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ТЕОРЕТИЧЕСКАЯ И ПРИКЛАДНАЯ МАТЕМАТИКА у
UDC 004.438
doi:10.31799/1684-8853-2022-2-2-10
Some new symmetric Hadamard matrices
D. Z. Dokovica, Dr. Sc., Tech., Distinguished Professor Emeritus, orcid.org/0000-0002-0176-2395,
djokovic@uwaterloo.ca
aUniversity of Waterloo, Department of Pure Mathematics and Institute for Quantum Computing, Waterloo,
Ontario, N2L 3G1, Canada
Introduction: It is conjectured that the symmetric Hadamard matrices of order 4v exist for all odd integers v > 0. In recent years, their existence has been proven for many new orders by using a special method known as the propus construction. This construction uses difference families Xk (k = 1, 2, 3, 4) over the cyclic group Zv (integers mod v) with parameters (v; kp k2, k3, k4; A) where X1 is symmetric, X2 = X3, and k1 + 2k2 + k4= v + A. It is also conjectured that such difference families (known as propus families) exist for all parameter sets mentioned above excluding the case when all the k are equal. This new conjecture has been verified for all odd v < 53. Purpose: To construct many new symmetric Hadamard matrices by using the propus construction and to provide further support for the above-mentioned conjecture. Results: The first examples of symmetric Hadamard matrices of orders 4v are presented for v = 127 and v = 191. The systematic computer search for symmetric Hadamard matrices based on the propus construction has been extended to cover the cases v=55, 57, 59, 61, 63. Practical relevance: Hadamard matrices are used extensively in the problems of error-free coding, and compression and masking of video information.
Keywords — symmetric Hadamard matrices, propus construction, propus difference families.
Articles
For citation: Dokovic D. Z. Some new symmetric Hadamard matrices. Informatsionno-upravliaiushchie sistemy [Information and Control Systems], 2022, no. 2, pp. 2-10. doi:10.31799/1684-8853-2022-2-2-10
Introduction
We fix some notation which will be used throughout this note. Let Xi, i = 1, 2, 3, 4, be a difference family (DF) in a finite abelian group G (written ad-ditively) and let
(v; k1, k2, k3, k4; X)
be its parameter set (PS). Thus v = |G|, |Xi| = ki and Ski(ki - 1) = X(v - 1), where |X| denotes the cardinality of a finite set X. If 1kt = X + v we say that this PS is a Goethals — Seidel parameter set (GSPS) and that this DF is a Goethals — Seidel difference family (GSDF). If the Xt form a GSDF and we replace one of the blocks by its set-theoretic complement in G, we again obtain a GSDF although the parameter set may change. For that reason we shall always assume that all the ki < v/2.
Each GSDF in G gives a Hadamard matrix H of order 4v. For more details about this construction see e.g. [1, 2]. Briefly, each Xi provides a G-invariant matrix Ai of order v, and H is obtained by plugging the Ai into the well known Goethals — Seidel array
GSA =
We recall that a matrix A = (ax y) with indices x, ye G is G-invariant if ax+z y+z = axy for all x,
Ai a2r A3R a4 r
- A2 R Ai - ra4 RA3
- A3R ra4 Ai - RA2
-a4 r -RA3 RA2 Ai
y, z e G. The matrix R = (rx y) may be defined by the formula rxy = 8x+y 0, x, y e G, where 8 is the Kronecker symbol.
For a subset X of G, we say that it is symmetric if -X = X, and we say that it is skew if G is a disjoint union of X, -X and {0}. If at least one of the blocks Xi of a GSDF is skew then, after rearranging the Xi so to have X1 skew, the Hadamard matrix H will be skew Hadamard, i.e. such that H + HT = 2I4v (T denotes the matrix transposition, and Ik is the identity matrix of order k).
In order to obtain a symmetric Hadamard matrix H we require that two of the blocks Xi are the same and that one of the other two blocks is symmetric. A propus parameter set (PPS) is a GSPS having ki = kj for some i ^ j. By permuting the ki's we may assume that k2 = k3 and k1 > k4. In that case we say that this PPS is normalized. Note that these conditions in general do not specify the ki's uniquely. For instance the PPSs (5; 1, 2, 2, 1; 1) and (5; 2, 1, 1, 2; 1) are both normalized but they become the same if we ignore the ordering of the ki's.
We say that a GSDF is a propus difference family (PDF) if Xi = Xj for some i ^ j and one of the other two blocks is symmetric. If the Xi's form a PDF then, after rearranging the blocks we may assume that X2 = X3 and that X1 is symmetric. Then we plugg the corresponding matrix blocks Ai into the so called propus array (PA) to construct a symmetric Hadamard matrix of order 4v. This construction is known as the propus construction. It has been first introduced in [3]. For the reader's convenience we display the propus array
2 У ИНФОРМАЦИОННО-УПРАВЛЯЮЩИЕ СИСТЕМЫ
X № 2, 2022
PA =
-Ai A2R A3 R a4 r
A3R ra4 Ai -RA2
A2R Ai -ra4 RA3
a4 r -RA3 RA2 Ai
Note that PA is obtained from GSA by multiplying the first column by -1 and interchanging the second and the third rows.
From now on we assume that G = Zv, a cyclic group of order v, and that v is odd. Under this assumption, the matrix blocks Ai will be circulants. All PPSs for v < 41 are listed in [4] together with the corresponding PDF's. There was only one case of a PPS having no PDF, namely (25; 10, 10, 10, 10; 15). Similarly, the cases 41 < v <51 were handled in [5], and the case v = 53 in [6]. Again there was one exceptional case, (49; 21, 21, 21, 21; 35). In the present note, for each PPS with 51 < v < 63 we exhibit at least one PDF. For more information on the exceptional cases see [5].
