Regular Hadamard matrices constructed from Hadamard 2-designs and conference graphs

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Regular Hadamard matrices constructed from Hadamard 2-designs and conference graphs Dean Crnković Department of Mathematics, University of Rijeka, Radmile Matejčić 2, 51000 Rijeka, Croatia

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Article history: Received 30 July 2016 Accepted 17 September 2017 Available online xxxx Keywords: Regular Hadamard matrix Hadamard 2-design Conference graph

a b s t r a c t 1 m −3 Suppose there exists a Hadamard 2-(m, m− , 4 ) design having skew incidence matrix. If 2 there exists a conference graph on 2m − 1 vertices, then there exists a regular Hadamard matrix of order 4m2 . A conference graph on 2m + 3 vertices yields a regular Hadamard matrix of order 4(m + 1)2 . © 2017 Elsevier B.V. All rights reserved.

1. Introduction A Hadamard matrix of order m is an (m × m)-matrix H = (hi,j ), hi,j ∈ {−1, 1}, satisfying HH T = H T H = mIm , where Im is the identity matrix of order m. The order of a Hadamard matrix must be 1, 2, or a multiple of 4. The Hadamard conjecture proposes that a Hadamard matrix of order 4k exists for every positive integer k. A Hadamard matrix is regular if the row and column sums are constant. A regular Hadamard matrix is necessarily of order 4k2 with constant row sum 2k. It was conjectured that a regular Hadamard matrix of order 4k2 exists for every positive integer k. Constructions of regular Hadamard matrices coming from the theory of Menon–Hadamard difference sets are given in [1,5]. Important results about the existence of regular Hadamard matrices are given by T. Xia, M. Xia and J. Seberry in [12,13]. In [13] they proved the following statement: When k = 47, 71, 151, 167, 199, 263, 359, 439, 599, 631, 727, 919, 5q1 , 5q2 N, 7q3 , where q1 , q2 and q3 are prime powers such that q1 ≡ 1 (mod 4), q2 ≡ 5 (mod 8) and q3 ≡ 3 (mod 8), N = 2a 3b t 2 , a, b=0 or 1, t ̸ = 0 is an arbitrary integer, then there exist regular Hadamard matrices of order 4k2 . The existence of some regular Hadamard matrices of order 4p2 , when p is a prime, p ≡ 7 (mod 16), is established in [8], and the following theorem is proved in [3]. Theorem 1. Let p and 2p − 1 be prime powers and p ≡ 3 (mod 4). Then there exists a regular Hadamard matrix of order 4p2 . The construction employed to prove Theorem 1 uses Paley graphs and Paley designs. In this paper, we generalize this construction using conference graphs and Hadamard designs having skew incidence matrix. In a similar way, we generalize the construction of regular Hadamard matrices given in [4]. 2. Hadamard 2-designs A 2-(v, k, λ) design is a finite incidence structure (P , B, I), where P and B are disjoint sets and I ⊆ P × B, with the following properties: E-mail address: [email protected]. https://doi.org/10.1016/j.disc.2017.09.020 0012-365X/© 2017 Elsevier B.V. All rights reserved.

Please cite this article in press as: D. Crnković, Regular Hadamard matrices constructed from Hadamard 2-designs and conference graphs, Discrete Mathematics (2017), https://doi.org/10.1016/j.disc.2017.09.020.

