We construct new TW -sequences, weighing matrices and orthogonal designs using near-Yang sequences. In particular we construct new OD(60(2m + 1) + 4t; 13(2m+ 1), 13(2m+ 1), 13(2m+ 1), 13(2m+ 1) and new W(60(2m+ 1) + 4t; 13s(2m+ 1))for all t ≥ O, m ≤ 30, s = 1,2,3,4

Gysin, Marc

Seberry, Jennifer

English

Research Online

University of Wollongong Research Online Faculty of Informatics - Papers (Archive) Faculty of Engineering and Information Sciences 1994 New results with near- Yang sequences Marc Gysin Jennifer Seberry University of Wollongong, jennie@uow.edu.au Follow this and additional works at: https://ro.uow.edu.au/infopapers  Part of the Physical Sciences and Mathematics Commons Recommended Citation Gysin, Marc and Seberry, Jennifer: New results with near- Yang sequences 1994. https://ro.uow.edu.au/infopapers/1087 Research Online is the open access institutional repository for the University of Wollongong. For further information contact the UOW Library: research-pubs@uow.edu.au New results with near- Yang sequences Abstract We construct new TW -sequences, weighing matrices and orthogonal designs using near-Yang sequences. In particular we construct new OD(60(2m + 1) + 4t; 13(2m+ 1), 13(2m+ 1), 13(2m+ 1), 13(2m+ 1) and new W(60(2m+ 1) + 4t; 13s(2m+ 1))for all t ≥ O, m ≤ 30, s = 1,2,3,4. Disciplines Physical Sciences and Mathematics Publication Details Marc Gysin and Jennifer Seberry, New results with near- Yang sequences, Utilitas Mathematica, 45, (1994), 85-89. This journal article is available at Research Online: https://ro.uow.edu.au/infopapers/1087 New Results with Near-Yang Sequences Marc Gysin and Jennifer Seberry The University of Wollongong Wollongong, NSW 2500 Australia AbstracL We construct new TW -sequences, weighing matrices and onbogooal de-signs using near-Yang sequences. In panicular we construct new OD(60(2m + 1) + 4t; 13(2m+ I), 13(2m+ I), 13(2m+ I), 13(2m+ 1» and new W(60(2m+ 1) + 4t; 13 .. (2m+ 1» forallt;;:: O,m ~ 30, .. = 1,2,3,4. 1. Introduction For definitions we refer the reader to [9, Introduction] and [11, Section 2]. We give one new definition. Definition 1. (near-Yang sequences) A triple (F, G, H) of sequences is said to be a set of near-Yang sequences for length n (abbreviated as NY( n» if the following conditions are satisfied. (i) F = (f/r.) is a (0, 1, -1) sequence of length n. (ii) G = (g/r.) and H = (h/r.) are sequences of length n with entries 0,1, -1, such that G+ H = (g/r.+ h/r.) and G - H = (g/r. - h/r.) are both (0,1, -1) sequences of length n. (iii) g" + g .... ,,+1 == 0 (mod 2) h" + h .... ,,+1 == 0 (mod 2) s= I,.:.,[r] (iv) NF(S) + No(s) + Nn{s) =0, s=I, ... ,n-l. where 2. Computational Results .... " Nx{s) = :L: XjXj+". jE1 In Koukouvinos, Kounias, Seberry, Yang and Yang [6] it is shown that if in (ii) of the definition G ± H are both ( 1, -1) sequences then conditions (i), (ii) and (iv) imply condition (iii) but this is not true for near-Yang sequences. These sequences are nonnal sequences N S( l) . We searched for normal sequences N S{ £). N S( £) do exist for the following lengths£ E {1,2, 3,4 ,5,7,8,9,10,11,12,13,15,16,18,19,20,25,26,29, 32 , ... } and they do not exist for £ E {6 , 14 , 17 , 21 , 22 , 23 , 30 ,46 , 56 , 62 ,78 , Utilitas Mathematica 45(1994), pp. 85-89 Thble 1: Nonnal sequences via Simulated Annealing. Length P. Sequences Weight 20 F = --++++++-++-+-+++-+- 20 G = 0+-000+000000+000-+0 6 H = -00+++0-++--+0---00+ 14 25 F--+-+-+++---++--+--+ 25 G = 0-0-0+0+0+0-0+0-0-0-0+0+0 12 H = -0-0-0+0+0-0+0-0+0+0-0-0+ 13 94, ... } [3,6, 16]. We note here that we found normal sequences of length 25 and 20 (Table 1) using simulated annealing; this is described in Gysin [3]. Se-quences of length 25 can be obtained from Twyn sequences of