aa r X i v : . [ m a t h . C O ] D ec HADAMARD MATRICES OF SMALL ORDER ANDYANG CONJECTURE
DRAGOMIR ˇZ. D– OKOVI ´C
Abstract.
We show that 138 odd values of n < n exists have been overlooked in the re-cent handbook of combinatorial designs. There are four additionalodd n = 191 , , , n exist. There is a unique equivalence class ofnear-normal sequences N N (36), and the same is true for
N N (38)and
N N (40). This means that the Yang conjecture on the ex-istence of near-normal sequences
N N ( n ) has been verified for alleven n ≤
40, but it still remains open.
Introduction
Recall that a Hadamard matrix of order m is a {± } -matrix A ofsize m × m such that AA T = mI m , where T denotes the transpose and I m the identity matrix. Let us denote by HM ( m ) the set of Hadamardmatrices of order m . By abuse of language, we say that HM ( m ) existif HM ( m ) = ∅ . If m > HM ( m ) exist, then m is divisible by 4.In the recent handbook [1, pp. 278–279] one finds a table of all oddintegers n < t (2 ≤ t ≤
8) for which it is knownthat HM (2 t n ) exist. The Hadamard conjecture asserts that we shouldalways have t = 2 (for n > t ≥ t = 2 are indicated inthe table by a dot. The table has 5000 entries of which 1006 are bad.In this note we point out that 138 of these 1006 cases are in fact good.Additional 4 cases can be also eliminated, reducing the number of badcases to 864.In the next section we recall a well known construction (see Propo-sition 2.1) of HM (4 n ) which uses Yang multiplication, T-sequences,orthogonal designs, and Williamson-type matrices. In section 3 we list138 bad cases and invoke Proposition 2.1 to show that they shouldhave been classified as good cases. We also mention four additional Key words and phrases.
Base sequences, normal and near-normal sequences, T-sequences, orthogonal designs, Williamson-type matrices, Yang conjecture. bad cases that we can eliminate. In section 4 we show that there existnear-normal sequences
N N ( n ) for n = 32 , , ,
38 and 40. Thus, de-spite of our efforts to find a counter-example, the Yang conjecture onnear-normal sequences still remains open.2.
Preliminaries
Let us introduce the following notation for the sets of some importantcombinatorial objects (for their definitions see [1, 3, 4, 10]): GS ( g ) Golay sequences of length gT S ( t ) T-sequences of length tBS ( r, s ) Base sequences of lengths r, sN S ( l ) Normal sequences, as a subset of BS ( l + 1 , l ) N N ( l ) Near-normal sequences, as a subset of BS ( l + 1 , l ) OD (4 d ) Orthogonal designs OD (4 d ; d, d, d, d ) W T ( w ) Williamson-type matrices of order wBHW (4 h ) Baumert–Hall–Welch arrays of order 4 h It is well known that there exist constructions (i.e., maps) GS ( g ) → N S ( g ) , (2.1) GS ( g ) → BS ( g, , (2.2) BS ( r, s ) → T S ( r + s ) , (2.3) N S ( l ) × BS ( r, s ) → T S ((2 l + 1)( r + s )) , (2.4) N N ( l ) × BS ( r, s ) → T S ((2 l + 1)( r + s )) , (2.5) BHW (4 h ) × T S ( t ) → OD (4 ht ) , (2.6) OD (4 d ) × W T ( w ) → HM (4 dw ) . (2.7)The first three constructions are elementary. They are given by( A ; B ) → ( A, +; A, − ; B ; B )( A ; B ) → ( A ; B ; +; +)( A ; B ; C ; D ) → (( A + B ) / , s ; ( A − B ) / , s ;0 r , ( C + D ) /
2; 0 r , ( C − D ) / , where + and − stand for +1 and −
