Cretan (4t + 1) matrices Текст научной статьи по специальности «Математика»

ТЕОРЕТИЧЕСКАЯ И ПРИКЛАДНАЯ МАТЕМАТИКА /

UDC 004.438

doi:10.15217/issn1684-8853.2016.4.2

CRETAN (4t + 1) MATRICES

N. A. Balonina, Dr. Sc., Tech., Professor, [email protected] Jennifer Seberryb, PhD, Emeritus Professor, [email protected] aSaint-Petersburg State University of Aerospace Instrumentation, 67, B. Morskaia St., 190000, Saint-Petersburg, Russian Federation

bDepartment of Computing and Information Techology, University of Wollongong, NSW 2522, Australia

Purpose: We tried to obtain a Cretan(4t +1) matrix of order 4t +1, i.e. an orthogonal matrix whose elements have moduli <1. The only Cretan(4t+1) matrices previously published were of orders 5, 9,13, 17 and 37. Results: In the paper, we give an infinite number of new Cretan(4t+1) matrices constructed by the use of regular Hadamard matrices, SBIBD(4t+1; k; A), weighing matrices, generalized Hadamard matrices and Kronecker product. We introduce an inequality for the matrix radius and give a construction for a Cretan matrix of any order n > 5. Practical relevance: Cretan(4t +1) matrices have direct practical applications to the problems of noise-immune coding, compression and masking of video information.

Keywords — Hadamard Matrices, Regular Hadamard Matrices, OrthogonalMatrices, Symmetric Balanced Incomplete Block Designs (SBIBD), Cretan Matrices,Weighing Matrices, Generalized Hadamard Matrices, 05B20.

Introduction

An application in image processing (compression, masking) led to the search for orthogonal matrices, all of whose elements have modulus <1 and which have maximal or high determinant.

Cretan matrices were first discussed, per se, during a conference in Crete in 2014. This paper follows closely the joint work of N. A. Balonin, Jennifer Seberry and M. B. Sergeev [1-3].

The orders 4t (Hadamard), 4t - 1 (Mersenne), 4t - 2 (Weighing) are discussed in [4-6]. This present work emphasizes the 4t + 1 (Fermat type) orders with real elements <1. Cretan matrices which are complex, based on the roots of unity or are just required to have at least one 1 are mentioned.

Preliminary Definitions

The absolute value of the determinant of any matrix is not altered by 1) interchanging any two rows, 2) interchanging any two columns, and/or 3) multiplying any row/or column by -1. These equivalence operations are called Hadamard equivalence operations. So the absolute value of the determinant of any matrix is not altered by the use of Hadamard equivalence operation.

Write In for the identity matrix of order n, J for the matrix of all 1's and let ra be a constant. An orthogonal matrix, S, of order n, is square, has real entries and satisfies SST = raIn. The core of a matrix is formed by removing the first row and column.

A Cretan matrix, S, of order n has entries with modulus <1 and at least one 1 per row and column. It satisfies SST = raIn and so it is an orthogonal matrix. A Cretan(n; x; ra) matrix, or CM(n; x; ra) has x levels or values for its entries [1].

An Hadamard matrix of order n has entries ±1 and satisfies HHT = nIn for n = 1, 2, 4t, t > 0 an integer. Any Hadamard matrix can be put into normalized form, that is having the first row and column all plus 1s using Hadamard equivalence operations: that is it can be written with a core. A regular Hadamard matrix of order 4m2 has 2m2 ± m elements 1 and 2m2 + m elements -1 in each row and column (see [7, 8]).

Hadamard matrices and weighing matrices are well known orthogonal matrices. We refer to [2, 7-10] for more details and other definitions. The reader is pointed to [11-13] for details of generalized Hadamard matrices, Butson — Hadamard matrices and generalized weighing matrices.

For the purposes of this paper we will consider an SBIBD(v, k, X), B, to be a vxv matrix, with entries 0 and 1, k ones per row and column, and the inner product of distinct pairs of rows and/or columns X. This is called the incidence matrix of the SBIBD. For these matrices X(v - 1) = k(k - 1),

v-1

BBT = (k - X)I + XJ and detB = k(k -XpT.