Symmetric Hadamard matrices of orders 508 and 764
The symmetry symbol (abc) written immediately after a PPS shows the symmetry types of the three blocks Xv X2 and X4. More precisely, the letter s means that we require that the corresponding block be symmetric, the letter k is used if we require that block to be skew, and the symbol * is used otherwise. In particular a = s means that we require X1 to be symmetric, a = k means that we require X1 to be skew, and a = * means that no symmetry condition is imposed on X1.
The group of units Zv acts on Zv by multiplication. It may happen that there is a nontrivial subgroup H of Zv such that some block Xt of a PDF is a union of orbits of H. In such case we may specify Xt by writing it as HYt, where Yt is a set of representatives of the H-orbits contained in Xv
For v = 127 we give five nonequivalent PDFs and for v = 191 only one.
v = 127, H = {1, 19, 107}
(127; 57, 61, 61, 55; 107) (**s)
= H{4, 5, 6, 9, 12, 15, 23, 24, 30, 33, 36, 39, 45, 52, 58, 59, 60, 64, 66} X2 = H{0, 4, 5, 6, 13, 15, 17, 26, 30, 32, 40, 46, 51, 53, 58, 59, 60, 64, 65, 66, 72} X2 = H{0, 2, 4, 8, 9, 12, 15, 23, 24, 26, 30, 33, 40, 46, 51, 52, 53, 65, 71}
(127; 60, 60, 60, 54; 107) (s**)
= H{1, 5, 6, 11, 13, 15, 16, 17, 20, 23, 24, 29, 32, 45, 46, 52, 58, 66, 71, 72} X2 = H{2, 4, 5, 11, 12, 15, 16, 18, 22, 23, 29, 33, 36, 39, 46, 51, 52, 53, 60, 71} X2 = H{6, 8, 17, 20, 22, 23, 30, 33, 36, 39, 45, 51, 58, 59, 60, 64, 66, 71}
(127; 60, 60, 60, 54; 107) (**s)
= H{1, 2, 3, 4, 5, 6, 9, 11, 12, 13, 15, 17, 23, 24, 32, 33, 39, 46, 64, 65} X2 = H{2, 5, 9, 10, 12, 13, 15, 16, 17, 29, 33, 36, 39, 40, 45, 51, 53, 58, 60, 66} X2 = H{1, 5, 6, 10, 11, 13, 16, 17, 20, 23, 30, 32, 45, 58, 64, 65, 66, 71}
(127; 58, 60, 60, 55; 106) (**s)
= H{0, 2, 3, 4, 5, 12, 13, 16, 17, 18, 20, 22, 29, 30, 46, 51, 53, 58, 59, 71} X2 = H{8, 9, 10, 16, 20, 22, 23, 24, 26, 29, 32, 36, 45, 46, 51, 52, 59, 60, 65, 78} X2 = H{0, 3, 5, 10, 11, 17, 18, 22, 24, 29, 32, 39, 45, 52, 58, 59, 60, 64, 72}
(127; 60, 57, 57, 58; 105) (s**)
= H{2, 4, 6, 10, 12, 13, 15, 17, 23, 24, 26, 36, 40, 46, 51, 52, 58, 64, 71, 78} X2 = H{1, 2, 3, 4, 10, 16, 17, 18, 20, 23, 29, 30, 45, 51, 52, 58, 64, 66, 72} X2 = H{0, 2, 5, 8, 9, 10, 11, 13, 15, 17, 18, 30, 39, 40, 46, 53, 58, 60, 66, 78}
v = 191, H = {1, 39, 49, 109, 184}
(191; 91, 90, 90, 85; 165) (s**)
= H{0, 1, 3, 4, 7, 9, 16, 17, 18, 21, 22, 28, 31, 36, 57, 61, 62, 68, 112}
№ 2, 2022 ^
MHfflOPMAIiMOHHO-ynPABAfiroWME CMCTEMbl 3
X2 = H{1, 4, 14, 16, 18, 19, 22, 23, 28, 29, 31, 32, 34, 36, 38, 61, 62, 68} X4 = H{1, 2, 9, 11, 12, 17, 18, 22, 28, 29, 31, 32, 38, 41, 56, 61, 66}
Small orders of symmetric Hadamard matrices
There are several known infinite series of PDFs [3, 7]. We shall use only two of them. The first one is essentially the Turyn series [8] with v = (q + 1)/2, q a prime power =1 (mod 4), and all four blocks Xi symmetric. The second one is essentially the series constructed in [9] (see also [7]) to which we refer as the XXSW-series. In this case v = (q + 1)/4, q a prime power =3 (mod 8), and we may arrange the blocks so that X1 is skew, X2 = X3 and X4 is symmetric.
In the handbook [10] published in 2007 it is indicated (see Table 1.52, p. 277) that, for odd v < 200, no symmetric Hadamard matrices of order 4v are known for
v = 23, 29, 39, 43, 47, 59, 65, 67, 73, 81, 89, 93, 101, 103, 107, 109, 113, 119, 127, 133, 149, 151, 153, 163, 167, 179, 183, 189, 191, 193
The cases v = 23 and v = 81 should not have been included. For the case v = 23 see [7]. For v = 81 note that symmetric Hadamard matrices of orders 4 ■ 9k, k > 1 integer, were constructed by Turyn [11] back in 1984. Moreover, the Bush-type Hadamard matrix of order 4 ■ 81 = 324 constructed in 2001 [12] is also symmetric.
The propus construction has been used in several recent papers [3, 4-7, 13] to construct symmetric Hadamard matrices of new orders. By taking into account these results and those from the previous section, the above list of undecided cases reduces to
v = 65, 89, 93, 101, 107, 119, 133, 149, 153, 163, 167, 179, 183, 189, 193.
