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68
ОБРАБОТКА ИНФОРМАЦИИ И УПРАВЛЕНИЕ у/
UDC 004.438
A REVIEW AND NEW SYMMETRIC CONFERENCE MATRICES
N. A. Balonina, Dr. Sc., Tech., Professor, [email protected] Jennifer Seberryb, PhD, Foundation Professor, [email protected] aSaint-Petersburg State University of Aerospace Instrumentation, 67, B. Morskaia St., 190000, Saint-Petersburg, Russian Federation
bCentre for Computer Security Research, EIS, University of Wollongong, NSW, 2522, Australia
Purpose: The paper deals with symmetric conference matrices which were first highlighted by Vitold Belevitch, who showed that such matrices mapped to lossless telephone connections. The goal of this paper is developing a theory of conference matrices using the preliminary research results. Methods: Extreme (by determinant) solutions were obtained by minimization of the maximum of matrix elements absolute values, followed by their subsequent classification. Results: We give the known properties of symmetric conference matrices, known orders and illustrations for some elementary and some interesting cases. We restrict our attention in this note to symmetric conference matrices. We give two symmetric conference matrices of order 46 which are inequivalent to those given by Rudi Mathon and show they lead to two new families of symmetric conference matrices of order 5 x 92t+1 + 1, t > 0 is an integer. Practical relevance: Web addresses are given for other illustrations and other matrices with similar properties. Algorithms of building symmetric conference matrices have been used for developing research software.
Keywords — Conference Matrices, Hadamard Matrices, Weighing Matrices, Symmetric Balanced Incomplete Block Designs (SBIBD), Circulant Difference Sets, Symmetric Difference Sets, Relative Difference Sets, Constructions, Telephony.
AMS Subject Classification: 05B20; 20B20.
Introduction
Symmetric conference matrices are a particularly important class of {0, ±1} matrices. Usually written as C, they are n x n matrices with elements 0, +1 or -1 which satisfy
CTC = CCT = (n-1) In,
where "T" - denotes the matrix transpose and In is the identity matrix of order n. We say that a conference matrix is an orthogonal matrix (after the column-normalization).
In this paper we use - for -1 which corresponds to the usual Hadamard or weighing matrix notation [1-20].
A circulant matrix Cn = (ci}) of order n satisfies
cij = c1, j-i+1 (mod n).
Properties of Symmetric Conference Matrices
We note the following properties of a conference matrix:
— the order of a conference matrix must be = 2 (mod 4);
— n - 1, where n is the order of a conference matrix, must be the sum of two squares;
— if there is a conference matrix of order n then there is a symmetric conference matrix of order n with zero diagonal. The two forms are equivalent as one can be transformed into the other by (i) interchanging rows (columns) or (ii) multiplying rows (columns) by -1;
— a conference matrix is said to be normalized if it has first row and column all plus ones;
— CT = (n - 1) C-1.
Known Conference Matrix Orders
Conference matrices are known [see Appendix] for the following orders:
Key Method Explanation References
cl pr + 1 pr = 1(mod 4) is a prime power [11, 6]
c2 q2(q + 2) + 1 q = 3(mod 4) is a prime power q + 2 is a prime power [10]
c3 46 [10]
c4 5 x 92i+1 + 1 t > 0 is an integer [15]
c5 (n - 1)8 + 1 s > 2 is an integer, n — the order of a conference matrix [17, 14]
c6 (h - 1)28 + 1 s > 1 is an integer, h — the order of a skew-Hadamard matrix [17, 14]
c7 4 circulant matrices with two borders Example below
c8 Certain relative difference sets with two borders [1]
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We now describe the examples of the C46 which differ from that of Mathon. We will observe and use three types of cells:
1) type 0: 0-circulant (zero shift, all rows are equal to each-other);
2) type 1: circulant (circulant shift every new row right);
3) type 2: back-circulant (circulant shift every new row left).
