03. Search in Google Scholar
[7] Jennifer Seberry and Mieko Yamada. Hadamard matrices, sequences, and block designs, Contemporary Design Theory: A Collection of Surveys, J. H. Dinitz and D. R. Stinson, eds., John Wiley and Sons, Inc., 1992. pp. 431–560. Search in Google Scholar
©2015 N. A. Balonin and Jennifer Seberry
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.