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ТЕОРЕТИЧЕСКАЯ И ПРИКЛАДНАЯ МАТЕМАТИКА /
UDC 004.438
TWO LEVEL CRETAN MATRICES CONSTRUCTED VIA SINGER DIFFERENCE SETS
N. A. Balonina, Dr. Sc., Tech., Professor, [email protected]
Jennifer Seberryb, PhD, Professor of Computer Science, [email protected] aSaint-Petersburg State University of Aerospace Instrumentation, 67, B. Morskaia St., 190000, Saint-Petersburg, Russian Federation
bSchool of Computer Science and Software Engineering, Faculty of Engineering and Information Science, University of Wollongong, NSW, 2522, Australia
Purpose: This note discusses two level quasi-orthogonal matrices which were first highlighted by J. J. Sylvester; Hadamard matrices, symmetric conference matrices, and weighing matrices are the best known of these matrices with entries from the unit disk. The goal of this note is to develop a theory of such matrices based on preliminary research results. Methods: Extreme solutions (using the determinant) have been established by minimization of the maximum of the absolute values of the elements of the matrices followed by their subsequent classification. Results: We show that if B is the incidence matrix of a (v, k, A) difference set, then there exists a two-level quasi-orthogonal matrix, S, a Cretan(v) matrix. We apply this result to the Singer family of difference sets obtaining a new infinite family of Cretan matrices. Practical relevance: Web addresses are given for other illustrations and other matrices with similar properties. Algorithms to construct Cretan matrices have been implemented in developing software of the research program-complex.
Keywords — Hadamard Matrices, Quasi-Orthogonal Matrices, Cretan Matrices, Difference Sets, Singer Difference Sets, Hadamard Difference Sets.
AMS Subject Classification: 05B20; 20B20.
Introduction
Difference sets are of considerable use and interest to image processing (compression, masking) to statisticians undertaking medical or agricultural research, to position smaller telescopes to make very large deep space telescopes and to spacing the tread on rubber tyres for vehicles.
In this and further papers we use some names, definitions, notation differently than we have in the past [1]. This, we hope, will cause less confusion, bring our nomenclature closer to common usage and conform for mathematical purists. We have chosen the use of the word level, instead of value for the entries of a matrix, to conform to earlier writings. We note that the strict definition of an orthogonal matrix, X, of order n, is that XTX = XXT = In where In is the identity matrix of order n. In this paper we consider STS = SST = raIn where ra is a constant. We call these quasi-orthogonal matrices [2, 3].
Definition 1. A real square matrix S of order n is called quasi-orthogonal if it satisfies STS = SST = raIn, where In is the nxn identity matrix, and ra is a constant real number. The values of the entries of a matrix are called levels.
There is a trivial one-level matrix for every order n; it is the zero matrix of order n. Hadamard matrices are two-level matrices and symmetric conference matrices and weighing matrices are three-level matrices. Quasi-orthogonal matrices with maximal determinant of odd orders have been discovered with a larger number of levels [1].
We will denote the level/values of two-level Cretan matrices as a, -b; for positive 0 < b < a = 1.
In this and future work we will only use quasiorthogonal to refer to matrices with moduli of real elements < 1 [2], where at least one entry in each row and column must be 1. Hadamard matrices [4], symmetric conference matrices [5], and weighing matrices [6] are the best known of these matrices with entries from the unit disk [7]. We refer to [2] for definitions of these matrices.
The matrix orthogonality equation STS = SST = = raIn, is a set of n2 scalar equations, giving two kinds of formulae: g(a, b) = ra, there are n such equations, and f(a, b) = 0, there are n2 - n such equations. We concentrate on two of them: g(a, b) = ra, f(a, b) = 0.
The entries in raIn which are on the diagonal, are given by the radius equation ra = g(a, b), they depend on the choice of a, b. If a=1, then ra < n.
The maximal weight ra = n arises from Hadamard matrices, symmetric conference matrices have ra = n - 1. Quasi-orthogonal matrices can have also irrational values for the weight.
The second equation f(a, b) = 0 we name the characteristic equation, as it allows us to find a formulae for level b < a.