For every SBIBD(v, k, X) there is a complementary SBIBD(v, v - k, v - 2k + X). One can be made from the other by interchanging the 0's of one with the 1's of the other. The usual SBIBD convention that v > 2k and k > 2X is followed.

We now define our important concepts the orthogonality equation, the radius equation(s), the characteristic equation(s) and the weight of our matrices.

Definition 1 (orthogonality equation, radius equation(s), characteristic equation(s), weight). Consider the matrix S = (si;) of order n comprising the variables x1, x2, ..., xx.

The matrix orthogonality equation

STS = SST = raIn (1)

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yields two types of equations: the n equations which arise from taking the inner product of each row/column with itself (which leads to the diagonal elements of raIn being ra) are called radius equation(s), g(xp x2, ..., xx) = ra, and the n2 - n equations, /(x1, x2, ..., xx) = 0, which arise from taking inner products of distinct rows of S (which leads to the zero off diagonal elements of raIn are called characteristic equation(s)). Cretan matrices must satisfy the three equations: the orthogonality equation (1), the radius equation and the characteristic equation(s).

Notation: We use CM(n; x; ra; det(optional); (t1, t2, ..., tx)), or just CM(n; x; ra), where t1, t2, ..., tx are the possible values (or levels) of the elements in CM.

Inequalities

Some inequalities are known for matrices which have real entries <1. Hadamard matrices, H = (hi}), which are orthogonal and with entries ±1 satisfy the equality of Hadamard's inequality (2) [9]

det (HHT )<n

(2)

i=1j=1

have determinant < n2 . Further Barba [14] showed that for matrices, B, of order n whose entries

are ±1:

n-1

det B <V2n -1 (n -1) 2

v ' n

or asymptotically « 0.858(n)2 .

For n = 9 Barba's inequality gives Vl7x84 = = 16 888.24. The Hadamard inequality gives 19 683 for the bound on the determinant of the ±1 matrix of order 9. So the Barba bound is better for odd orders. We thank Professor Christos Koukouvinos for pointing out to us that the literature, see Ehlich and Zeller, [15], yields a ±1 matrix of order 9 with determinant 14 336. These bounds have not been met for n = 9.

Koukouvinos also pointed out that in Raghavarao [16] a ±1 matrix of order 13 with determinant 14 929 920 « 1.49 x 107 is given. This is the same value given for n = 13 given by Barba's inequality. The Hadamard inequality gives 1.74 x 107 for the bound on the determinant of the ±1 matrix of order 13.

These bounds have been significantly improved

(n-l)

by Brent and Osborn [17] to give < (n +1) 2 .

Wojtas [18] showed that for matrices, B, whose entries are ±1, of order n = 2 (mod 4) we have

n-2

det B < 2(n - l))n - 2) 2 or asymptotically « 0.736 (n)

This gives a determinant bound <73 728 for order 10 whereas the weighing matrix of order 10 has determinant 95 = 59 049.

We observe that the determinant of a CM(n; x;

n

ra; det) is always ra 2.

Hence we can rewrite the known inequalities of this subsection noting that only the Hadamard in equality applies generally for elements with modulus <1. Thus we have:

Theorem 1. Hadamard — Cretan Inequality. The radius of a Cretan matrix of order n is <n.

Two Trivial Cretan(n) Families

The next two families are included for completeness.

The Basic Family

Lemma 1. Consider C=aI + b(J - I) of order n, a, b

( 4(n -1)) i; 2; 1 -1

variables. This gives a CM

(n - 2)2

matrix

of order n, i.e. a CM

4(n -1) ( -2 n; 2; 1 -2; det; I 1,--

(n - 2)2 I n - 2

Proof. Writing C with a on the diagonal and other elements b, the radius and characteristic equations become

a2 + (n - 1)b2 = ra and 2a + (n - 2)b = 0.

Hence with a = 1 and b =

-2 n - 2

we have

ra = 1 +

for the required CM(n) matrix.