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1. |P | = v ; 2. every element of B is incident with exactly k elements of P ; 3. every pair of distinct elements of P is incident with exactly λ elements of B. The elements of the set P are called points and the elements of the set B are called blocks. If |P | = |B| = v and 2 ≤ k ≤ v − 2, then a 2-(v, k, λ) design is called a symmetric design. Hadamard matrices of order 4k can be used to create symmetric designs with parameters (4k − 1, 2k − 1, k − 1) or (4k − 1, 2k, k), which are called Hadamard 2-designs (see [2,11]). The construction is reversible, so that symmetric designs with these parameters can be used to construct Hadamard matrices. It is also well known that the existence of a symmetric design with parameters (4u2 , 2u2 − u, u2 − u) or (4u2 , 2u2 + u, u2 + u) is equivalent to the existence of a regular Hadamard matrix of order 4u2 (see [11, Theorem 1.4]). Such symmetric designs are called Menon designs. A matrix A is skew-symmetric if AT = −A. A Hadamard matrix H of order 4k is skew-type if H = A + I4k , where AT = −A. A (0, 1)-matrix D is skew if D + DT = J − I, where J the all-one matrix. A skew-type Hadamard matrix corresponds to a Hadamard 2-design with skew incidence matrix, and vice versa. Suppose D is the incidence matrix of a symmetric design. D + I is again the incidence matrix of a symmetric design if and 1 m−3 , 4 ) design. Further, let D = (dij ) only if D is skew (see [9]). Let D be a skew incidence matrix of a Hadamard 2-(m, m− 2 m+1 m+1 be an incidence matrix of a complementary Hadamard design with parameters (m, 2 , 4 ). Then D + Im and D − Im are 1 m+1 1 m−3 incidence matrices of symmetric designs with parameters (m, m+ , 4 ) and (m, m− , 4 ), respectively. 2 2 For v ∈ N we denote by jv the all-one vector of dimension v , by 0v the zero-vector of dimension v , by Jv the all-one matrix of dimension (v × v ), and by 0v×v the zero-matrix of dimension (v × v ). Further, for a, b ∈ N we denote by Ja×b the all-one matrix of dimension (a × b). 1 m−3 , 4 ) design and let D = (dij ) be an incidence matrix of Lemma 1. Let D be a skew incidence matrix of a Hadamard 2-(m, m− 2 m+1 m+1 a complementary Hadamard (m, 2 , 4 ) design. The matrices D and D have the following properties: T

D · D = (D − Im )(D + Im )T =

m+1

[ D | D − Im ] · [ D − Im | D ] T = [ D | D ] · [ D + Im | D − Im ]T = [ D | D ] · [ D − Im | D − Im ]T =

4

Jm −

m−1 2 m−1 2 m−1 2

Proof. Each block of a symmetric (m,

m+1

Jm −

4

Im ,

m−1 2

Im ,

Jm , Jm .

3 1 , m− ) design meets the complement of any other block in m+ points. Hence, 4 4 m+1 m+1 m+1 m+1 D · D = 4 Jm − 4 Im . Further, D + Im is the incidence matrix of a symmetric (m, 2 , 4 ) design and D + Im = D − Im , 1 1 so (D − Im )(D + Im )T = m+ Jm − m+ Im . 4 4 T

m−1 2

The other equalities follow from the properties listed below:

Di · (D − Im )Tj =

⎧ 0, ⎪ ⎪ ⎪ ⎪ ⎨m + 1

if i = j,

− 1, if dij = 1, 4 ⎪ ⎪ ⎪ ⎪ ⎩m + 1, if dij = 0, i ̸ = j, 4

(D − Im )i · DTj =

⎧ 0, ⎪ ⎪ ⎪ ⎪ ⎨m + 1

if i = j,

− 1, if dji = 1, 4 ⎪ ⎪ ⎪ ⎪ ⎩m + 1, if dji = 0, i ̸ = j, 4

⎧ m−1 ⎪ ⎪ , if i = j, ⎪ ⎪ 2 ⎪ ⎪ ⎨ m−3 Di · (D + Im )Tj = + 1, if dij = 1, ⎪ 4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩m − 3, if dij = 0, i ̸ = j, 4

Please cite this article in press as: D. Crnković, Regular Hadamard matrices constructed from Hadamard 2-designs and conference graphs, Discrete Mathematics (2017), https://doi.org/10.1016/j.disc.2017.09.020.

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⎧ m−1 ⎪ ⎪ , if i = j, ⎪ ⎪ 2 ⎪ ⎪ ⎨ m+1 Di · (D − Im )Tj = − 1, if dij = 1, i ̸= j, ⎪ 4 ⎪ ⎪ ⎪ ⎪m + 1 ⎪ ⎩ , if dij = 0, 4

where Xi denotes the ith row of a matrix X .