lengths 13 and 12 and a complete search for these was carried out over 20 years ago. It is known that there are eight inequivalent sets of Twyn sequences of lengths 13 and 12 [2] and hence by the construction discussed in [6] probably at least sixteen inequivalent sets of normal sequences of length 25. It would be interesting to know if there are N S( 25) which cannot be made from Turyn sequences. There exist N S( 20) which cannot be made from Turyn sequences, an example is given in Table 1. In those small cases where N S(p') do not exist we searched for NY ( n), which con-tain more zeros in appropiate positions. We obtained the follo~ing new results: NY( n) with weight u = 12 exist for the following lengths: n E {7, 11,13, IS}. In Table 2 two conditions were imposed in counting the number of inequivalent triples of sequences: two triples of sequences were considered equivalent if one triple of sequences can be changed into the other triple of sequences by reversing and/or negating one or more sequences of the triple; if the three sequences F, G and H all started or ended with '0' they were considered to be of smaller length and not counted for this length n. This allows us to find new 4-complementary sequences of lengths I5( 2 m + 1), 23(2m + 1), 27(2m + I), 31(2m + 1) and weights 13(2m + 1), m ::;; 30. 3. Construction Definition 2. (suitable sequences) [5,8, 11] A. B. C, D are suitable sequences SS(m + p, m; w) with elements 0.1. -1 0/ lengths m + p. m + p. m. m and total total weight w if A and B are disjoint. C and D are disjoint and A. B. C. and D have zero non-periodic autocorrelation/unction. We use a modified version of Yang's [5,8, 16] theorem 86 Table 2: New near-Yang ·sequences. Length n No of Seq. Sequence Examples Weight 7 2 F= +++0-+- 6 G = 0++0-+0 4 H = +00000+ 2 F = +++0-+- 6 G = 0+000+0 2 H = +0+0-0+ 4 11 12 F = -0+0+00000+ 4 G = 0+0+000+0-0 4 H = +0 + 00000-0+ 4 13 24 F = +0+0+0-0+000- 6 G = 0 + ()()()()()()OO( + 0 2 H = +00+00000-00+ 4 15 26 F = +0+00000+00000- 4 G = 0+0000+0-0000+0 4 H = +00+0000000-00+ 4 Theorem 1. Let A. B. O. D be SSe m+ p, m; w) and F. G. H be NY( n) with total weight uand 0',0 be sequences of zeros of length m+pand mrespectively and X· be the reverse sequence of X then Q = {AI ... CgI-Dhl;O'.O;AI ...... I,Cil -Dhl; o',O; ... ;Aft .Cg• - Dh.;O'.O; BO.O} R= {BI .. ,Dg .. + Ch .. ; 0',0; BI ...... I.Dgro-l +Ch ...... 1 ;0',0; ... ; Bft ,Dg1 +CI>I ;0',0; -AO;O} S = {O',O; Ag .. + Bhl, -C/I ; 0',0; Ag ...... 1 + Bhz, -C/z; ... ; 0' ,0; Agl + Bh., -C,.; 0', DO} T = {O'.O; -Bgi +Ah ... Dh;O',O; -Bgz + Ah ...... 1 ,D/z; . .. ; 0'.0; -Bg .. +Ahl,DI .. ; O',CO} are TW-sequences ofle'!gth (2m + p)(2n+ 1) and total weight (u + l)w. This gives many new TW -sequences, weighing matrices and orthogonal de-signs. Many other corollaries are also possible. Example 1. LetF= {+++O-+-},G= {O+ooO+O},H= {+o+O-O+}and A, B, 0, D be suitable sequences of length m + p and m and total weight w. Then 87 with 0 I and 0 zero vectors of length m + p' and m respectively we have Q = {- A,-D; O',O;A,C; 0',0; -A,-D; 0',0; 0',0; 0',0; A,D;O',O; A,C;O',O;A, -D; 0',0; B· ,O} R = {- B,C;O',O;B,D;O',O; -B,-C; 0',0;0',0;0',0; B,C;O',O; B,D;O',O; B,C; 0',0; -A·,O} S = {O',O; B,-C; 0',0; A,-C; 0',0; B, -C; 0',0; 0',0; 0',0; -B,C; O',O;A,-C; 0',0; B,C; O',D·} T = {O',O;A,D; 0',0; -B,D; 0',0; -A,D;O',O; 0',0; O',O;A, -D; 0',0; -B,D;O',O; A,-D;O',C*} are TW -sequences of of length 15(2 m + p) and total weight 13 w. Corollary 1. Suppose there are suitable sequences of length m + p, m + p, m, m and total weight w, SS( m + p, m; w) and near-Yang sequences of length n and total weight u. Then there are TW-sequences of length (2 n + I) (2 m + p) and total weigh/ (u + I) w. Corollary 2. Suppose there exist