4(n -1)

1 + ~-2

(n - 2)2

Remark 1. For n = 7, 9, 11, 13 this gives

24 32 40 48

ra=1—, 1—,1 and 1 respectively. These

25 49 81 121

determinants are very small. However they do give a CM(n; 2) for all integers n > 0.

Known Families

The following results may be found in [19] and [6]. Proposition 1. [Cretan(4t)]. There is a Cretan(4t; 2; 4t) for every integer 4t for which there exists an Hadamard matrix.

Proposition 2. [Cretan(4t - 1)]. There are Cretan(4t - 1; 2; ra), ra = 4t + 1 -Jt and

2t3 +1 - 2t (2t - 1))t ra = for every integer 4t for

(t -1)2

which there exists an Hadamard matrix.

The next two results are easy for the knowled-gable reader and merely mentioned here.

Proposition 3. [Cretan(4t - 2)]. There are Cretan(41 - 2; 3; k) whenever there is a W(41 - 2, k)

2

n

y TEOPETMHECKÄS M nPMKAÄAHÄl MATEMÄTMKÄ

weighing matrix. For k = 4t - 3, the sum of two squares, and a W(4t - 2,4t - 3) is known, the complex Cretan matrix CM(41 - 2; 3; 4t - 2) has elements i = V—1,1 or -1.

Proposition 4. [Cretan(np)]. There are complex Cretan(np; p; n), when ever there exists a generalized Hadamard matrix based on the p th roots of unity.

The Additive Families

We will illustrate this construction using two Cretan matrices to give a Cretan matrix whose order is the sum of their orders. This shows how many possible matrices we might find for any n but again all the determinants are small.

Lemma 2. Let A and B be CM(n1; 3; ra1) and CM(n2; 3; ra2) respectively. Then A © B given by

A 0"

0 B

is a CM(n1 + n2; 4; ra) matrix of order n1 + n2 with ra = min(ra1, ra2). (Note it does not have one 1 per row and column.)

Remark 2. We note using smaller CM(ni; x; rai) gives many inequivalent CM(n; x; ra) for any order n = ^ ,n,i but the elements of all but the smallest sub matrix will not contribute 1 to the resulting Cretan matrix.

Now with n = n1 + n2 for 21 = 4 + 17, 5 + 16, 6 + 15, 7 + 14, 8 + 13, 9 + 12, 10 + 11 plus other combinations, the sub matrices of orders n1 and n2 contribute differently to x and ra. This means

Proposition 5. There is a Cretan(n; x; ra) for every integer n.

In the section on Kronecker product of Cretan matrices we explore the same Proposition 5 for more interesting x.

Constructions for Cretan(4t + 1; x) Matrices

We now describe a number of constructions for Cretan(4t + 1) matrices.

Constructions using SBIBD

• 2-level Cretan(4t + 1) matrices via SBIBD(v = 4t + 1, k, X)

The following Theorem is a special case of the construction for 2-level Cretan(v = 4t + 1) given in [6]. It also yields a valid CM(37; 2).

Theorem 2 [6]. Let S be a CM(v = 4t + 1; 2; ra; (a, b)) based on SBIBD(v = 4t + 1, k, X) then a = 1,

(k-X)+V k-X

b = -

v - 2k + k

and ra = ka2 + (v - k)b2, provi-

ded |b| < 1.

Example 1. Using the La Jolla Repository http:// www.ccrwest.org/ds.html of difference sets we obtain an SBIBD(37, 9, 2). Using Theorem 2 we obtain CM(37; 2; 12.325; (1, 0.345)) and CM(37; 2; 9.485;

(1, 0.132)). The complementary SBIBD(37, 28, 21) does not give any Cretan matrix as |b|is >1.

We especially note the (45, 12, 3) difference set,

where the occurrence of the Cretan I 45; 2; 20—

( 1 ï ^ 4 matrix and the Cretan I 45; 2; 14— I matrices both

I 16 )

arise from the SBIBD(45, 12, 3): the complementary SBIBD(45, 33, 24) does not yield any Cretan matrix.

Example 2. Orthogonal matrices of orders 13 and 21 may be constructed by using the SBIBD(13, 4, 1) and SBIBD(21, 5, 1) ( ( 3 + /3 ^

13; 2; 9; 60; 1

given in [20]. CM

1,-

and

)

CM ^21; 2; 10; --JJ are given in Fig. 1, a, b.