□

3. Conference graphs A graph is regular if all the vertices have the same degree. A regular graph is strongly regular of type (v, k, λ, µ) if it has v vertices, degree k, and if any two adjacent vertices are together adjacent to λ vertices, while any two non-adjacent vertices are together adjacent to µ vertices. A conference matrix of order n is a (n × n) (0, ±1)-matrix W satisfying WW T = (n − 1)In . If W is a conference matrix of order n, then either n ≡ 0 (mod 4) and W is equivalent to a skew-symmetric matrix, or W is equivalent to a symmetric matrix (see [6]). If W is a symmetric conference matrix of order n, then n ≡ 2 (mod 4) and n − 1 is the sum of two squares. W is a skew-symmetric conference matrix if and only if H = W + In is a skew-type Hadamard matrix. The property of being a conference matrix is unchanged under changing the sign of any row or column. Let W be a symmetric conference matrix of order n. We can assume that all entries in the first row and column (apart from their intersection) are +1. Let S be the matrix obtained from W by deleting the first row and column. Further, let C be obtained from S by replacing +1 by 0 and −1 by 1. Then C is the adjacency matrix of a strongly regular graph with parameters 1 v−5 v−1 (v, v− , 4 , 4 ), where v = n − 1. Strongly regular graphs with parameters (v, v−2 1 , v−4 5 , v−4 1 ) are called conference graphs. 2 Let C be the adjacency matrix of a conference graph with v vertices, and C = Jv − C . Then v ≡ 1 (mod 4) and v is the sum of two squares. Let Ci and Cj , i ̸ = j, be ith and jth rows of the matrix C , respectively. Then

Ci · CjT =

⎧v − 1 ⎪ , ⎨ 4

if cij = cji = 0,

⎪ ⎩ v − 1 − 1, if c = c = 1. ij ji 4

The matrix C − Iv has the same property. Let C i and C j , i ̸ = j, be ith and jth rows of the matrix C = (c ij ), respectively. Then T

Ci · Cj =

⎧v − 1 ⎪ , ⎨ 4

if c ij = c ji = 0,

⎪ ⎩ v − 1 + 1, if c = c = 1. ij ji 4

The matrix C + Iv has the same property. Lemma 2. Let C be the adjacency matrix of a conference graph with parameters (v, following properties hold: C · (C + Iv )T = C · (C − Iv )T =

C · (C − Iv )T = T

(C + Iv ) · C =

v−1 4

v+3 4

Jv −

Jv −

v−1 4

v−1 4

v−1 4

[ C | C + Iv ] · [ C | C + Iv ]T = [ C | C − Iv ] · [ C | C − Iv ]T = [ C | C + Iv ] · [ C | C − Iv ]T =

Jv +

v−1 4

v−1 2

, v−4 5 , v−4 1 ) and C = Jv − C . Then the

Iv ,

Iv , Iv ,

v−1 2

v−1 2

v+1 2

Jv +

Jv +

Jv −

v+1 2

v+1 2

v+1 2

Iv , Iv , Iv .

Please cite this article in press as: D. Crnković, Regular Hadamard matrices constructed from Hadamard 2-designs and conference graphs, Discrete Mathematics (2017), https://doi.org/10.1016/j.disc.2017.09.020.

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Proof. The equalities follow from the properties of the matrices C , C + Iv , C , and C − Iv listed above, and the following two properties:

T

Ci · C j =

⎧ 0, ⎪ ⎪ ⎪ ⎪ ⎨v − 1

if i = j,

, if cij = cji = 0, i ̸ = j, 4 ⎪ ⎪ ⎪ ⎪ ⎩ v − 1 + 1, if cij = cji = 1, 4

(C + Iv )i · (C − Iv )Tj =

⎧ 0, ⎪ ⎪ ⎪ ⎪ ⎨v − 1

if i = j,

, if cij = cji = 1, 4 ⎪ ⎪ ⎪ ⎪ ⎩ v − 1 + 1, if cij = cji = 0, i ̸= j.