We will say that a matrix has a rich structure, if it consists of several different types of cells. Such notation allows us to describe special matrix structures for different C46.
Rich Structures and Families of Mathon Structure
The Mathon C46 [10] has as its core the usual block-circulant matrix, every block has 9 little 3x3-cells. We write it as
W = circ(A, B, C, CT, BT), where all cells of type 0 are situated inside of C.
The Basic Mathon C46 has cells only of types 0 and 1
Cells (Fig. 1) have
1) type 1: inside of A = circ(a, b, bT);
2) type 1: inside of B = backcirc(c, d, cT);
3) type 0: inside of C = crosscirc(e).
The C = crosscirc(e) consists of m = 3 columns (m — size of e), every column has m = 3 rows — circulant shifted cell of type 0. We will call it a cross-shifted matrix (or cross-matrix, for short).
The new Balonin — Seberry C46 is based on cells of all types 0, 1 and 2 (that is there are richer cells)
The different structures that appear have cells (Fig. 2) with
1) type 1: inside of A = circ(a, b, bT);
2) type 2: inside of B = circ(c, d, d*);
3) type 0: inside of C = crosscirc(e).
■ Fig. 1. Matrices A, B, C of original cell-structure
V
L П
■ Fig. 2. Matrices A, B, C of new cell-structure
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■ Fig. 3. Matrices C46 of original (poor) and new (rich) cell-structures
Now let d = [d4 d2 d3] then cell d* = [d3 d1 d2] is used instead of dT with back-circulant cells.
This the most compact description of Mathon's matrix based on the term: "rich structure" (Fig. 3).
The old structure has 2 types of cells and 3 types of matrices A, B, C. The new structure has 3 types of cells and 2 types of matrices. There is an important structural invariant: the common quantity of types (cells and matrices) is equal to 5.
To show the inequivalence of these C46 we would start by using permutations of size 5 to try to transform the blocks from the second matrix into the form of the first. This is carried out for both the row blocks and column blocks. However, when we look at the resulting structure we see it is not symmetric. To force it to be symmetric we have to reverse the operations we have just carried out. Hence we can not permute one structure into the other. About the inequivalence of rich and poor structures, we can say the following: there are "inequivalence by structure" (ornamental inequivalence) and "inequivalence by permutations". Among Hadamard matrices (for example) there are well known Sylvester and Walsh constructions, they have the first type of difference: ornamental inequivalence.
Easy to use Conference Matrix Forms
When used in real world mechanical applications it may be useful to have them in one of a few main forms: a conference matrix with circulant core, this is cla below, or a conference matrix constructed from two circulant matrices, this is clb below, the latter matrices will not be normalized. The type described as c7, for which we give
an example, but not an infinite class may also be useful.
Key Method Explanation References
cla P + 1 p = 1 (mod 4) is a prime [11, 6]
clb P + 1 p = 1 (mod 4) is a prime [5]
c7 4 circulant matrices with two borders
The conference matrix (actually an OD(13; 4, 9)) found by D. Gregory of Queens University, Kingston, Canada1 given here is of the type c7.
0 1 1 1 1 1 1 1 1 1 1 1 1 1
1 0 1 1 1 - - - 1 1 1 - - -
1 1 0 - - 1 - - 1 1 1 1 -
1 1 - 0 - - 1 - - 1 1 - 1 1
1 1 - - 0 - - 1 1 - 1 1 - 1
1 — 1 - - 0 1 1 - 1 1 1 - -
1 - - 1 - 1 0 1 1 - 1 - 1 -
1 - - - 1 1 1 0 1 1 - - - 1
1 1 1 - 1 - 1 1 0 - - - 1 -
1 1 1 1 1 - 1 - 0 - - - 1
1 1 - 1 1 1 1 - - - 0 1 - -
1 — 1 - 1 1 - - - - 1 0 1 1
1 - 1 1 - 1 - 1 - - 1 0 1
1 - - 1 1 - - 1 - 1 - 1 1 0
1 D. Gregory, private communication, 1973.
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Families of Conference Matrices
Seberry and Whiteman [15] showed how to extend the symmetric conference matrix C46 of Mathon to an infinite familiy of symmetric conference matrices of order 5x 92i+1 + 1, t > 0 is an integer. That paper carefully calculated all the interactions between the basic blocks of the 9x9 original blocks.