All the examples of SBIBD(4t + 1, k, X) that we have given from the La Jolla Repository have been constructed using difference sets. Most of those we give arise from Singer difference sets and finite geometries: these SBIBD((pn+1 - 1)/(p - 1), (pn - 1)/(p - 1), (pn-1 - 1)/(p - 1)) difference sets are denoted as PG(n, p). The bi-quadratic type constructions are due to Marshall Hall [21]. There are many SBIBD constructed without using difference sets.

• Bordered Constructions

We do not elaborate on the next theorem here but note it gives many Cretan matrices CM(v + 1).

Theorem 3. The matrix C below can be used to construct many CM(v + 1; x; ra) with borders by replacing the matrix B by an SBIBD(v, k, X).

When a matrix C is written in the following form

C=

x

B

B is said to be the core of C and the s's are the borders of B in C. C is said to be in bordered form. The variables s and x can be realized in the cases described below.

■ Fig. 1. 2-level Cretan matrices of order 13 and 21: a — CM(13; 2; 9.60); b — CM(21; 2; 10)

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• Using Regular Hadamard Matrices

For details and constructions many of the known Regular Hadamard Matrices the interested reader is referred to [8, 7, 22].

Lemma 3. Let M be a regular Hadamard matrix of order 4m2 with 2m2 + m positive elements per row and column. Then forming C as follows

C =

1 s

s •••

2m

M

gives

1) matrix or

Cretan(4 n2 + 1; 4

CM( 4m2 +1; 4; 1; |0,1,-^, 7=1 ^ ^ 2m 2m

Proof. For C to be a Cretan matrix it must satisfy the orthogonality, radius and characteristic equations. These are

CCT =(l + 4m2s2)l, 2 1=(s2 + 4m2)l, 2 n =raL 2 , V I 4m2+1 V I 4m2+1 4m2+1

44 = 22 , ..., there is a Cretan = |

for the orthogonality equation, giving s = 0, ra = 1 for the radius equation and 0 for the characteristic equations.

Hence we have a matrix of order 4m2 + 1 with

elements 0, 1, + satisfying the required Cretan 2m

equations.

Corollary 1. Since there exists a regular (symmetric) Hadamard matrix of order 4 = 22,42 = 22 ,

= (222 ... +1;4; 1) forn a Fermat number.

Proof. Let S be the regular symmetric Hadamard matrix of order 4. Then the Kronecker product

S x S x ... x S

is the required core for the construction in Lemma 3.

Example 3. Purported examples of pure Fermat matrices in Fig. 2, a, b for orders 5 and 17: levels a, b are white and black colours, the border level s is given in grey. However the reader is cautioned that though the figures appear to be Cretan matrices

b)

V\

V > ■■■■

■ Fig. 2. Orthogonal Cretan(Fermat) matrices for Fermat numbers 5 (a) and 17 (b)

■ Fig. 3. Regular Hadamard matrix of order 36 (a) and a 3-level Cretan(37) (b)

they are not. They are based on SBIBD, including the regular Hadamard matrix SBIBD(4m2, 2m ± m, m ± m) and require c = a. We note though that when c = a ^ 1 the radius and characteristic equations do not give meaningful real solutions.

Example 4. See Fig. 3, a, b for examples of a regular Hadamard matrix of order 36 and a purported new Balonin — Seberry type of 3-level Cretan(37) with complex entries that is a orthogonal matrix of order 37. A real Cretan(37; 2) does exist from Theorem 2 above (see example).

Using Normalized Weighing Matrix Cores

The next construction is not valid in the real numbers. However we can allow Cretan matrices to have complex elements and choose the diagonal to be i =>/-1.

Lemma 4. Suppose there exists a normalized conference matrix, B, of order 4t + 2, that is a W(4t + 2, 4t + 1). Then B may be written as

i 1 - r 1 ...... ;

i ••• F i

1 ...... :

B =

This is a Cretan matrix.

Removing the first row and column of B to study the core F is unproductive.