□

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4. Construction of regular Hadamard matrices Let H = (hij ) and K be m × n and m1 × n1 matrices, respectively. Their Kronecker product is a mm1 × nn1 matrix. h12 K h22 K

h1n K h2n K ⎥

.

.. .

... ...

hm1 K

hm2 K

...

hmn K

h11 K ⎢h21 K

⎡

H ⊗K =⎢ ⎣..

⎤

.. .

⎥. ⎦

1 m−3 Theorem 2. Let there exist a Hadamard 2-(m, m− , 4 ) design with skew incidence matrix, and a conference graph with v 2 vertices, where v = 2m − 1. Then there exists a regular Hadamard matrix of order 4m2 . 1 m−3 , 4 ) design, and let C be the adjacency matrix of Proof. Let D be the skew incidence matrix of a Hadamard 2-(m, m− 2 a conference graph with v = 2m − 1 vertices. We will show that then there exists a Menon design with parameters (4m2 , 2m2 − m, m2 − m). Let D and C be defined as above. Define a (4m2 × 4m2 ) matrix M in the following way:

⎡

0 0v

⎢ ⎢ ⎢ ⎢ j ⎢ m·v M=⎢ ⎢ ⎢ ⎢ ⎣ 0

m·v

0Tv 0v×v C ⊗ jm

(C + Iv ) ⊗ jm

jTm·v (C − Iv ) ⊗ jTm (C + Iv ) ⊗ D + C ⊗ (D − Im ) C ⊗ (D + Im ) + (C − Iv ) ⊗ (D − Im )

0Tm·v C ⊗ jTm C ⊗D + (C − Iv ) ⊗ D (C + Iv ) ⊗ (D − Im ) + C ⊗D

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

It is easy to see that M · J4m2 = (2m2 − m)J4m2 . Using properties of the matrices D, D, C and C one can verify that M · M T = (m2 − m)J4m2 + m2 I4m2 , which means that M is an incidence matrix of a symmetric design with parameters (4m2 , 2m2 − m, m2 − m). □ The following theorem generalizes the construction given in [4]. 1 m−3 Theorem 3. Let there exist a Hadamard 2-(m, m− , 4 ) design with skew incidence matrix, and a conference graph with v 2 vertices, where v = 2m + 3. Then there exists a regular Hadamard matrix of order 4(m + 1)2 . 1 m−3 Proof. Let D be the skew incidence matrix of a Hadamard 2-(m, m− , 4 ) design, and let C be the adjacency matrix of a 2 conference graph with v = 2m + 3 vertices. Further, let us define the matrices D, C and M as in the proof of Theorem 2, and let M1 = M + I4(m+1)2 . Using similar calculations as in the proof of Theorem 2, one concludes that M1 is the incidence matrix of a symmetric (4(m + 1)2 , 2m2 + 3m + 1, m2 + m) design. □

In [10] Szekeres proved that a regular Hadamard matrix of order n2 exists whenever there is a Hadamard matrix of order n. Thus, Theorem 3 does not yield new parameters, but gives another class of regular Hadamard matrices. The construction from Theorems 2 and 3 fails for other values of v = 2m − 1 + 4n, n a nonnegative integer. For example, for v = 2m + 7 the number of rows (and columns) of the matrix M is 4(m2 + 4m + 2), and m2 + 4m + 2 is not a square for any integer m. Please cite this article in press as: D. Crnković, Regular Hadamard matrices constructed from Hadamard 2-designs and conference graphs, Discrete Mathematics (2017), https://doi.org/10.1016/j.disc.2017.09.020.