1, respectively, comma denotes theconcatenation of sequences, and the symbol 0 d denotes the sequenceof d zeros. The arithmetic operations on sequences are performedcomponent-wise. The constructions (2.4) and (2.5) are due to Yang[11]. For the “plug in” constructions (2.6) and (2.7) see [10] Theorems3.10 and 3.8, respectively.Recall that BHW (4 h ) exist for h ∈ { , , } (see [10]). An integer g is a Golay number if GS ( g ) exist. It is known that 2 a b c , a, b, c ≥ ADAMARD MATRICES OF SMALL ORDER 3 integers, are Golay numbers. An odd integer y = 2 l + 1 is a Yangnumber if N S ( l ) or N N ( l ) exist. The following well known fact is animmediate consequence. Proposition 2.1. If y is a Yang number and BHW (4 h ) , BS ( r, s ) and W T ( w ) exist, then HM (4 n ) exist for n = yh ( r + s ) w .Proof. We first apply the construction (2.4) or (2.5), whichever is ap-propriate, to obtain T-sequences of length t = y ( r + s ). Next, we applythe construction (2.6) to obtain an OD (4 ht ). Finally, the construction(2.7) produces an HM (4 n ). (cid:3) An update of the list of bad cases
Let ∆ be the following set of 138 odd integers < { , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , } . According to [1, pp 278–279] all these cases are bad, i.e., they allhave t ≥ HM (4 n ) for n ∈ ∆ has been knownfor some time and that all these cases should have been classified asgood. To prove this claim, it suffices to apply the above proposition.We used only the facts known for a few years prior to the publicationof [1]. In more detail, we used the existence of BS ( l + 1 , l ) for l ≤ BS (2 l − , l ) for even l ≤ BS ( g , g ) for g and g Golay numbers,and
N N ( l ) for even l ≤
30. For Williamson-type matrices
W T ( w ), weused the listing for odd w < W T ( w ) exist for w = 35 D.ˇZ. D– OKOVI ´C and w = 127, proven in [2] and recorded in [1, Table V.1.50, p. 277] aswell.Since the verification is of routine nature and tedious, we shall justgive a few examples (see Table 1) and list the acceptable choices for theparameters y, h, ( r, s ) , w . In some cases there are several such choices,which may give different constructions for HM (4 n ).Since 2048 and 2600 are Golay numbers, it follows from (2.1) that4097 = 2 · · BS ( r, s ) that occur in Table 1 can be found in many places, e.g., [4, 10],except for the case ( r, s ) = (34 , BS (100 , n = 7373, exist because of (2.2). Table 1: Parameters for the construction of HM (4 n ) n y h ( r, s ) w n y h ( r, s ) w ,
23) 1 4495 31 5 (15 ,
14) 13953 59 1 (34 ,
33) 1 31 5 (1 ,
0) 294097 4097 1 (1 ,
0) 1 31 1 (15 ,
14) 51 1 (2049 , ,
2) 294389 19 1 (17 ,
16) 7 29 5 (16 ,
15) 119 1 (11 ,
10) 11 29 5 (1 ,
0) 3119 1 (6 ,
5) 21 29 1 (3 ,
2) 3119 1 (4 ,
3) 33 29 1 (16 ,
15) 511 1 (29 ,
28) 7 5201 5201 1 (1 ,
0) 111 1 (11 ,
10) 19 1 1 (2601 , ,
9) 21 5875 25 5 (24 ,
23) 111 1 (4 ,
3) 57 25 1 (24 ,
23) 511 1 (1 ,
0) 399 5 1 (24 ,
23) 257 1 (29 ,
28) 11 5 5 (24 ,
23) 57 1 (17 ,
16) 19 1 5 (24 ,
23) 257 1 (10 ,
9) 33 5913 1 1 (41 ,
40) 737 1 (6 ,
5) 57 7373 1 1 (100 ,
1) 737 1 (1 ,
0) 627 9065 49 5 (19 ,
18) 11 1 (6 ,
5) 399 49 5 (1 ,
0) 374453 1 1 (31 ,
30) 73 49 1 (19 ,
18) 549 1 (3 ,
2) 3737 5 (25 ,
24) 137 5 (1 ,
0) 4937 1 (25 ,
24) 537 1 (3 ,
2) 49
ADAMARD MATRICES OF SMALL ORDER 5
There are four additional bad cases n = 191 , , , HM (4 n )for n = 191 appeared in [5]. For the remaining three cases we againapply Proposition 2.1. The acceptable choices for the parameters are y = h = 1, ( r, s ) = (37 ,
36) and w = 79 , , N N (36) exist, and consequently BS (37 ,
36) exist (see the next section).4.