Generalized Hadamard Matrices and Generalized Weighing Matrices

We first note that the matrices we study here have elements from groups, abelian and non-abeli-an, (see [11-13, 23, 24] for more information) and may be written in additive or multiplicative notation. The matrices may have real elements, elements {1, -1}, elements |n| < 1, elements {1, i, i2 = -1}, elements {1, i, -1, -i, i2 = -1}, integer elements {a + ib, i2 = -1}, n-th roots of unity, the quaternions {1 and i, j, k, i2 = j2 = k2 = -1, ijk = -1}, (a + ib) + j(c + id), a, b, c, d, integer and i, j, k quaternions or otherwise as specified.

We use the notations BT for the transpose of G, BH for the group transpose, BC for the complex conjugate of BT, BQ for the quaternion conjugate and BV for the quaternion conjugate transpose.

In all of these matrices the inner product of distinct rows a and b is a - b or ab-1 depending on whether the group is written in additive or multiplicative form.

• Generalized orthogonality: A generalized Hadamard matrix, or difference matrix, GH(gn, g), of order h = gn, over a group of order g, has the inner product of distinct rows the whole group the same number of n times. The inner product is {agl1 >• • • > gihgjh} • Forexample

G =

1111 1 a b ab 1 b ab a 1 ab a b

; GGH = (group)I4 = (Z2 x Z2)I

orthogonality is because of the definition of the inner product.

• Butson Hadamard matrix [11]

"1 1 1 "

B = 1 ra ra2 ; BBC = 3I3, ra3 = 1, 1 + ra + ra2 = 0

1 ra2 ra

is said to be a Butson Hadamard matrix. Orthogonality depends on the fact that the n nth roots of unity add to zero.

• A generalized Hadamard matrix [11, 12, 13], GH(np, G), where G is a group of order p,

can also be written in additive form for example:

0 0 0 0 0 0

0 0 0 1 0 2 0 2 0 1

is a GH(6, Z3).

* 0 0 0 0

0 * 1 2 0

0 1 * 0 2

0 2 0 * 1

0 0 2 1 *

• A generalized weighing matrix, W = GW(np, G, k) [23], where G is a group of order p, has w nonzero elements in each column and W is orthogonal over G. The following two matrices are additive and multiplicative GW(5, Z3), respectively:

"0 1

1 0

1 ra 1 ra2 1

* is zero but not the zero of the group.

Theorem 4. Any generalized Hadamard matrix or generalized weighing matrix is a CM(n; g) over the group G, of order g, which may be the roots of unity.

The Kronecker Product of Cretan Matrices

Lemma 5. Suppose A and B are CM(n1; x1; ra1) and CM(n2; x2; ra2) then the Kronecker product of A and B written AxB is a CM(n1n2; x; ra1ra2) where x depends on x1 and x2.

1 1 1

ra ra2 1

0 1 ra2

1 0 ra

ra2 ra 0

■ Table 1. Some Cretan CM(4t + 1), 3 < 4t + 1 < 199

From Regular Hadamard Matrices (ra = 1) 5 17 37 65 101 145 197

From Difference Sets (ds)

v k X Existence Difference set Comment

13 4 1 All known PG(2, 3) Unique Hall [28]

21 5 1 All known PG(2, 4) Unique Hall [28]

37 9 2 Exists Biquadratic residue ds Hall [28]

45 12 3 All known — La Jolla [20]

57 8 1 All known PG(2, 7) Unique Hall [28]

73 9 1 All known PG(2, 8) Unique Hall [28]

85 21 5 Exists PG(3, 4) [20]

101 25 6 Exists Biquadratic residue ds Hall [28]

109 28 7 Exists Biquadratic residue ds Hall [28]

121 40 13 Exists PG(4, 3) [20]

133 33 8 Exists — La Jolla [20]

197 49 12 Exists Biquadratic residue ds Hall [28]

Kronecker Product All orders which are the product a known order and of prime power = 3 (mod 4)

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Example 5. From [6, 25] we see that CM(3; 2; 2.25), CM(7; 2; 5.03) and CM(7; 2; 3.34) exist so there exist CM(21; 3; 11.32) and CM(21; 3; 7.52). The Hadamard — Cretan bound gives, for n = 21, radius <21.