List of PPSs and PDFs for odd v, 53 < v < 63
The following conventions and notation will be used in the listings below. We have Zv = {0, 1, 2, ..., v - 1} and recall that v is odd. Let X ç Zv and k = |X|. Define X' = Xn{1, 2, ..., (v - 1)/2}. In particular ZV={1, 2,..., (v-1)/2}.
If X is skew then k = (v - 1)/2 and
x = x' u (-(zv \ x ')).
If X is symmetric then
fx' U (-X'), for k even; = [{0} U X' U (-X'), for k odd.
Hence, a skew X can be recovered uniquely from X'. This is also true for symmetric X provided we know the parity of k.
For a PDF Xi, i = 1, 2, 3, 4, with normalized PPS (v; k1, k2, k3, k4; X) we always assume that X2 = X3. Thus it suffices to specify only the blocks X1, X2 and X4. We say that a PPS is exceptional if all the ki are equal. The following conjecture is implicit in [4-6]. It has been verified there for odd v < 53.
Conjecture 1. For each normalized and non-exceptional PPS (v; k1, k2, k3, k4; X) there exist PDFs with symmetry symbols (s**) and (**s).
The list below shows that the conjecture is true also for v = 55, 57, 59, 61, 63.
If a block Xiis symmetric or skew, in order to save space we record only X-. As the ki are specified by the PPS, Xi can be recovered uniquely from X[.
Example 1. For the first PDF below, the symmetry symbol (s**) shows that X1 must be symmetric. As Xi = {5,6,7, 9,10,13,15,16,19, 21, 23, 25, 27} we have' - Xi = {28, 30, 32, 34, 36, 39, 40, 42, 45, 46,48, 49, 50}. As k1 = 27 is odd we have X1 = {0} U Xi U (-Xi).
v = 55
(55; 27, 25, 25, 21; 43) (s**)
Xi = {5,6,7, 9,10,13,15,16,19, 21, 23, 25, 27}
4 y MHOOPMAIiMOHHO-ynPABAfiroWME CMCTEMbl
X № 2, 2022
X2 = {О, 1, 2, В, 4, 6, 9, 1О, 14, 17, 19, 24, 26, 29, ВО, В4, В7, В8, В9, 4О, 41, 47, 48, б2, бВ} X2 = {О, В, 4, б, 1О, 11, 12, 14, 16, 17, 18, 19, 21, 22, 24, ВО, В6, 4В, 44, 46, 47}
(бб; 27, 2б, 2б, 21; 4В) (**s)
= {О, 4, б, 6, 7, 9, 1О, 12, 1б, 2О, 21, 24, 2б, 26, 28, В2, ВВ, В4, В8, В9, 41, 44, 4б, б1, б2, бВ, б4} X2 = {О, В , б, 7, 8, 9, 1О, 19, 2В, 2б, 28, В1, В2, ВВ, В4, В6, В7, 4О, 41, 4В, 44, 4б, 47, 48, бВ} X4 = {2,4, б, g, 12,13, is, ig, 20,23}
(бб; 27, 24, 24, 22; 42) (s**)
Xi = {1, 4, б, 7, g, 11,14,15,1S, 20, 23, 24, 27}
X2 = {О, 4, 6, 7, 12, 14, 1б, 16, 18, 2B, 2б, 26, ВО, Bl, ВВ, В4, Вб, В6, В7, 4О, 41, 46, 47, бВ} X2 = {О, 1, 4, 6, 16, 17, 18, 19, 2О, 21, 22, 2В, 24, 27, 29, В1, ВВ, В6, 42, 4В, 44, 46}
(бб; 27, 24, 24, 22; 42) (**s)
= {О, 1, б, 8, 1О, 12, 1б, 16, 17, 24, 2б, 26, 29, ВО, В4, В7, В9, 4О, 41, 42, 44 ,47, 48, бО, б2, бВ, б4} X2 = {О, 1О, 1В, 14, 1б, 16, 17, 2О, 21, 22, 24, 26, ВО, В1, ВВ, В4, Вб, 41, 4В, 44, 47, 49, бО, бВ} X4 = {1, б, 11,12,13,1б, ig, 20, 22, 24, 27}
(бб; 26, 2В, 2В, 24; 41) (s**)
Xi = {1, 2, б, S, 10,11,13,14,15, ig, 21, 23, 27}
X2 = {О, 2, В, 4, б, 6, 7, 1О, 11, 14, 18, 24, 2б, ВО, Вб, В7, В9, 4О, 41, 42, бО, б1, б2} X2 = {О, 2, б, 6, 8, 14, 16, 19, 2О, 21, 22, 24, 2б, 28, Bl, В2, ВВ, В7, В8, 4О, 4б, 47, 49, б2}
(бб; 26, 2В, 2В, 24; 41) (**s)
= {О, 2, 7, 11, 14, 1б, 17, 18, 2В, 24, 2б, 28, 29, ВО, В1, В6, В7, В8, В9, 42, 44, 4б, 46, 48, бО, бВ} X2 = {О, В, 8, 12, 1В, 14, 18, 19, 2В, 26, ВВ, В4, В6, В8, 42, 4б, 46, 47, 48, 49, бО, б1, б2} X4 = {1, 2, 7,10,11,13,17, ig, 21, 24, 2б, 27}
(бб; 24, 27, 27, 21; 44) (s**)
