aa r X i v : . [ m a t h . C O ] J u l A NEW YANG NUMBER AND CONSEQUENCES
DRAGOMIR ˇZ. D– OKOVI ´C
Abstract.
Base sequences BS ( m, n ) are quadruples ( A ; B ; C ; D )of {± } -sequences, A and B of length m and C and D of length n ,the sum of whose non-periodic auto-correlation functions is zero.Base sequences and some special subclasses of BS ( n + 1 , n ) knownas normal and near-normal sequences, N S ( n ) and N N ( n ), as wellas T-sequences and orthogonal designs play a prominent role inmodern constructions of Hadamard matrices. In our previous pa-pers [3, 4] we have classified the near-normal sequences N N ( s ) forall even integers s ≤
32 (they do not exist for odd s > s = 34. Moreover we con-struct the first example of near-normal sequences N N (36). Con-sequently, we construct for the first time T-sequences of length 73.For all smaller lengths, T-sequences were already known. Anotherconsequence is that 73 is a Yang number, and a few importantconsequences of this fact are given.
Preliminaries
A sequence A = a , a , . . . , a m , of length m , is binary respectively ternary if a i ∈ {± } respectively a i ∈ { , ± } . We identify A withthe polynomial A ( z ) = a + a z + · · · + a m z m − . The norm of a Lau-rent polynomial f ( z ) is defined as N ( f ) = f ( z ) f ( z − ). A quadruple( A ; B ; C ; D ) of binary sequences, with A and B of length m and C and D of length n , are base sequences if N ( A ) + N ( B ) + N ( C ) + N ( D ) = 2( m + n ) . The set of such base sequences is denoted by BS ( m, n ). The BaseSequence Conjecture (BSC) asserts that BS ( n + 1 , n ) = ∅ for all non-negative integers n (see [2]). It has been confirmed for n ≤
35. It isalso well known that it holds when n is a Golay number , i.e., a numberof the form 2 a b c where a, b, c are nonnegative integers. Key words and phrases.
Base sequences, near-normal sequences, T-sequences,Hadamard matrices, orthogonal designs.Supported in part by an NSERC Discovery Grant.
The base sequences ( A ; B ; C ; D ) ∈ BS ( n + 1 , n ) are normal respec-tively near-normal if a i = b i respectively a i = ( − i − b i for all indexes i in the interval 1 ≤ i ≤ n . Let N S ( n ) resp. N N ( n ) be the subsetof BS ( n + 1 , n ) consisting of all normal resp. near-normal sequences.It is known that N S ( n ) = ∅ when n is a Golay number. For positiveintegers n ≤ N S ( n ) = ∅ exactly for n ∈ { , , , , , , , , , , , } (see [4]). It is known that N N ( n ) = ∅ when n is odd. Yang Conjecture (YC) asserts that
N N ( n ) = ∅ when n is even. This has been confirmedfor n ≤
34 (see [4]).A quadruple of ternary sequences ( A ; B ; C ; D ), all four of length n ,are T -sequences if N ( A ) + N ( B ) + N ( C ) + N ( D ) = n and for each index i , 1 ≤ i ≤ n , exactly one of a i , b i , c i , d i is nonzero.Let x i , i = 1 , , . . . , u , be independent commuting variables. An n × n matrix S all of whose entries belong to { , ± x , . . . , ± x u } andsuch that SS T = ( s x + · · · + s u x u ) I n is an orthogonal design . Here the superscript T denotes transpositionof matrices and I n is the identity matrix. The symbols s , . . . , s u arepositive integers. The set of all such orthogonal designs is denoted by OD ( n ; s , . . . , s u ).Yang [6, Theorems 1 and 3] constructed two maps, known as Yangmultiplications , N S ( s ) × BS ( m, n ) → T S ((2 s + 1)( m + n )) ,N N ( s ) × BS ( m, n ) → T S ((2 s + 1)( m + n )) . (As pointed out in [4], there are two misprints in his Theorem 1.) A Yang number is an odd integer 2 s + 1 such that N S ( s ) or N N ( s ) is notempty. 2. Main result and consequences
We have carried out an exhaustive search for
N S ( n ) when n = 34 , N N ( n ) for even integers n ≤
30. For the classification in the case n = 32 see [4]. We have nowcompleted the classification in the case n = 34. The representatives( A ; B ; C ; D ) of the NN-equivalence classes of N N (34) are given in Table
NEW YANG NUMBER 3
1. The sums a, b, c, d of the sequences
A, B, C, D are also recorded.The sequences are given in encoded form for the sake of compactness.The encoding scheme is explained in our papers [2, 3]. This table alsocontains the unique known example (up to NN-equivalence) of
N N (36),which we have constructed recently.