SS( m + p, m; w). Then since there are near-Yang sequences of length 7 and total weight 12 there are TW -sequences of length 15(2 m + p) and total weight 13 w. From [11] we see SS( m+ I, m; 2 m+ I) exist for all m ~ 30 hence there exist TW -sequences of length 15( 2 m + I) and weight 13( 2 m + I) for all m ~ 30. USing theorems 3.6, 3.7, 3.8 of [11] we have OD(60(2m + I) + 4t; 13(2m + I), 13(2m + 1),13(2 m + I), 13(2m + I» forall t ~ 0 and m ~ 30. Further-more Q, R, S, T can be used in the Goethals-Seidel array to form OD( 4t; 13w, 13w, 13w, 13w) foreveryt > 15(2m+ I). Recalling that variables in an OD can be set equal or set zero to give weighing matrices, we obtain W(60(2 m + I) + 4t; 13 s(2 m + 1) and W( 4t; 13 sw), s = 1,2,3,4. Since N S( n) are NY( n) with total weight 2 n. CorOllary 3. Suppose there exist SS( m+ p, m; w). Then since there are normal sequences of length 25 and total weight 50 there are TW -sequences of length 51(2m + p) and total weight 51w. If w = 2m + p then we have T-sequences. Again using theorems 3.6,3.7,3.8 of [11] we have OD( 104(2m + I) + 4t; 51(2m+ 1),51(2m+ 1),51(2m+ 1),51(2m+ I» for all t ~ 0 andm ~ 30. Furthermore Q, R, S, T can be used in the Goethals-Seidel array to form OD(4t;5Iw, 51w, 5Iw,5Iw) for every t > (2n+ 1)(2m+ I). Acknowledgement We wish to thank the University ofTrondheim, Norway for letting us use hundreds of CPU-hours on their machines "fiolilla" and "lise". 88 References [1] Joan Cooper and Jennifer Wallis, A constructionfor Hadamard arrays, Bull. of the Austr. Math. Soc. 7, No 2 (1972),269-277. [2] G. M. Edmondson, J. Seberry, M. R. Anderson, On the existence ofTuryn sequences of length less than 43, Math. Compo to appear. [3] M. Gysin, On Normal and Near-Yang Sequences. M.Sc. Thesis, University of Wollongong, to be published. [4] A.V. Geramita and J. Seberry, "Orthogonal Designs: Quadratic Forms and Hadamard Matrices", Marcel Dekker, New York - Basel, 1979. [5] C. Koukouvinos, S. Kounias, and J. Seberry, Further results on base se-quences. disjoint complementary sequences. OD( 4 t; t, t, t, t) and the excess of Hadamard matrices, Ars Combinatoria 30 (1990), 241-256. [6] C. Koukouvinos, S. Kounias, J. Seberry, C.H. Yang and J. Yang. Multipli-cation of sequences with zero autocorrelation, to appear. [7] C. Koukouvinos, S. Kounias, J. Seberry, C.H. Yang and J. Yang. On se-quences with zero autocorrelation, to appear. [8] C. Koukouvinos and J. Seberry, Addendum to further results on base se-quences. disjoint complementary sequences. OD( 4t; t, t, t, t) and the excess of Hadamard matrices, Congr. Num. 82 (1991),97-103. [9] C.Koukouvinos and J.Seberry. On weighing matrices, Utilitas Math., to appear. [10] C.Koukouvinos and J.Seberry. On sequences with zero autocorrelation function, submitted. [11] Jennifer Seberry and Mieko Yamada, Hadamard matrices. sequences and block designs, in "Contemporary Design Theory - a Collection of Surveys, eds J. Dinitz and DJ. Stinson", John Wiley and Sons, New York, 1992, pp. 431-560. [12] C.H. Yang, Hadamard matrices and 8-codes of length 371, Proc. Amer. Math. Soc. 85 (1982), 480-482. [13] C.H. Yang, A composition theoremfor 8-codes, Proc. Amer. Math. Soc. 89 (1983),375-378. [14] C.H. Yang, Lagrange identity for polynomials and 8-codes of lengths 7t and 13t, Proc. Amer. Math. Soc. 88 (1983),746-750. [15] C.H. Yang, On composition offour-symboI8-codes and Hadamard matri-ces, Proc. Amer. Math. Soc. 107 (1989),763-776. [16] C.H. Yang. On Golay, near normal and base sequences, in preparation. 89 

New results with near- Yang sequences

https://ro.uow.edu.au/cgi/viewcontent.cgi?article=2104&amp;context=infopapers

New results with near- Yang sequences

Abstract

Similar works

Full text

Available Versions

Research Online