Definition 2. A Cretan(n) matrix, CM, is a quasiorthogonal matrix of order n with entries < 1, where there must be at least one 1 per row and column. The inner product of a row of CM(n) with itself is the weight ra. The inner product of distinct rows of CM(n) is zero. A x-level Cretan(n; x; ra) matrix, CM(n; x; ra), has x levels or values for its entries. Level a = 1 is pre-determined for all Cretan matrices.
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Cretan(n), or CM(n) quasi-orthogonal matrices are studied in [2, 3]. In more general notation these are can be CM(order), CM(order; number of levels = x), CM(order; number of levels = x; occurrences of levels = Lv L2, ..., Lx), CM(order; number of levels = x; weight = ra), and CM(order; number of levels = x; weight; occurrences of levels in whole matrix), etc. etc. etc.
The definition of Cretan is not that each variable occurs some number of times per row and column but L1, L 2, ..., Lx times in the whole matrix. So we have CM(n; x; ra; L1, L 2, ..., Lx) so
(-0.5 1 1
A
-0.5 1 1 -0.5
is a CM(3), a CM(3;2), a CM(3;2;2.1), a CM(3;2;2.25), a CM(3;2;2.25;6.3) depending on which numbers (in brackets) are currently of interest. We call them Cretan matrices because they were first discussed in this generality at a conference in Crete in July, 2014.
The over-riding aim is to seek CM(n) with absolute or relative (local) maximal determinants as they have many applications in image processing and masking [3].
Definitions
This paper studies the construction of some Cretan matrices made here using Singer difference sets.
Definition 3. Let D = {d1, d2
dk} be a subset
of the integers 0, 1, 2, ... , v - 1. If the collection A = {di - dji i, j, 1...k, i ^ j} contains each element 1, 2, ..., v - 1 exactly X times, D will be called a (v, k, X) difference set.
This is said to be additive notation. Equivalently, (v, k, X) difference set in a multiplicative group G of order v is k-subset D of G such, that every element g ^ 1 of G has exactly X representations g = d1d2-1 with d1, d2 from G. The parameter v is called the order of the difference set.
Difference sets due to James Singer (1938, [8]) appeared first. Marshall Hall [9] wrote an extensive survey in 1956. Difference sets of regular Hadamard matrices were discussed in [10, 11].
We note that for every (v, k, X) difference set there is a complementary (v, v - k, v - 2k + X) difference set made by choosing the subset of 1, 2, ., v not in D.
Definition 4. Let D = {d1, d2, ..., dk} be a difference set. Then the v x v, matrix B = (bi;) is said to be the incidence matrix of D if bij = 1 for j - i in D and 0 if j - i is not in D.
Example 1. Let D be the subset {1, 3, 4, 5, 9} of the integers 0, 1, 2, ..., 10. Hence we take all the dif-
ferences modulo 11. Then A contains 1 - 3 = -2 = 9 1 - 4 = -3 = 8; 1 - 5 = -4 = 7; 1 - 9 = -8 = 3; 3 - 1 = 2
3 - 4 = -1 = 10; 3 - 5 = -2 = 9; 3 - 9 = -6 = 5; 4 - 1 = 3
4 - 3 = 1; 4 - 5 = -1 = 10; 4 - 9 = -5 = 6; 5 - 1 = 4
5 - 3 = 2; 5 - 4 = 1; 9 - 1 = 8; 9 - 3 = 6 = 5; 9 - 4 = 5
9 - 5 = 4; which is each non-zero integer 0, 1, 2, ...,
10 exactly twice.
The incidence matrix of D is B = circ(0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0).
Definition 5. The parameters of a PG(q, m) projective geometry or Singer difference set are
(v, k, X) =
( -1 qm -1 qm-1 -1 ^
q -1 ' q -1 ' q -1
q a prime power.
Examples of matrices with these parameters are given in the survey of Marshall Hall [9], they were generalized in [11, 12].
Construction for Two-Level Cretan Matrices
We now use difference sets to construct two-level Cretan (quasi-orthogonal) matrices. We use the
notation y =
qm-1 -1
qm-1 (q -1)'
Construction 1. Consider the Singer difference sets with parameters
(v, k, X) =
( qm+1 -1 qm -1 qm1 -1 ^ q -1 ' q -1 ' q -1
q a prime power.