From Balonin and Seberry [6] we have that since

(

an SBIBD

pr -1 pr - 3

exists for all prime

powers pr = 3 (mod 4) there exist CM(pr; 2; ra) for all these prime powers (see Proposition 2). Hence using Kronecker products in the previous theorem and writing n as a product of prime powers we have.

Theorem 5. There exists a CM(n; x; ra) ra > 1 for all odd orders n, n = npx pl1 p2..., where p is an order for which a Cretan CM(p = 4t + 1) is known and pl pi2,... are any prime powers =3 (mod 4), for some x and ra.

Table 1 gives the integers for which p is presently known. Similar theorems can be obtained for all even n.

Remark 3. We note that x depends on the actual

(a, b))

(a2,

12

construction used. Combining CM(n1; 2; and CM(n2; 2; ra1 : (a, b)) gives CM(n1n2; 3 ab, b2)). General formulae for x from CM with different levels are left as an exercise.

The Difference between Cretan(4t + 1; x) Matrices and Fermat Matrices

The first few pure Fermat numbers are v = 3, 5, 17, 257, 65 537, 4 294 967 297,... . We note these are all =1 (mod 4) and may be constructed using Corollary 1. Fig. 4 gives an early example of a Fermat matrix.

Finding 3-level orthogonal matrices of order =1 (mod 4) for non-pure Fermat numbers has proved challenging. Orders n = 9 and n = 13 are given in [4].

Orders v = 2even + 1 called Fermat type matrices, pose an interesting class to study.

■ Fig. 4. Core of Russian Fermat Matrix from mathscinet.ru

Orders 4t + 1, t is odd, are Cretan(41 + 1) — matrices; their order is neither a Fermat number (2 + 1 = 3, 22 + 1 = 4 + 1, 222 +1 = 16 +1,

222

22 +1 = 256 +1, ...) nor a Fermat type number

(2even + 1). Examples of regular Hadamard matrices of order 36, giving the first CM(37; 3; 1) matrix of order 37 [3] where 37 is not a Fermat number or Fermat type number, have been placed at [26]. They use regular Hadamard matrices as a core and have the same, as any other Hadamard matrix, level functions. We call them Cretan(4t + 1) matrices and will consider them further in our future work.

Matrices of the Cretan(4t + 1) family made from Singer difference sets (see [21]) also have orders belonging to the set of numbers 4t + 1, t odd: these are different from the three-level matrices of Balonin — Sergeev (Fermat) family [27, 19] with orders 4t + 1, t is 1 or even.

Summary

In this paper we have given new constructions for CM(4t + 1). These are summarised in Table 1 for 4t + 1 < 200. Table 2 gives 2-level and 3-level CM(4t ± 1).

