MERSENNE, EULER, SEIDEL, FERMAT LEVELS
Hadamard matrix family catalogue and online algorithms © Nickolay A. Balonin, 1.05.2014 TABLE OF LEVEL FUNCTIONS There are 5 main families (Mersenne, Euler, Hadamard, Fermat, Seidel) of absolute and local maximum determinat matrices: det(X)→max; x_{ij}≤1; X'X=ω(n)I. Local maximum determinant matrices [8], [15] can be described by functions of their levels. Varying levels a=1, –b; 0≤b≤1 (can be irrational) allow to construct such simple structures (imposible with integer levels for some orders) as circulant and twocirculant matrices. Definition 1. The values of the entries of a matrix are called levels [1]. Hadamard matrices are optimal twolevel matrices [2, 3], symmetric conference matrices [4] and weighing matrices are threelevel matrices [5]. Quasiorthogonal matrices with maximal determinant of odd orders have been discovered with a larger number of levels [6]. Definition 2. Quasiorthogonal matrix, X, of order n, satisfies X^{T}X = ω(n)I, where ω(n)≤n is the weight, and defined by a function ω(n) or functions a(n)=1, b(n), .. of its levels. We will denote the levels of twolevel matrices as a, –b; for positive 0 ≤ b ≤ a, a = 1. Determinant det(X)=ω(n)^{n/2}, ω(n)=nψ(n)^{2}, where ψ – a normalized determinant, ψ(n)=1 for Hadamard matrices.
Hadamard inequality det(X)≤n^{n/2} gives an estimation of determinant, while det(X)=n^{n/2}×ψ(n)^{n} gives a possible value.
MAIN MATRIX FAMILIES Hadamard matrices H_{12} and H_{16}
Hadamard matrix. A quasiorthogonal matrix, named Hadamard matrix [1] H, is a twolevel quasiorthogonal matrix of order n with level functions a=1,–b; b=1. Hadamard conjecture [3]: Hadamard matrices exist for n=1,2 and 4t, t is integer. Their orders cover n = 2^{k}, k is integer, for so called elementary Hadamard matrices set. Normalized determinant ψ = 1. 
Mersenne matrices M_{11} and M_{15}
Mersenne matrix. A quasiorthogonal matrix, named Mersenne matrix M [8], is a twolevel quasiorthogonal matrix of order n with level functions a=1, –b; where b=t/(t+sqrt(t)), t=(n+1)/4. Conjecture [8,12,15]: Mersenne matrices exist for n=4t–1, t is integer. Their orders cover Mersenne numbers n = 2k–1, k is integer, for so called elementary Mersenne matrices set (or pure Mersenne matrices). Invariant of Mersenne matrices is difference 1 between numbers of positive and negative entries in every column (row), so weight ω = (n+1)/2+(n–1)b^{2}/2=2t+(2t–1)b^{2}. Normalized determinant ψ = sqrt((2t+(2t–1)b^{2}))/(4t–1). 
Euler matrices E_{10} and E_{14}
Euler matrix. A quasiorthogonal matrix, named Euler matrix E [8], is a fourlevel quasiorthogonal matrix of order n, it can be observed as twocirculant (or two blocks A, B) matrices with two level functions a=1, –b; where b=t/(t+sqrt(2t)), t=(n+2)/4. Conjecture [8,12,15]: Euler matrices exist for n=4t–2, t is integer. Their orders cover singular orders for the symmetric conference matrices: the latests do not exist if n–1 is not sum of two squares. Invariant of Euler matrices is difference 2 between numbers of ±1 and ±b entries in every column (row), so weight ω = (n+2)/2+(n–2)b^{2}/2=2t+2t–1)b^{2}. Normalized determinant ψ = sqrt((2t+2(t–1)b^{2}))/(4t–2). 