Then, when its incidence matrix has its ones replaced by a and zeros with -b, we obtain a two-level quasi-orthogonal matrix S satisfying the orthogonality equation
STS = SST = fflT , (1)
giving the radius equation k a2 +(v - k)b2 = ra or
„m 1
q -1 a2 + qmb2 =ra,
q-1
(2)
and the characteristic equation Xa2 - 2(k - X)ab + + (v - 2k +X)b2 = 0 or
ya2 - 2ab + (q - 1)b2 = 0,
thus so we have solutions b =
1 ±V 1 - y(q -1) '
(3)
(4)
The level a=1 is pre-determined for all quasi-orthogonal matrices; if b>a we have to choose the second level to be 1/b for the complementary difference set, to ensure entries are from the unit disk. v The determinant of Cretan matrices det(S) = ra 2.
y
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We are particularly interested in the projective planes PG(q, 2) with y= —.
q
Corollary 1 (Singer Difference Sets and Projective Planes). Let q be a prime power.
Then there exists a projective plane (q2 + q + 1, q + 1, 1). Hence we have a two-level quasi-orthogonal matrix, S, satisfying (1)-(3) with b = - 1
ra =
(q ±jq )2
- + q + 1, , and det(S) = ra
q +q+i 2
For the "—" sign, choosing a = 1, we have a principal solution with bigger b and bigger determinant of Cretan matrix.
Corollary 2 (Singer Difference Sets and PG(2, m) with y =1 - 21-m). Let q be a prime power.
Then there exists a projective plane (2m+1 - 1, 2m - 1, 2m-1 - 1). Hence we have a two-level quasi-orthogonal matrix, S, satisfying (1)-(3) with
2m+i-1
b
X
r, ra=k + (v-k)b2, and det(S) = ra 2
x+i±VX+i
For the "-" sign, choosing a = 1, we have b > a, it leads to a principal solution with the complementary
difference set (v, v - k, v - 2k + X) and modulus of level 1/b.
Example 2. Consider PG(7, 2) with parameters (57,8,1) and y = - X 1
m-1
q
7
From [13] we have difference set {0, 1 , 5, 7, 17, 35, 38, 49} to generate Cretan matrix CM(57) with modu-
y 1
li of levels a = 1, b = -
1 -V1 -y(q-1) 7-V7
= 0.2297
57
and weight ra=k + (v-k)b2 =10.5845, det(S) = ra 2 = = 1.6 1029 for the principal solution Figure, a. Non principal solution will have the same structure of
matrix S with smaller b =
Y
1+ V1 -Y(q -1)
=0.1037
and smaller determinant 3.7x1026.
Example 3. Consider PG(5, 3) with parameters
(156,31,6) and y = - X 6
q
m-1
25
From [13] we have difference set {0,1,2,4,14,18, 21,22,30,31,37,42,45,49,51,55,56,60,76,82, 85,87,88,93,95,98,108,110,117,134,142} to generate Cretan matrix CM(156) with moduli of levels a = 1,
y _ 6
b =
1 -V1 -Y(q -1) 25-V25
= 0.3 and weight
2
q
■ Table 1. The CM given by PG(q, 2)
q (v, k, X) b ra det(S)
2 (7,4,2)* (7,3,1) 0.5858* 0.2929 5.0294 3.3431 2.85x102 68
3 (13,4,1) 0.7887 0.2113 5.8660 4.4019 9.87x104 1.53x104
5 (31,6,1) 0.3618 0.1382 6.6545 6.4775 5.73x1012 3.77x1012
7 (57,8,1) 0.2297 0.1037 8.3692 8.5267 1.97x1026 3.36x1026
11 (133,12,1) 0.1302 0.0698 12.1863 12.5903 1.6x1072 1.4x1073
13 (183,14,1) 0.1064 0.0602 14.1473 14.6129 1.9x10105 3.6x10106
Signs "-" or "+" (two solutions for b) give two columns.
For case, denoted by *, b > a, it leads to the complementary difference set (v, v - k, v - 2k + X) and level 1/b = 0.5858 by an analogue of the equivalent version for an Hadamard matrix.