Since this calculation is arithmetical and not instructive we do not copy it here. However exactly the same techniques can be used to find new, in-equivalent families, c4bswa and c4bswb from our two new C46. This technique is also similar to that in Seberry [13].
Conference matrices with cores and from two block matrices
We particularly identify conferences matrices, of order n, which are normalized and can be written in one of the two forms: conference matrices with core or conference matrices made from two blocks.
These two forms look like
0
and
A B "
BT —AT
It is not necessary for A or B in either case to be circulant. However, in the form written they must commute. A variation of the second matrix can be used if A and B are amicable.
Then we say we have a conference matrix with circulant core or a conference matrix constructed from two circulant matrices the latter matrices will not be normalized.
Example.
0 1 1 1 1 1 '
1 0 1 — — 1
1 1 0 1 — —
1 — 1 0 1 —
1 — — 1 0 1
1 1 — — 1 0
and
' 0 11 10 1 110 1—1 1—1 1 1—1 1 1—1 1 1—1 1—1
1—1 1—1 1 1—1 1 1—1 1 1—1 1—1 0 — — — 0 — — — 0
In this example the two matrices are in fact equivalent [13, 15].
A Classification to Differentiate between Symmetric Conference Matrices
We classify these by whether they:
1) have a circulant core;
2) are constructed from two circulant blocks;
3) have a core but it is not circulant;
4) are constructed from two blocks but they are not circulant;
5) Mathon's type;
6) from skew Hadamard matrices;
7) are constructed from four blocks with two borders;
8) any other pattern we see;
9) ad hoc.
Useful URLs and Webpages Related to This Study
Some useful url's include:
1) http://mathscinet.ru/catalogue/OD/
2) http://mathscinet.ru/catalogue/artifact22/
3) http://mathscinet.ru/catalogue/conference/ blocks/
4) http://mathscinet.ru/catalogue/belevitch3646/
5) http://www.indiana.edu/~maxdet/
6) http://www.math.ntua.gr/~ckoukouv/
7) http://www.uow.edu.au/~jennie/Hadamard. html/
8) http://tomas.rokicki.com/newrec.html
We also note a very useful package for Latin to Cyrillic conversion: package[utf 8]inputenc
Acknowledgements
The authors would like to acknowledge the great effort of Mr Max Norden BB (Bmgt) (Wollongong), and Mme Tamara Vladimirovna Balonina (Gerasi-mova), who greatly helped with the LaTeX, design, layout and presentation of this paper.
Conclusion and Future Work
Comment. In order to consider other matrices with these kinds of cells we consider the condition n = p2 (q + 2) +1 as this allows many more little cells.
Version n = 9 x 9 + 1 is very well known and class c1 [11, 19]; versions n = 5 x 9 x 9 x 9 + 1 and in general n = 5 x 92t+1 is class c4 [15]: c4bswa and c4bswb, given above, are also this type. Version n = 9 x 9 x 9 x 9 + 1 is very well known and class c1 [11]. To continue to look at the versions mpr + 1 we would next have to consider version n = 13 x 9 x 9 + 1 and so on.
Henceforth we consider the Mathon matrix as oscillations motivated by the Fourier basis. Then the new Balonin-Seberry C46 reflects phases "shift right"-"0-shift"-"shift-left"-"shift-left"-"0-shift".
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References
1. Arasu K. T., Chen Yu. Q., Pott A. Hadamard and Conference Matrices. Journal Algebraic Combinatorics, 2001, no. 14, pp. 103-117.