Seidel matrices S_{9} and S_{13}
Seidel matrix. A quasiorthogonal matrix, named Seidel matrix S [8], is a quasiorthogonal matrix of order n with level functions a=1, –b, d; where b=1–2d, d=1/(1+sqrt(n)) belongs to diagonal entries, t=(n+3)/4. Their orders cover singular points of symmetric conference matrices; Seidel matrices do not exist if n is not sum of two squares [8]. Invariant of Seidel matrices is difference 1 between numbers of positive and negative entries in every column (row), so weight ω = (n–1)(1+b^{2})/2+d^{2}=2(t–1)(1+b^{2})+d^{2}. Normalized determinant ψ = sqrt((2(t–1)(1+b^{2})+d^{2})/(4t–3).  Hadamard H_{16} anf Fermat F_{17}
Fermat matrix. A quasiorthogonal matrix, named Fermat matrix F [8], is a quasiorthogonal matrix of order n with level functions a=1, –b, s; for n=3, 2a=b=s=1; for n>3, q=n–1=4u^{2}, p=q+sqrt(q), we have b≤s<a; where b=(2n–p)/p=(2u^{2}–u+1)/(2u^{2}+u)=1–(2u–1)/u(2u+1), s=sqrt(nq–2sqrt(q))/p=sqrt(nu^{2}–u)/(2u^{2}+u) is border. Their orders cover Fermat numbers n = 2^{2t}+1, t is integer, for so called elementary Fermat matrices set (or pure Fermat matrices). Invariant of Fermat matrices: every non border column has k=(q–sqrt(q))/2=2u^{2}–u of entries a=1, so weight ω(n)=k+(q–k)b^{2}+s^{2}. Normalized determinant ψ = sqrt(ω(n)/n).  MERSENNE MATRIX LEVEL b b=(q–(4q)^{1/2})/(q–4), q=n+1, n=4k–1.
EULER MATRIX LEVEL b b=(q–(8q)^{1/2})/(q–8), q=n+2, n=4k–2.
DETERMINANT GRAPH D=1/m^{n}=n^{n/2}/h^{n}Hadamard H, Belevitch C, Euler E, Mersenne M, Fermat F matrices, Cyan is Proclus. References 1. Balonin N.A., Seberry, Jennifer. Remarks on Extremal and Maximum Determinant Matrices with Moduli of Real Entries ≤ 1, Informatsionnoupravliaiushchie sistemy, 2014, ¹ 5, pp. 2–7. 2. Seberry, Jennifer, Yamada, Mieko. 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. 3. Hadamard J. Résolution d'une question relative aux déterminants. Bulletin des Sciences Mathématiques. 1893. Vol. 17. pp. 240–246. 4. Balonin N. A., Seberry, Jennifer. A review and new symmetric conference matrices. Informatsionnoupravliaiushchie sistemy, 2014. ¹ 4 (71), pp. 2–7. 5. Wallis (Seberry), Jennifer. Orthogonal (0,1,–1) matrices, Proceedings of First Australian Conference on Combinatorial Mathematics, TUNRA, Newcastle, 1972. pp. 61–84. 6. Balonin N. A., Mironovski L.A. Hadamard matrices of odd order, Informatsionnoupravliaiushchie sistemy, 2006. ¹ 3, pp. 46–50 (In Russian). 7. Balonin N. A., Sergeev M. B. Mmatrices. Informatsionnoupravliaiushchie sistemy, 2011, ¹ 1, pp. 1421 (In Russian) 8. Balonin N. A., Sergeev M. B. Local Maximum Determinant Matrices. Informatsionnoupravliaiushchie sistemy, 2014. ¹ 1 (68), pp. 2–15 (In Russian). 9. Balonin Yu. N., Sergeev M. B. Mmatrix of 22nd order. Informatsionnoupravliaiushchie sistemy, 2011. ¹ 5 (54), pp. 87–90 (In Russian). 10. Balonin Yu. N., Sergeev M. B. The algorithm and program for searching and studying of Mmatrices. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2013. ¹ 3, pp.82 – 86 (In Russian). 11. Balonin N. A. Quasiorthogonal matrix with maximal determinant, order 58, http://mathscinet.ru/catalogue/artifact58 12. Balonin N. A. Existence of Mersenne Matrices of 11th and 19th Orders. Informatsionnoupravliaiushchie sistemy, 2013. ¹ 2, pp. 89 – 90 (In Russian). 13. Balonin N. A., Sergeev M. Â. Two Ways to Construct HadamardEuler Matrices. Informatsionnoupravliaiushchie sistemy, 2013. ¹ 1(62), pp. 7–10 (In Russian). 14. Sergeev A.M. Generalized Mersenne Matrices and Balonin’s Conjecture. Automatic Control and Computer Sciences, 2014. Vol. 48, ¹ 4. pp. 214–220. 15. Balonin N. A., Sergeev M. B. On the Issue of Existence of Hadamard and Mersenne Matrices. Informatsionnoupravliaiushchie sistemy, 2013. ¹ 5 (66), pp. 2–8 (In Russian). 15. Balonin N. A., Djokovic D. Z., Mironovski L.A., Seberry Jennifer, Sergeev M. B. Hadamard type Matrices Catalogue, http://mathscinet.ru/catalogue