Xi = {б, S, 10,13,1б, 1б, 1S, ig, 20, 22, 2б, 2б}
X2 = {О, 2, 4, б, 8, 1В, 14, 16, 17, 19, 2б, 26, 27, В2, ВВ, В4, В7, В8, В9, 4О, 41, 42, 44, 49, бО, бВ, б4} X2 = {О, 4, б, 8, 11, 12, 1В, 16, 18, 2О, 22, 24, В1, ВВ, В4, В6, В7, 41, 42, 44, б1}
(бб; 24, 27, 27, 21; 44) (**s)
= {О, В, 11, 12, 1В, 14, 16, 17, 2В, 24, 26, 27, 29, ВО, Вб, В8, В9, 4О, 41, 42, 47, 48, 49, бО} X2 = {О, 1, В, 4, б, 6, 8, 9, 14, 16, 18, 21, 22, 27, 28, В2, Вб, В6, В7, В9, 4В, 47, 49, б1, б2, бВ, б4} X4 = {3, б, g, 10,14,1б, 17, 20, 23, 2б}
(бб; 24, 2б, 2б, 22; 41) (s**)
Xi = {1, 3, б, б, 7,11,14,1б, 1б, 1S, 21, 2б}
X2 = {О, 9, 11, 12, 14, 17, 2О, 22, 2В, 24, 2б, 26, 27, ВО, В1, ВВ, В7, В8, 42, 46, 47, 48, 49, б2, б4} X2 = {О, 4, б, 7, 8, 11, 12, 16, 18, 19, 21, 24, 2б, 26, 28, В4, Вб, В9, 41, 4В, бВ, б4}
(бб; 24, 2б, 2б, 22; 41) (**s)
= {О, 2, В, б, 1В, 14, 16, 17, 21, 22, 2В, 26, 29, В2, В7, В8, 42, 4В, 44, 4б, 46, 48, 49, б1} X2 = {О, 1, 2, В, 4, 1О, 11, 1б, 17, 18, 19, 21, 22, 24, 2б, 27, 28, ВО, В2, В4, Вб, В9, 4О, 44, 46} X4 = {2, 3, б, 7,10,11,1б, 21, 23, 2б, 2б}
(бб; 2В, 26, 26, 22; 42) (sss), Turyn series Xi = {б, 7,10,11,1б, 17,1S, ig, 21, 24, 2б} X2 = {i, 2, 4, S, 14,1б, 17,1S, ig, 23, 24, 2б, 27} X4 = {б, 7,10,11,1б, 17,1S, ig, 21, 24, 2б}
v = б7, H = {1, 7, 49}
(б7; 28, 28, 28, 21; 48) (s**)
Xi = {1, 2, б, S, 10,11,14,1б, 17,1S, 21, 22, 2б, 27}
X2 = {О, 2, В, б, 6, 7, 11, 12, 1B, 1б, 18, 2B, 24, 2б, 26, 27, 28, 29, ВО, B2, B4, B9, 4О, 42, 46, 49, бО, бб} X2 = {О, 1, 2, В, б, 6, 8, 9, 12, 1В, 14, 1б, 19, 2В, ВО, В1, В2, В9, 4б, 48, б1}
№ 2, 2022 ^
ИНФаРМ^ Б
(57; 28, 28, 28, 21; 48) (k*s), XXSW series Xi = {2,4,12,13,15, 21, 23,24, 25, 27, 28}
X2 = {1, 3, 5, 8, 9, 12, 15, 20, 23, 24, 26, 27, 29, 31, 32, 33, 35, 36, 37, 41, 42, 45, 49, 50, 51, 52, 53, 55} X4 = {1,4, 6,13,14,15,19, 20, 21, 26}
(57; 27, 26, 26, 22; 44) (s**), all Xi are H-invariant Xi = {3, 6, 9,10,11,13,15,19, 20, 21, 23, 24,26}
X2 = {4, 6, 8, 9, 15, 16, 19, 23, 25, 28, 30, 31, 37, 38, 39, 42, 43, 44, 45, 46, 47, 48, 50, 51, 55, 56} X2 = {2, 4, 6, 9, 10, 11, 13, 14, 20, 22, 25, 26, 28, 30, 34, 38, 39, 40, 41, 42, 45, 52}
(57; 27, 26, 26, 22; 44) (**s)
= {0, 8, 9, 10, 12, 13, 15, 18, 19, 20, 25, 30, 33, 34, 37, 40, 41, 43, 47, 48, 49, 51, 52, 53, 54, 55, 56} X2 = {0, 4, 8, 9, 11, 16, 17, 18, 22, 24, 25, 26, 27, 28, 30, 31, 33, 34, 37, 38, 42, 43, 51, 53, 54, 55} X4 = {2, 4, 7,9,11,12,14,17, 21, 23, 28}
(57; 27, 25, 25, 23; 43) (s**), all Xi are H-invariant Xi = {3, 6, 9,10,11,13,15,19, 20, 21, 23, 24,26}
X2 = {2, 3, 4, 14, 16, 21, 22, 24, 25, 28, 29, 30, 32, 33, 36, 38, 39, 40, 41, 43, 45, 52, 53, 54, 55} X2 = {1, 2, 5, 7, 10, 13, 14, 17, 19, 24, 29, 30, 32, 34, 35, 36, 38, 39, 41, 45, 49, 53, 54}
(57; 27, 25, 25, 23; 43) (**s)
= {0, 1, 7, 11, 15, 16, 17, 19, 20, 21, 24, 25, 26, 28, 30, 35, 36, 37, 38, 40, 44, 46, 47, 48, 49, 51, 54} X2 = {0, 1, 2, 3, 7, 10, 14, 17, 19, 20, 23, 26, 31, 32, 34, 36, 37, 38, 41, 42, 43, 44, 45, 46, 49} X4 = {2, 4, 6,9,10,11,15,16,18, 23, 26}
(57; 25, 25, 25, 24; 42) (sss), Turyn series Xi = {2, 3,8, 9,10,18, 20, 22, 23, 24, 26, 27} X2 = {6, 7, 9,10,14,16,19, 21, 24, 25, 27, 28} X4 = {2, 3, 8, 9,10,18, 20, 22, 23, 24, 26, 27}
v = 59
The third and the sixth PDF below are taken from [6].