The current status of the Yang conjecture
In his paper [11] Yang said that ”it is likely” that
N N ( n ) exist forall even integers n >
0. This assertion has become known as “Yangconjecture”, see e.g., [1, 4]. Note that, in our notation which is differentfrom that of Yang, the set
N N ( n ) is empty for odd n >
1. In ourrecent paper [6] we have introduced an equivalence relation for near-normal sequences, to which we refer as
N N -equivalence. This leadsto a canonical form for
N N -equivalence which is too technical to begiven here. By using this canonical form we were able to enumeratethe
N N -equivalence classes in
N N ( n ) for even n ≤
30. Subsequentlythese exhaustive computations were extended to cover the cases of alleven n ≤
40. For the cases n = 32 and n = 34 see our notes [7] and [8],respectively. After finding out that there is only one N N -equivalenceclass for n = 36, we lost any hope that Yang conjecture may be truein general. However, to our great surprise, it turned out that there isagain a single N N -equivalence class in
N N (38), and the same holdstrue for
N N (40). The computations in the last two cases were carriedout on SHARCNET’s machines running at 3.0 GHz. The CPU timefor the case n = 40 was about 1300 days.Here we give examples of N N ( n ) for n = 32 , , , ,
40 in ourencoded form: n = 32 : [07651732153537650 , ,n = 34 : [076417646512321462 , ,n = 36 : [0764841234846532153 , ,n = 38 : [07641237828515856281 , ,n = 40 : [058214351717346462170 , . For the reader’s convenience we also give an example of BS (34 , , . The encoding scheme is explained in our papers [3, 4, 6].
D.ˇZ. D– OKOVI ´C Acknowledgments
References [1] C.J. Colbourn and J.H. Dinitz, Editors, Handbook of Combinatorial Designs,2nd edition, Chapman & Hall, Boca Raton/London/New York, 2007.[2] D.ˇZ. D– okovi´c, Good matrices of orders 33, 35 and 127 exist, J. Comb. Math.Comb. Comput. (1993), 145–152.[3] , Base sequences, complementary ternary sequences, and orthogonaldesigns, J. Combinatorial Designs (1996), 339–351.[4] , Aperiodic complementary quadruples of binary sequences, JCMCC (1998), 3–31. Correction: ibid (1999), p. 254.[5] , Hadamard matrices of order 764 exist, Combinatorica (4) (2008),487–489.[6] , Classification of near-normal sequences, Discrete Mathematics, Algo-rithms and Applications, , No. 3 (2009), 389–399. Available as a preprint onarXiv:0903.4390v2 [math.CO] 1 Sep 2009.[7] , Some new near-normal sequences, arXiv:0907.31290v1 [math.CO] 17Jul 2009.[8] , A new Yang number and consequences, Des. Codes Cryptogr. (toappear).[9] S. Kounias and K. Sotirakoglu, Construction of orthogonal sequences, Proc.14-th Greek Stat. Conf. 2001, 229–236 (in Greek).[10] J. Seberry and M. Yamada, Hadamard matrices, sequences and block designs,in “Contemporary Design Theory, A Collection of Surveys”, J.H. Dinitz andD.R. Stinson, Eds., J. Wiley, New York, 1992.[11] C. H. Yang, On composition of four-symbol δ -codes and Hadamard matrices,Proc. Amer. Math. Soc. (1989), 763–776. Department of Pure Mathematics, University of Waterloo, Water-loo, Ontario, N2L 3G1, Canada
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