Mersenne and Hadamard Matrices

Балонин, Н. А.; Sergeev, Mikhail

doi:10.15217/issn1684-8853.2016.1.2

Cited by 11 publications

(5 citation statements)

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“…The opposite is also true. Extreme quasi-orthogonal matrices with a small number of entries are called Cretan (see more precise definitions in [24,27]). The interrelation of extremal problems allows us to use the same numerical method to find both Cretan and maximum determinant matrices [20][21][22][23].…”

Section: Numerical Sequences and Number Theorymentioning

confidence: 99%

“…The definition of Fermat matrices can be expanded to "quadratic" orders 4u 2 1. These families are discussed more specifically in [18,24,27]. Orders of Mersenne and Fermat [18,28]…”

Section: Numerical Sequences and Number Theorymentioning

confidence: 99%

“…Based on the preliminary study, we identified that: -skew-symmetric circulant Mersenne matrices correspond to prime Mersenne numbers [27,28];…”

Section: Maximum Determinant Matrix Conjecturementioning

confidence: 99%

“…Generalised Gauss сonjecture. Quasi-orthogonal local maximum determinant matrices [24,27] [18,24], the latter of which is shown on Fig. 3, can be rounded to {a1, b1, entries of s are 1} and checked to validate the conjecture.…”

Section: Maximum Determinant Matrix Conjecturementioning

confidence: 99%

“…Extremal matrices of orders 257 and 65 537 are very difficult to calculate and check the values of their determinants (to match with determinants of Fermat matrices) due to the large number of possible permutations. However, we have calculated the Fermat matrix of order 257 using a modified Sylvester algorithm [27][28][29]. The correspondence of symmetries to matrix orders was studied in [31,32].…”

Section: Maximum Determinant Matrix Conjecturementioning

confidence: 99%

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Prime Fermat numbers and maximum determinant matrix conjecture

Балонин

Sergeev

Vostrikov

2020

ICS

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Purpose: Solution to the problem of optimizing the determinants of matrices with a modulus of entries < 1. Developing a theory of such matrices based on preliminary research results. Methods: Extreme solutions (in terms of the determinant) are found by minimizing the absolute values of orthogonal matrix elements, and their subsequent classification. Results: Matrices of orders equal to prime Fermat numbers have been found. They are special, as their absolute determinant maximums can be reached on a simple structure. We provide a precise evaluation of the determinant maximum for these matrices and formulate a conjecture about it. We discuss the close relation between the solutions of extremal problems with the limitation on the matrix column orthogonality and without it. It has been shown that relative maximums of orthogonality-limited matrix determinants correspond to absolute maximums of orthogonality-unlimited matrix determinants. We also discuss the ways to build extremal matrix families for the orders equal to Mersenne numbers. Practical relevance: Maximum determinant matrices are used extensively in the problems of error-free coding, compression and masking of video information. Programs for maximum determinant matrix search and a library of constructed matrices are used in the mathematical network “mathscinet.ru” along with executable online algorithms.

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Section: Numerical Sequences and Number Theorymentioning

confidence: 99%