■ Table 2. Cretan 2-level and 3-level CM(4t ± 1), 3 < 4t + 1 < 199

v Method v Method v Method

3 BM [4] + Prop. 2 5 BM [4] 7 BM [4] + Prop. 2

9 BM [4] 11 BM [4] + Prop. 2 13 BM [4]

15 Kronecker 17 — 19 Prop. 2

21 From SBIBD Table 1 23 Prop. 2 25 Kronecker

27 Prop. 2 29 — 31 Prop. 2

33 Kronecker 35 Kronecker 37 —

39 Kronecker 41 — 43 Prop. 2

45 From SBIBD Table 1 47 Prop. 2 49 Kronecker

51 — 53 — 55 Kronecker

57 From SBIBD Table 1 59 Prop. 2 61 —

У ТЕОРЕТИЧЕСКАЯ И ПРИКЛАДНАЯ МАТЕМАТИКА "7

■ Finish of table 2

v Method v Method v Method

63 Kronecker 65 Kronecker 67 Prop. 2

69 Kronecker 71 Prop. 2 73 From SBIBD Table 1

75 Kronecker 77 Kronecker 79 Prop. 2

81 Prop. 2 83 — 85 From SBIBD Table 1

87 — 89 — 91 Kronecker

93 Kronecker 95 Kronecker 97 —

99 Kronecker 101 From SBIBD Table 1 103 Prop. 2

105 Kronecker 107 Prop. 2 109 From SBIBD Table 1

111 — 113 — 115 Kronecker

117 Kronecker 119 — 121 From SBIBD Table 1

123 — 125 Kronecker 127 Prop. 2

129 Kronecker 131 Prop. 2 133 From SBIBD Table 1

135 Kronecker 137 — 139 Prop. 2

141 Kronecker 143 — 145 —

147 Kronecker 149 — 151 Prop. 2

153 — 155 Kronecker 157 —

159 — 161 Kronecker 163 Prop. 2

165 Kronecker 167 Prop. 2 169 Kronecker

171 Prop. 2 173 — 175 Kronecker

177 Kronecker 179 Prop. 2 181 —

183 — 185 — 187 —

189 Kronecker 191 Prop. 2 193 —

195 Prop. 2 197 From SBIBD Table 1 199 Prop. 2

Conclusions

Cretan matrices are a very new area of study. They have many research lines open: what is the minimum number of variables that can be used; what are the determinants and radii that can be found for Cretan(n; x) matrices; why do the congruence classes of the orders make such a difference to the proliferation of Cretan matrices for a given order; find the Cretan matrix with maximum and minimum determinant for a given order; can one be found with fewer levels?

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2. Balonin N. A., Seberry J. A Review and New Symmetric Conference Matrices. Informatsionno-upravli-aiushchie sistemy [Information and Control Systems], 2014, no. 4 (71), pp. 2-7.

Acknowledgements

We thank Professors Richard Brent, Christos Koukouvinos and Ilias Kotsireas for their valuable input to this paper. The authors also wish to sincerely thank Mr Max Norden, BBMgt(C.S.U.), for his work preparing the layout and LaTeX version of this article, and Mrs Tamara Balonin for her work preparing the text of the 'word' version. We acknowledge http://www.wolframalpha. com for the number calculations in this paper and http://www.mathscinet.ru for the graphics.

3. Balonin N. A., Seberry J., Sergeev M. B. Three-Level Cretan Matrices of Order 37. Informatsionno-upravliaiush-chie sistemy [Information and Control Systems], 2015, no. 2(75), pp. 2-3. doi: 10.15217/issn1684-8853.2015.2.2

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^ ТЕОРЕТИЧЕСКАЯ И ПРИКЛАДНАЯ МАТЕМАТИКА "V

Control Systems], 2012, no. 6(61), pp. 90-93 (In Russian).

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УДК 004.438

doi:10.15217/issn1684-8853.2016.4.2 Критские матрицы порядков 4t + 1

Н. А. Балонина, доктор техн. наук, профессор Дженнифер Себерри6, PhD, профессор

^анкт-Петербургский государственный университет аэрокосмического приборостроения, Санкт-Петербург, РФ ^Университет Вуллонгонг, Вуллонгонг, Новый Южный Уэльс, Австралия

Цель: дать критские матрицы Cretan(4t + 1) порядков 4t + 1 — ортогональные матрицы с элементами, ограниченными по модулю <1 (ранее публиковались критские матрицы типа Cretan(4t + 1) определенных порядков 5, 9, 13, 17 и 37). Результаты: приведено неограниченно много новых критских матриц Cretan(4t + 1), конструируемых при помощи регулярных матриц Адама-ра, симметричного сбалансированного блочного дизайна SBIBD(4t + 1; к; X), взвешенных матриц, обобщенных матриц Адамара и произведения Кронекера. Предложено неравенство для радиуса матриц и дана конструкция критской матрицы для каждого порядка n>5. Практическая значимость: критские матрицы Cretan(4t + 1) имеют непосредственное практическое применение к проблемам помехоустойчивого кодирования, сжатия и маскирования видеоинформации.

Ключевые слова — матрицы Адамара, регулярные матрицы Адамара, ортогональные матрицы, симметричный сбалансированный блочный дизайн (SBIBD), критские матрицы, взвешенные матрицы, обобщенные матрицы Адамара, 05B20.