Theorem 2.1.
N N (36) = ∅ . Proof.
Direct verification. (cid:3)
Table 1: Near-normal sequences
N N ( n ) A & B C & D a, b, c, dn = 341 076417646512321462 16738541372344337 7 , , − ,
62 076535878535141762 17677852174231455 − , , ,
83 076782178767646231 17621532262576812 − , , , −
24 058214353712141461 11868756376664254 11 , , − ,
25 053765656464871261 17765746348615187 1 , , − , n = 361 0764841234846532153 165154775335162126 3 , − , , N N (36) and write it as a quadruple ( A ; B ; C ; D ) of binary sequences.In fact we write + for +1 and − for − A = + − + + − + + + − + − + + − − + + − − ++ + + + − − − − + + + − − − − − +; B = + + + − − − + − − − − − + + − − + + − − + − + − − + − + + − + + − + − + − ; C = + + − + − + − − − − − − + + + + + + − ++ + − + + + + + − − − + + + − +; D = + + + + + − + + + − − + + + − + − + − − + − − + − + + − + + + − + − − + . Let us list some consequences of the above results.(i) Since
N N ( n ) ⊆ BS ( n + 1 , n ), we can update the status of theBSC and YC: BS ( n + 1 , n ) = ∅ for n ≤
36, and
N N ( n ) = ∅ for even n ≤ D.ˇZ. D– OKOVI ´C (ii) An odd (positive) integer n ≤
73 is a Yang number if and onlyif n = 35 , , , , , , . In order to rule out the integer 71, weneed to use the fact that
N S (35) = ∅ mentioned above.(iii) Our main result implies that there exist T-sequences T S (73) (see[5, Lemma 5.21]). Consequently, by [5, Theorem 3.6], there exists anorthogonal design OD (4 t ; t, t, t, t ) for t = 73. Neither T-sequences norT-matrices for n = 73 were known previously, see the Remarks V 2.51and V 8.47 in [1]. In spite of the claim made in Remark V 2.119.5 thatan OD with the above parameters is known, we believe that this is notthe case as we could not find such a result anywhere in the literatureand our request for a reference failed.(iv) One can plug into our OD any Williamson-type matrices ofsome order n to obtain Hadamard matrices of order 4 · · n . SinceWilliamson-type matrices are known for infinitely many odd orders n ,we obtain infinitely many Hadamard matrices of order an odd multi-ple of 4. In particular, there exist Williamson-type matrices of order n = 61 , , ,
101 (see [5, Table A.1]). Hence, there exist Hadamardmatrices of order 4 · · n for the same values of n . These ordersfall in the range covered by Table V 1.53 of [1], where it is indicatedthat no Hadamard matrices of these orders are known. However, wehave recently discovered over 100 errors in this table. It turns out thatHadamard matrices of these four orders are in fact known. This willbe discussed in more details elsewhere.(v) For s = 2 , , . . . ,
34, the number of NN-equivalence classes in
N N ( s ) is 1 , , , , , , , , , , , , , , , , . Our computer program performs an exhaustive search for near-normalsequences
N N ( s ) for fixed s . The search is divided into 12 cases whichcan be run separately. For s = 36, we started to run them at differenttimes according to the availability of machines. Six of the cases havecompleted within a month or two without finding any near-normal se-quences. Only one of the remaining cases produced (after about 40days) the example given in the above table.It is a pleasure to thank an anonymous referee for his valuable com-ments. 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, Aperiodic complementary quadruples of binary sequences,JCMCC (1998), 3–31. Correction: ibid (1999), p. 254. NEW YANG NUMBER 5 [3] , Classification of near-normal sequences, Discrete Mathematics, Al-gorithms and Applications, Vol. , No. 3 (2009), 389–399. arXiv:0903.4390v2[math.CO] 1 Sep 2009.[4] , Some new near-normal sequences, International Mathematical Forum , No. 32 (2010), 1559–1565. arXiv:0907.3129v2 [math.CO] 14 Feb 2010.[5] J. Seberry and M. Yamada, Hadamard matrices, sequences and block designs,in Contemporary Design Theory: A Collection of Surveys, Eds. J.H. Dinitzand D.R. Stinson, J. Wiley, New York, 1992, pp. 431–560.[6] 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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