■ Table 2. The CM given by PG(2, m)
m (v, k, X) b ra det(S)
1 (3,2,1)* (3,1,0) 0.5 0 2.25 1 3.375 1
2 (7,4,2)* (7,3,1) 0.5858* 0.2929 5.0294 3.3431 2.85x102 68
3 (15,8,4)* (15,7,3) 0.6667* 0.5 11.1111 9 6.97x107 1.43x107
4 (31,16,8)* (31,15,7) 0.7388* 0.6464 24.1873 21.6863 2.79x1021 0.51x1021
5 (63,32,16)* (63,31,15) 0.8* 0.75 51.84 49 1.0x1054 1.7x1053
6 (127,64,32)* (127,63,31) 0.8498* 0.8232 109.4938 106.3726 3.2x10129 5.1x10128
Signs "-" or "+" (two solutions for b) give two columns.
For case, denoted by *, b > a, it leads to the complementary difference set (v, v - k, v - 2k + X) and level 1/b = 0.5858, the same, as we have seen before.
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■ Cretan matrix CM(57) for PG(7, 2) (a) and Cretan matrix CM(156) for pG(5, 3) (b) of Singer Family
-,126
156
ra = k + (v-k)b2 =42.25, det(S) = ra 2 =6.5xlO1 for the principal solution Figure, b. Non principal solution will have the same structure of matrix S
with smaller b = -
Y
= 0.2 and smaller
1+ V1 - y(q -1) determinant 2.5x10121.
Acknowledgements
The authors would like to acknowledge Professor Mikhael Sergeev for his advice regarding the content of this paper. The authors also wish to sincerely thank Tamara Balonina for converting this paper into printing format. We acknowledge the use of the http://www.mathscinet.ru and http://www. wolframalpha.com sites for the number and symbol calculations in this paper.
Conclusion
We note that there exist (v, k, X) Singer difference sets for v = 4t + 1, 4t, 4t - 1, 4t - 2.
The La Jolla Difference Set Repository [13] gives many parameter sets which can make circulant incidence matrices from difference sets. It opens new possibilities for image processing (compression, masking) and other areas we mentioned in our introduction.
References
1. Balonin N. A., Mironovskii L. A. Hadamard Matrices of Odd Order. Informatsionno-upravliaiushchie siste-my, 2006, no. 3(22), pp. 46-50 (In Russian).
2. Balonin N. A., Seberry Jennifer. Remarks on Extremal and Maximum Determinant Matrices with Real Entries < 1. Informatsionno-upravliaiushchie siste-my, 2014, no. 5(71), pp. 2-4.
3. Balonin N. A., Sergeev M. B. Local Maximum Determinant Matrices. Informatsionno-upravliaiushchie sistemy, 2014, no. 1(68), pp. 2-15 (In Russian).
4. Hadamard J. Résolution d'une Question Relative aux Déterminants. Bulletin des Sciences Mathématiques, 1893, vol. 17, pp. 240-246 (In French).
5. Balonin N. A., Seberry Jennifer. A Review and New Symmetric Conference Matrices. Informatsionno-up-ravliaiushchie sistemy, 2014, no. 4(71), pp. 2-7.
6. Wallis (Seberry) Jennifer. Orthogonal (0, 1, -1) Matrices. Proc. of First Australian Conf. on Combinatorial Mathematics, TUNRA, Newcastle, 1972, pp. 61-84.
7. Seberry J., Yamada M. 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, 1992, pp. 431-560.
8. Singer J. A Theorem in Finite Projectie Geometry and Some Applications to Number Theory. Transactions of the American Mathematical Society, 1938, vol. 43, pp. 377-385.
9. Hall Jr. M. A Survey of Difference Sets. Proc. American Mathematical Society, 1956, vol. 7, pp. 975-986.
10. Seberry Jennifer. Regular Hadamard Matrices of Order 36. Available at: http://www.uow.edu.au/ jennie/matri-ces/H36/36R.html (accessed 1 December 2014).
11. Jonathan Jedwab, James Davis. A Survey of Hadamard Difference Sets. Hewlett Packard Technical Reports. Richmond, University of Richmond, 1994. 16 p.
12. McFarland, Robert L. A Family of Difference Sets in Non-cyclic Groups. Journal of Combinatorial Theory. Ser. A15, 1973, no. 1, pp. 1-10.
13. La Jolla Difference Set Repository. Available at: www. ccrwest.org/ds.html (accessed 1 December 2014).
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