(59; 28, 29, 29, 22; 49) (s**)
Xi = {1, 3, 5,8,10,11,13,15,16, 20, 21, 22, 26, 29}
X2 = {0, 1, 3, 7, 8, 9, 10, 12, 13, 15, 16, 19, 21, 22, 25, 27, 29, 34, 35, 36, 37, 38, 39, 40, 44, 51, 54, 55, 58} X2 = {0, 3, 4, 5, 7, 8, 14, 15, 16, 17, 19, 22, 24, 27, 28, 33, 39, 49, 53, 54, 55, 56}
(59; 28, 29, 29, 22; 49) (**s)
= {0, 5, 6, 7, 8, 9, 10, 12, 17, 18, 20, 25, 26, 27, 34, 35, 39, 42, 44, 45, 47, 48, 49, 50, 51, 54, 55, 58} X2 = {0, 1, 2, 3, 4, 5, 6, 9, 11, 13, 14, 18, 20, 24, 25, 26, 29, 31, 32, 35, 37, 44, 45, 47, 48, 51, 55, 56, 58} X4 = {1, 5,7,11,15,16,18, 20, 21, 23, 29}
(59; 27, 25, 25, 26; 44) (s**)
Xi = {2, 4,7,8,12,13,15,16,17,18, 20, 23, 29}
X2 = {1, 2, 4, 5, 12, 13, 17, 19, 20, 21, 22, 23, 26, 27, 31, 35, 37, 38, 40, 44, 47, 49, 50, 55, 57} X2 = {3, 7, 12, 13, 14, 16, 18, 19, 20, 22, 23, 24, 25, 26, 31, 32, 33, 34, 36, 38, 43, 45, 46, 50, 51, 53}
(59; 27, 25, 25, 26; 44) (**s)
= {0, 1, 3, 5, 6, 7, 8, 9, 12, 15, 18, 20, 28, 29, 31, 33, 34, 35, 38, 42, 44, 47, 48, 49, 55, 56, 58} X2 = {0, 3, 4, 5, 7, 10, 16, 21, 22, 24, 25, 26, 28, 29, 32, 33, 34, 38, 39, 40, 41, 43, 48, 49, 52} X4 = {1, 3, 5, 7,10,12,13,15,18,19, 20, 27,28}
(59; 26, 28, 28, 23; 46) (s**)
Xi = {4, 6,10,12,13,15,17, 21, 22, 24, 25, 27, 29}
X2 = {0, 1, 4, 5, 6, 7, 10, 11, 12, 13, 14, 17, 21, 22, 23, 27, 33, 34, 36, 37, 39, 41, 42, 45, 52, 55, 56, 57} X2 = {0, 7, 9, 12, 13, 14, 22, 25, 26, 27, 28, 31, 33, 35, 39, 40, 46, 47, 49, 51, 55, 57, 58}
6 y MHfflOPMAIiMOHHO-ynPABAiroWME CMCTEMb
f № 2, 2022
(б9; 26, 28, 28, 2В; 46) (**s)
= {2, В, 1О, 12, 1В, 14, 16, 18, 19, 26, 28, 29, В6, В8, В9, 4О, 42, 44, 46, 47, 49, бО, бВ, б4, бб, б7} X2 = {4, б, 7, 11, 12, 16, 17, 24, 2б, 26, 27, 28, 29, ВВ, В4, В7, В9, 4О, 42, 4В, 44, 4б, 47, 49, б1, бВ, б6, б8} X4 = {1,4, б, 7, S, 11,14, 20, 2б, 2S, 2g}
v = 61, Hx = {1, 1В, 47}, H2 = {1, 9, 2О, В4, б8}
(61; ВО, 29, 29, 2В; бО) (s**)
Xi = {2, 3, S, 10,11,13,14,1б, 1S, 20, 22, 23, 24, 2S, 30}
X2 = {О, 1, 2, 4, 9, 1О, 11, 1B, 18, 19, 22, 24, 26, 27, B7, B8, B9, 4О, 42, 4B, 44, 48, 49, б1, б2, бб, б6, б8, б9} X2 = {О, 6, 7, 8, 11, 12, 14, 16, 17, 2В, 26, ВО, В1, ВВ, В4, Вб, В7, В8, 4О, 4В, 4б, 49, бО}
(61; ВО, 29, 29, 2В; бО) (**s)
= {1, 2, б, 6, 7, 8, 11, 12, 14, 1б, 19, 2О, 21, 22, 24, 27, 28, ВО, В1, В2, В8, В9, 4О, 41, 4В, 4б, 46, бб, б8, 6О} X2 = {О, 2, 4, 8, 9, 1О, 1В, 14, 17, 18, 2О, 2В, 24, 2б, 26, 27, В2, В7, В8, 44, 4б, 47, 49, бВ, бб, б6, б8, б9, 6О} X4 = {2,10,13,1б, 17,1S, ig, 21, 22, 2б, 2g}
(61; ВО, 26, 26, 26; 47) (s**), all Xi are H2-invariant Xi = {1, 3, 4, б, g, 12,13,14,1б, 1б, ig, 20, 22, 2б, 27}
X2 = {О, 1, 6, 8, 9, 11, 12, 2О, 21, 2б, 26, 28, ВО, B2, B4, B7, B8, 42, 4B, 44, 47, б1, б4, б7, б8, б9} X2 = {О, В, б, 6, 21, 2В, 24, 26, 27, ВО, В2, ВВ, В9, 41, 4В, 44, 4б, 46, 48, бО, б1, б2, бВ, б4, б9, 6О}
(61; ВО, 26, 26, 26; 47) (**s)
= {О, 2, 4, б, 6, 8, 9, 1О, 12, 18, 19, 21, 22, 2В, 2б, 26, 27, 28, ВО, В2, В4, В6, В7, 42, 49, бб, б6, б7, б8, б9} X2 = {О, 2, б, 8, 9, 1О, 1б, 18, 24, 27, 28, В1, ВВ, Вб, В8, В9, 44, 4б, 46, 47, 49, бО, б1, б2, б9, 6О} X4 = {2,3, б, g, 11,14, is, ig, 21,23,24,2б, 2g}