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5. Conclusion In [13] the authors give a list of 126 values of k < 1000 for which the existence of a regular Hadamard matrix of order 4k2 is undetermined. The paper [3] solves six of these 126 cases, namely the cases k = 79, 271, 279, 367, 607 or 967, and [8] solves the case k = 311. The cases solved in this paper by Theorem 2 are k = 231, 255, 399, 639, 651, 775, 799, 987 (for the existence of the corresponding skew-type Hadamard matrices we refer the reader to [7], the corresponding conference graphs are Paley graphs). Thus, based on the facts given in [8,13] and this paper, we get to the conclusion that there are 111 values of k < 1000 for which the existence of a regular Hadamard matrix of order 4k2 is still undetermined. These values are as follows: 103, 127, 141, 191, 209, 213, 217, 223, 237, 239, 253, 309, 329, 341, 355, 357, 369, 377, 381, 383, 385, 395, 403, 423, 425, 431, 437, 453, 455, 463, 465, 473, 479, 481, 483, 487, 493, 497, 501, 503, 515, 517, 527, 553, 561, 573, 589, 595, 597, 611, 615, 627, 629, 635, 647, 649, 657, 663, 665, 669, 689, 693, 697, 705, 711, 713, 715, 717, 719, 721, 737, 743, 751, 755, 759, 765, 781, 789, 793, 801, 805, 813, 817, 823, 833, 835, 837, 839, 861, 863, 869, 873, 887, 889, 893, 899, 901, 903, 911, 913, 923, 927, 933, 935, 949, 955, 969, 983, 989, 991, 995. It was conjectured by J. Seberry that a skew-type Hadamard matrix exists if and only if n = 1,2, or 4k, where k is a positive integer. In case that this conjecture holds true, Theorem 2 would imply the existence of a regular Hadamard matrix of order 4k2 for k = 715, or k= 835. For these values of k the existence of a skew-type Hadamard matrix of order k + 1 is still undetermined (see [7]). Acknowledgment This work has been fully supported by Croatian Science Foundation under the project 1637. References [1] T. Beth, D. Jungnickel, H. Lenz, Design Theory, second ed, Cambridge University Press, Cambridge, 1999. [2] R. Craigen, H. Kharaghani, in: C.J. Colbourn, J.H. Dinitz (Eds.), Hadamard Matrices and Hadamard Designs, second ed., in: Handbook of Combinatorial Designs, Chapman & Hall/CRC Press, Boca Raton, 2007, pp. 273–280. [3] D. Crnković, A series of regular Hadamard matrices, Des. Codes Cryptogr. 39 (2006) 247–251. [4] D. Crnković, A series of Menon designs and 1-rotational designs, Finite Fields Appl. 13 (2007) 1001–1005. [5] J.A. Davis, J. Jedwab, A unifying construction for difference sets, J. Combin. Theory Ser. A 80 (1997) 13–78. [6] Y.J. Ionin, H. Kharaghani, in: C.J. Colbourn, J.H. Dinitz (Eds.), Balanced Generalized Weighing Matrices and Conference Matrices, second ed, in: Handbook of Combinatorial Designs, Chapman & Hall/CRC Press, Boca Raton, 2007, pp. 306–313. [7] C. Koukouvinos, Summary results for the existences of Hadamard matrices, 2016. http://www.math.ntua.gr/~ckoukouv/ (Accessed 01.07.16). [8] K.H. Leung, S.L. Ma, B. Schmidt, New Hadamard matrices of order 4p2 obtained from Jacobi sums of order 16, J. Combin. Theory Ser. A 113 (2006) 822–838. [9] P. Ó Catháin, Nesting symmetric designs, Irish Math. Soc. Bull. 72 (2013) 71–74. [10] G. Szekeres, A new class of symmetric block designs, J. Combin. Theory 6 (1969) 219–221. [11] W.D. Wallis, A.P. Street, J.S. Wallis, Combinatorics: Room Squares, Sum-Free Sets, Hadamard Matrices, Springer-Verlag, Berlin-Heidelberg-New York, 1972. [12] T. Xia, M. Xia, J. Seberry, Regular Hadamard matrices, maximum excess and SBIBD, Australas. J. Combin. 27 (2003) 263–275. [13] T. Xia, M. Xia, J. Seberry, Some new results of regular Hadamard matrices and SBIBD.II., Australas. J. Combin. 37 (2007) 117–125.

Please cite this article in press as: D. Crnković, Regular Hadamard matrices constructed from Hadamard 2-designs and conference graphs, Discrete Mathematics (2017), https://doi.org/10.1016/j.disc.2017.09.020.