2. Belevitch V. Conference Networks and Hadamard Matrices. Ann. Soc. Scientifique Bruxelles, 1968, no. 82, pp. 13-32.
3. Belevitch V. Theorem of 2n-terminal Networks with Application to Conference Telephony. Electrical Communication, 1950, no. 26, pp. 231-244.
4. Balonin N. A., Sergeev M. B. Local Maximum Determinant Matrices. Informatsionno-upravliaiushchie sistemy, 2014, no. 1(68), pp. 2-15 (In Russian).
5. Delsarte P., Goethals J.-M., Seidel J. J. Orthogonal Matrices with Zero Diagonal. II. Canadian Journal of Mathematics, 1971, no. 23, pp. 816-832.
6. Geramita A. V., Seberry J. Orthogonal Designs: Quadratic forms and Hadamard matri-ces. New York — Basel, Marcel Dekker, 1979. 460 p.
7. Goethals J.-M., Seidel J. J. Orthogonal Matrices with Zero Diagonal. Canadian Journal of Mathematics, 1967, no. 19, pp. 1001-1010.
8. Horton J., Koukouvinos C., Seberry Jennifer. A Search for Hadamard Matrices Constructed from Williamson Matrices. Bull. Inst. Combin. Appl. 2002, no. 35, pp. 75-88.
9. Koukouvinos C., Seberry J. New Weighing Matrices Constructed Using Two Se-quences with Zero Autocorrelation Function — a Review. Journal Stat. Planning and Inf., 1999, no. 81, pp. 153-182.
10. Mathon R. Symmetric Conference Matrices of Order pq2 + 1. Canadian Journal of Mathematics, 1978, no. 30, pp. 321-331.
11. Paley R. E. A. C. On Orthogonal Matrices. J. Math. Phys, 1933, no. 12, pp. 311-320.
12. Neil J. A. Sloane. Online Encyclopedia of Integer Sequences ®, OEIS ®. Available at: http://oeis.org (accessed 5 Juny 2014).
13. Seberry J. New Families of Amicable Hadamard Matrices. J Statistical Theory and Practice, in memory of Jagdish N Srivastava, 2013, iss. 4, no. 7, pp. 650-657.
14. Seberry J. W. Combinatorial Matrices, PhD Thesis, La Trobe University, 1971.
15. Seberry J., Whiteman A. L. New Hadamard Matrices and Conference Matrices Obtained via Mathon's Construction. Graphs Combin., 1988, no. 4, pp. 355-377.
16. Turyn R. J. An Infinite Class of Williamson Matrices. Journal Combin. Theory. Ser. A, 1972, no. 12, pp. 391-321.
17. Turyn R. J. On C-matrices of Arbitrary Powers. Bull. Canad. Math. Soc., 1971, no. 23, pp. 531-535.
18. Van Lint J. H., Seidel J. J. Equilateral Point Sets in Elliptic Geometry. Indagationes Mathematicae, 1966, no. 28, pp. 335-348.
19. Scarpis U. Sui Determinanti di Valore Massimo. Ren-diconti della R. Istituto Lombardo di Scienze e Let-tere, 1898, 31, pp. 1441-1446 (In Italian).
20. Wallis W. D., Street A. P., Seberry J. W. Combinatorics: Room Squares, Sum-free Sets, Hadamard Matrices. Lecture Notes in Mathematics. Vol. 292. Berlin-Heidelberg-New York, Springer-Verlag, 1972. 508 p.