(61; ВО, 2б, 2б, ВО; 49) (sss), Turyn series
Xi = {1, 2, б, S, g, 12,13,14,1б, 1б, 17, ig, 24, 2б, 2S}
X2 = {i, б, б, S, 10,11,12,14, 20, 24, 27, 2g}
X4 = {3, 4, б, 7,10,11,1S, 20, 21, 22, 23, 2б, 27, 2g, 30}
(61; ВО, 2б, 2б, ВО; 49) (k*s), XXSW series
Xi = {3,4, б, 13,14,1б, 1б, ig, 21, 22, 23, 24, 2б, 27, 30}
X2 = {б, 6, 8, 12, 14, 1б, 17, 2О, Bl, B2, ВВ, B6, 4О, 44, 4б, 46, 48, 49, б1, бВ, б4, бб, б6, б9, 6О} X4 = {1, 3, б, g, 10,13,1б, 1б, 17, 20, 22,2б, 27,2g, 30}
(61; 28, 28, 28, 24; 47) (s**)
Xi = {1,4, б, 7, S, 10,11,14, ig, 20, 22, 2б, 2S, 30}
X2 = {О, 1, 6, 1О, 12, 1B, 17, 21, 22, 2B, 26, 27, 29, B2, B4, Вб, В6, В9, 4О, 42, 4В, бО, б2, б4, б7, б8, б9, 6О} X2 = {О, б, 6, 8, 18, 2В, 24, 28, В2, В4, В7, 42, 4В, 44, 48, 49, бО, б1, б2, б4, бб, б6, б7, б8}
(61; 28, 28, 28, 24; 47) (**s), all Xi are Hrinvariant
= {О, 1, B, 4, б, 9, 12, 1B, 14, 1б, 18, 19, 27, B2, B4, B9, 4О, 46, 47, 48, 49, бО, б1, б2, бВ, б6, б7, 6О} X2 = {О, 1, 2, 4, б, 7, 11, 1В, 14, 21, 22, 24, 26, 29, ВО, В1, ВВ, В6, В7, 41, 42, 4б, 47, 48, б2, б4, б8, 6О} X4 = {1, 3,11,13,14,1б, ig, 20, 21, 22, 2б, 2g}
(61; 28, 27, 27, 2б; 46) (s**)
Xi = {1, 2, 3,4, б, 7, g, 11,17,1S, 20, 24, 27, 2g}
X2 = {О, 4, 6, 7, 11, 12, 1б, 18, 19, 21, 26, ВО, Bl, B2, ВВ, Вб, В6, 41, 4В, 44, 4б, 48, 49, б1, б2, бВ, б4} X2 = {О, 4, 1О, 11, 1В, 16, 17, 21, 22, 27, ВО, В2, В7, В8, 4О, 42, 4В, 44, 46, 47, б1, бВ, б4, бб, б6}
(61; 28, 27, 27, 2б; 46) (**s), all Xi are Hrinvariant
= {О, 7, 11, 18, 21, 22, 2B, 24, 28, 29, ВО, Bl, B2, Вб, B6, B7, 4О, 41, 42, 44, 4б, бО, б1, бВ, б4, бб, б8, б9} X2 = {1, В, 7, 8, 9, 1О, 1В, 19, 2В, 24, 27, 28, ВО, В1, Вб, В7, В9, 4В, 44, 46, 47, 49, б4, бб, б6, б7, б9} X4 = {i, 3, s, 10,13,14,1б, is, ig, 20,22,2б}
(61; 2б, ВО, ВО, 2б; 49) (s**), all Xi are Hrinvariant Xi = {1, 3, S, 10,13,14,1б, 1S, ig, 20, 22, 2б}
№ 2, 2022 ^
ЙНФ0РМ^ 7
X2 = {2, 6, 7, 14, 17, 18, 22, 23, 24, 26, 27, 28, 30, 33, 35, 36, 38, 41, 42, 44, 45, 46, 48, 49, 51, 53, 55, 58, 59, 60}
X4 = {0, 1, 2, 3, 6, 13, 14, 17, 19, 26, 27, 31, 32, 33, 37, 38, 39, 40, 46, 47, 48, 49, 50, 54, 60}
v = 63, H1 = {1, 4, 16}, H2 = {1, 25, 58} (63; 31, 26, 26, 30; 50) (sss), Turyn series, all Xi are H2-invariant Xi = {1, 3, 4, б, 7,12,14,1б, 19, 2G, 2б, 2б, 28, 29, 31} X2 = {7,8, 9,11,14,18,19, 21, 23, 27, 28, 29, 31} X4 = {i, 3,4, б, 7,12,14,1б, 19, 2G, 2б, 2б, 28, 29, 31}
(63; 30, 30, 30, 24; 51) (s**), all Xi are H1-invariant Xi = {1, 2, 4, 8, 9,11,13,1б, 18,19, 22,2б, 2б, 27, 31}
X2 = {7, 9, 11, 13, 14, 15, 18, 19, 22, 25, 26, 27, 28, 30, 35, 36, 37, 38, 39, 41, 44, 45, 49, 50, 51, 52, 54, 56, 57, 60}
X4 = {5, 9, 13, 15, 17, 18, 19, 20, 22, 23, 25, 26, 29, 31, 36, 37, 38, 41, 51, 52, 53, 55, 60, 61} (63; 30, 30, 30, 24; 51) (**s), all Xi are H1-invariant
X1 = {3, 5, 6, 9, 10, 12, 13, 14, 17, 18, 19, 20, 23, 24, 26, 29, 30, 33, 34, 35, 36, 38, 39, 40, 41, 48, 52, 53, 56, 57} X2 = {3, 5, 7, 9, 10, 12, 13, 15, 17, 18, 19, 20, 23, 26, 27, 28, 29, 34, 36, 38, 40, 41, 45, 48, 49, 51, 52, 53, 54, 60} X4 = {б, 7, 9,1G, 14,17,18, 2G, 23, 27, 28, 29}
(63; 30, 27, 27, 27; 48) (s**), all Xi are H2-invariant Xi = {1, б, 8, 9,11,1б, 17,18,19, 22, 23, 2б, 27,29, 31}
X2 = {3, 4, 7, 12, 15, 16, 17, 20, 22, 26, 27, 28, 29, 32, 37, 41, 43, 44, 45, 46, 47, 48, 49, 51, 54, 59, 60} X2 = {4, 6, 9, 10, 13, 17, 18, 19, 24, 27, 29, 31, 32, 33, 34, 36, 37, 40, 41, 43, 44, 45, 47, 52, 54, 55, 61}
(63; 30, 27, 27, 27; 48) (**s), all Xi are H1-invariant
X1 = {3, 6, 11, 12, 13, 19, 22, 23, 24, 25, 26, 27, 29, 30, 33, 37, 38, 39, 41, 43, 44, 45, 46, 48, 50, 52, 53, 54, 57, 58}
X2 = {3, 11, 12, 13, 14, 15, 19, 22, 25, 26, 31, 35, 37, 38, 41, 43, 44, 46, 48, 50, 51, 52, 55, 56, 58, 60, 61} X4 = {1, 4, 7, 9,14,1б, 18, 21, 22, 2б, 2б, 27, 28}
(63; 29, 31, 31, 24; 52) (s**)
Xi = {2, 3, б, 8,9,1G, 12,1б, 1б, 22, 24, 2б, 28, 31}
X2 = {0, 1, 2, 3, 4, 6, 7, 8, 10, 11, 12, 15, 20, 23, 24, 28, 29, 30, 31, 34, 40, 42, 43, 45, 49, 50, 52, 58, 59, 602, 61}
X4 = {0, 2, 8, 9, 10, 12, 15, 16, 20, 25, 26, 30, 33, 34, 37, 39, 45, 46, 48, 50, 57, 60, 61, 62} (63; 29, 31, 31, 24; 52) (**s)
X1 = {1, 2, 3, 11, 15, 18, 19, 21, 22, 26, 27, 28, 29, 30, 33, 34, 35, 36, 38, 39, 45, 48, 49, 51, 52, 55, 59, 60, 61} X2 = {0, 2, 3, 5, 10, 11, 17, 18, 19, 21, 23, 24, 28, 30, 32, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 48, 51, 52, 53,
562, 62}
X4 = {2, б, 1G, 11,12,14,17, 2G, 22, 2б, 27,29}
(63; 27, 31, 31, 25; 51) (s**), all Xi are H1-invariant Xi = {2, 3, 7, 8,1G, 12,14,1б, 21, 23, 28, 29, 31}
X2 = {3, 6, 7, 11, 12, 13, 14, 19, 22, 23, 24, 25, 26, 28, 29, 33, 35, 37, 38, 41, 42, 43, 44, 46, 48, 49, 50, 52, 53, 56, 58}
X4 = {3, 5, 12, 13, 15, 17, 19, 20, 22, 23, 25, 26, 29, 30, 37, 38, 39, 41, 42, 48, 51, 52, 53, 57, 60} (63; 27, 31, 31, 25; 51) (**s)
X1 = {0, 1, 2, 3, 4, 6, 7, 10, 11, 12, 15, 19, 23, 25, 28, 31, 32, 39, 40, 47, 49, 51, 52, 53, 58, 59, 62} X2 = {0, 1, 6, 7, 9, 10, 13, 14, 16, 18, 19, 23, 24, 28, 30, 35, 37, 38, 40, 41, 43, 44, 46, 47, 48, 49, 50, 54, 59, 61, 62}
X4 = {1, 2, 4, б, 1G, 14,1б, 17,19, 21, 27, 28}
(63; 27, 29, 29, 26; 48) (s**), all Xi are H2-invariant Xi = {1, 2, 3, б, 1G, 12,13,1б, 19, 21, 2б, 29, 31}
8 У ИHФOPMАЦИOHHO-УПPАВЛЯЮЩИE ^CTEMbl
У № 2, 2022
X2 = {2, 7, 9, 10, 13, 14, 18, 20, 21, 26, 27, 28, 29, 32, 35, 36, 40, 42, 44, 45, 49, 50, 52, 53, 54, 55, 56, 59, 61} X2 = {4, 7, 8, 9, 11, 16, 17, 18, 20, 21, 22, 23, 26, 28, 29, 32, 36, 37, 41, 42, 43, 44, 46, 47, 49, 59}
(63; 27, 29, 29, 26; 48) (**s)
= {0, 1, 3, 5, 6, 8, 9, 12, 15, 17, 21, 25, 26, 27, 28, 29, 31, 35, 36, 41, 43, 46, 48, 53, 54, 57, 61} X2 = {0, 2, 3, 5, 9, 10, 21, 25, 26, 27, 28, 30, 31, 33, 34, 35, 41, 42, 43, 44, 45, 46, 49, 53, 54, 55, 57, 59, 60} X4 = {1, 3, 4,10,11,14,16,18, 20, 23,26, 27, 31}
In the case v = 57 our list contains two PDFs having different parameter sets and sharing the same symmetric block. The same is true for v = 61.
Acknowledgement
The author is indebted to Tamara Balonina for converting the manuscript into the printing format of the journal.