■ Known Conference Matrix Orders Less than 1000
Appendix
Order Exist? Type Order Exist? Type Order Exist? Type Order Exist? Type
6 V c1,c1a 254 NE 506 ? 758 V c1,c1a
10 V c1a, c6 258 V c1,c1a 510 V c1,c1a 762 V c1,c1a
14 V c1,c1a 262 ? 514 NE 766 ?
18 V c1,c1a 266 ? 518 NE 770 V c1,c1a
22 NE 270 V c1,c1a 522 V c1,c1a 774 V c1,c1a
26 V c1 274 NE 526 NE 778 NE
30 V c1,c1a 278 V c1,c1a 530 V c1, c6 782 NE
34 NE 282 V c1,c1a 534 ? 786 ?
38 V c1,c1a 286 NE 538 NE 790 NE
42 V c1,c1a 290 V c1 542 V c1,c1a 794 ?
46 V c2, c3,c4 294 V c1,c1a 546 ? 798 V c1,c1a
50 V c1, c6 298 NE 550 ? 802 ?
54 V c1,c1a 302 NE 554 NE 806 NE
58 NE 306 ? 558 V c1,c1a 810 V c1,c1a
62 V c1,c1a 310 NE 562 NE 814 NE
66 ? 314 V c1,c1a 566 ? 818 NE
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Order Exist? Type Order Exist? Type Order Exist? Type Order Exist? Type
70 NE 318 V c1,c1a 570 V c1,c1a 822 V c1,c1a
74 V c1,c1a 322 NE 574 NE 826 NE
78 NE 326 ? 578 V c1,c1a 830 V c1,c1a
82 V c1, c6 330 NE 582 NE 834 ?
86 ? 334 ? 586 ? 838 NE
90 V c1,c1a 338 V c1,c1a 590 NE 842 V c1
94 NE 346 NE 594 V c1,c1a 846 ?
98 V c1,c1a 350 V c1,c1a 598 NE 850 NE
102 V c1,c1a 354 V c1,c1a 602 V c1,c1a 854 V c1,c1a
106 NE 358 NE 606 ? 858 V c1,c1a
110 V c1,c1a 362 V c1, c6 610 NE 862 NE
114 V c1,c1a 366 ? 614 V c1,c1a 866 ?
118 ? 370 ? 618 V c1,c1a 870 NE
122 V c1, c6 374 V c1,c1a 622 NE 874 ?
126 V c1 378 ? 626 V c1 878 V c1,c1a
130 NE 382 NE 630 ? 882 V c1,c1a
134 NE 386 NE 634 NE 886 NE
138 V c1,c1a 390 V c1,c1a 638 ? 890 NE
142 NE 394 NE 642 V c1,c1a 894 NE
146 ? 398 V c1,c1a 646 NE 898 NE
150 V c1,c1a 402 V c1,c1a 650 NE 902 ?
154 ? 406 ? 654 V c1,c1a 906 ?
158 V c1,c1a 410 V c1,c1a 658 ? 910 ?
162 NE 414 NE 662 V c1,c1a 914 NE
166 NE 418 NE 666 NE 918 NE
170 V c1 422 V c1,c1a 670 NE 922 NE
174 V c1,c1a 426 ? 674 V c1,c1a 926 ?
178 NE 430 NE 682 NE 930 V c1,c1a
182 V c1,c1a 434 V c1,c1a 686 ? 934 NE
186 ? 438 NE 690 ? 938 V c1,c1a
190 NE 442 V c2 694 NE 942 V c1,c1a
194 V c1,c1a 446 ? 698 ? 946 NE
198 V c1,c1a 450 V c1,c1a 702 V c1,c1a 950 ?
202 NE 454 NE 706 NE 954 V c1,c1a
206 ? 458 V c1,c1a 710 V c1,c1a 958 NE
210 NE 462 V c1,c1a 714 NE 962 V c1, c6
214 NE 466 NE 718 NE 966 ?
218 NE 470 NE 722 NE 970 NE
222 ? 474 NE 726 ? 974 NE
226 ? 478 ? 730 V c1, c6 978 V c1,c1a
230 V c1,c1a 482 ? 734 V c1,c1a 982 ?
234 V c1,c1a 486 ? 738 NE 986 ?
238 NE 490 NE 742 NE 990 NE
242 V c1,c1a 494 ? 746 ? 994 NE
246 ? 498 NE 750 NE 998 V c1,c1a
250 NE 502 NE 754 NE 1002 ?
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