Financial support
This research was enabled in part by support provided by SHARCNET (http://www.sharcnet.ca) and Compute Canada (http://www.computecanada.ca).
References
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2. Seberry J., Yamada M. Hadamard matrices, sequences, and block designs. In: Contemporary design theory. A Collection of Surveys. D. J. Stinson and J. Din-itz (eds.). John Wiley and Sons, New York, 1992. Pp.431-560.
3. Seberry J., and Balonin N. A. Two infinite families of symmetric Hadamard matrices. Australasian Journal of Combinatorics, 2017, vol. 69(3), pp. 349-357.
4. Balonin N. A., Balonin Y. N., Bokovic D. Z., Karbovs-kiy D. A., and Sergeev M. B. Construction of symmetric Hadamard matrices. Informatsionno-upravliai-ushchie sistemy [Information and Control Systems], 2017, no. 5, pp. 2-11. doi:10.15217/issn1684-8853.2017.5.2
5. Balonin N. A., Bokovic D. Z., and Karbovskiy D. A. Construction of symmetric Hadamard matrices of order 4v for v = 47, 73, 113. Spec. Matrices, 2018, vol. 6, no. 1, pp. 11-22. doi.org/10.1515/spma-2018-0002
6. Abuzin L. V., Balonin N. A., Bokovic D. Z., Kotsireas I. S. Hadamard matrices from Goethals — Seidel difference families with a repeated block. Informatsion-no-upravliaiushchie sistemy [Information and Control Systems], 2019, no. 5, pp. 2-9. doi:10.31799/1684-88532019-5-2-9
7. Mateo O. Di, Bokovic D. Z., Kotsireas I. S. Symmetric Hadamard matrices of order 116 and 172 exist. Spec. Matrices, 2015, vol. 3, pp. 227-234.
8. Turyn R. J. An infinite class of Williamson matrices. J. Combinatorial Theory Ser. A, 1972, no. 12, pp. 319321.
9. Xia M., Xia T., Seberry J., and Wu J. An infinite series of Goethals — Seidel arrays. Discrete Applied Mathematics, 2005, vol. 145, pp. 498-504.
10. Craigen R., and Kharaghani H. Hadamard matrices and Hadamard designs. In: Handbook of Combinatorial Designs. 2nd ed. C. J. Colbourn, J. H. Dinitz (eds). Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC Press, Boca Raton, FL, 2007. Pp. 273-280.
11. Turyn R. J. An infinite class of Williamson matrices. J. Combinatorial Theory Ser. A, 1972, no. 12, pp. 319321.
12. Janko Z., Kharaghani H., Tonchev V. The existence of a Bush-type Hadamard matrix of order 324 and two new infinite classes of symmetric designs. Designs, Codes and Cryptography, 2001, vol. 24, iss. 2, pp. 225-232.
13. Balonin N. A., and Bokovic D. Z. Symmetric Hadamard matrices of orders 268, 412, 436 and 604. In-formatsionno-upravliaiushchie sistemy [Information and Control Systems], 2018, no. 4, pp. 2-8. doi:10.31799/1684-8853-2018-4-2-8
№ 2, 2022 ^
MHOOPMAIiMOHHO-ynPABAfiroWME CMCTEMbl 9
УДК 004.438
doi:10.31799/1684-8853-2022-2-2-10
Некоторые новые симметричные матрицы Адамара
Джокович Д. Ж.а, доктор наук, профессор, orcid.org/0000-0002-0176-2395, djokovic@uwaterloo.ca аУниверситет Ватерлоо, кафедра теоретической математики и Институт квантовых вычислений, Ватерлоо, Онтарио, N2L 3G1, Канада
Введение: предполагается, что симметричные матрицы Адамара порядка 4v существуют для всех нечетных целых чисел v > 0. В последние годы их наличие было доказано для многих новых порядков с помощью специального метода, известного как конструкция пропус. В этой конструкции используются разностные семейства Хк (к = 1, 2, 3, 4) над циклической группой Zv (целые числа по модулю v) с параметрами (v; к1, к2, к3, к4; X), где Хх симметрично, Х2 = Х3 и к1 + 2к2 + к4 = v + X. Также предполагается, что такие разностные семейства (известные как пропус-семейства) существуют для всех наборов параметров, упомянутых выше, за исключением случая, когда все к1 равны. Эта новая гипотеза была проверена для всех нечетных v < 53. Цель: построить новые симметричные матрицы Адамара, используя конструкцию пропус, и обеспечить дальнейшее подтверждение вышеупомянутой гипотезы. Результаты: представлены первые примеры симметричных матриц Адамара порядков 4v для v = 127 и v = 191. Систематический компьютерный поиск симметричных матриц Адамара, основанный на конструкции пропус, был расширен на случаи v = 55, 57, 59, 61, 63. Практическая значимость: матрицы Адамара широко используются в задачах безошибочного кодирования и сжатия и маскирования видеоинформации.
Ключевые слова — симметричные матрицы Адамара, конструкция пропус, разностные семейства пропусов.
Для цитирования: Dokovic D. Z. Some new symmetric Hadamard matrices. Информационно-управляющие системы, 2022, № 2, с. 2-10. doi:10.31799/1684-8853-2022-2-2-10
For citation: Dokovic D. Z. Some new symmetric Hadamard matrices. Informatsionno-upravliaiushchie sistemy [Information and Control Systems], 2022, no. 2, pp. 2-10. doi:10.31799/1684-8853-2022-2-2-10
Финансовая поддержка
Исследование частично поддержано SHARCNET (http://www.sharcnet.ca) и Compute Canada (http://www.computecanada.ca).
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У ИНФОРМАЦИОННО-УПРАВЛЯЮЩИЕ СИСТЕМЫ
У № 2, 2022
