Some improved upper and lower bounds are given for the excess of Hadamard matrices. The excess of orthogonal designs is defined and discussed

Hammer, J

Levingston, R

Seberry, Jennifer

English

J Hammer

R Levingston

Jennifer Seberry

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A remark on the excess of Hadamard matrices and orthogonal designs

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University of Wollongong Research Online Faculty of Informatics - Papers (Archive) Faculty of Engineering and Information Sciences 1978 A remark on the excess of Hadamard matrices and orthogonal designs J Hammer R Levingston 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 Hammer, J; Levingston, R; and Seberry, Jennifer: A remark on the excess of Hadamard matrices and orthogonal designs 1978. https://ro.uow.edu.au/infopapers/984 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 A remark on the excess of Hadamard matrices and orthogonal designs Abstract Some improved upper and lower bounds are given for the excess of Hadamard matrices. The excess of orthogonal designs is defined and discussed. Disciplines Physical Sciences and Mathematics Publication Details Hammer, J, Levingston, R and Seberry, J, A remark on the excess of Hadamard matrices and orthogonal designs, Ars Combinatoria, 5, 1978, 237-254. This journal article is available at Research Online: https://ro.uow.edu.au/infopapers/984 A REMARK ON THE EXCESS OF HADAMARD MATRICES AND ORTHOGONAL DESIGNS J. Hammer, R. Levingston, Jennifer Seberry ABSTRAC'l'. Some improved upper and lower bounds are given for the excess of Hadamard matrices. The excess of orthogonal designs is defined and discussed. An D~thDgonaL de8ign D = YlYl + Y2Y2 + ••• + YtYt of order n and type (51' ••• , St) on the commuting variables Yl ' •.• , Yt is a square matrix satisfying Each is a (O,l,-l)-matrix satisfying and is called a "Hlighillf} matri:c of weight Sj' A weighing matrix of order n and weight n is called an Hadama~d ma:t~i:c. We call the sum of the entries of an Hadamard matrix, H, the e:cae88 of H. We denote by o(n) the maximum possible sum of an Hadamard matrix of order n. We define the e:casss of the orthogona,l. duign D a!l where o(Yi ) is the sum of the entries of Yi • ARS COMBlNATORIA, Vol. 5 (1978), pp. 237-254. Wallis [4} and Schmidt and Wang !2] defined the weight of an Hadamard matrix H f denoted w(H), to be the number of ones in H. W(n) is used for the maximum value of w(H) for Hadamard matrices of order n. Wen) ~ "(n2 +0(n)). The excess was studied by Best [1]. War.g and Schmidt found (ll o(n) ,.. 2("n{2n+1I - n • This was improved by Best who found a(n) ,.. nrn. 1. A canstructian. Bere are four slightly different ways to construct an orth09'onal design of type (t,t,t,t) in order 4t using 4 circulant (O,l,-l)-matrices Xl' X2, X3, X4 which satisfy , • (i) ! XiXi tIt • i=l (ii) X.J x.J • • (2) (iii) Xi",X j 0 • i • j . , (iv) ! X. i. a (l,-l)-matrix, 1 .. 1 • (v) , + x, , + x J , + x, , t x, . - 238 -u = [" " " 'J "" (Uij ) . v = [' " " 'J = (v,;) . " " " -', . " -', -Y, " " -', Y, " " Y, -', -', " " -', " " -', Y, -Y, Write G -A, A,' A,' A,' H A, A,' A,' A,' A,. A, AT. • _A'l'R , -A2R A, AT. • _ATR , A,' _A'l'R , A, AT. , -AJR _ATR • A, AT. , A,' AT. , _A'l'R , A, -A4R AT. , _A'l'R , A, where R is the back diagonal matrix and AI' A2 , A3 , A4 are circulant matrices. Now we can form A. • b, either choosing , , A. 1 uik~ or A. 1 VD?k , , • 1,2,3,4 • k=' • k=1 . The different orthogonal designs now arise by using G and H. We further observe that if circulant Xl' X2 ' XJ• X4 of order t, exist with first rows ail' a i2 , ... , a it respectively. then the circulant matrices !l' !2' .!3'!.c of order 2t with first - 239 -where (i) (ii) (iii) (iv) a ij denotes -aij , satisfy the • . T ! lii!1 - 2tI2t '=1 lil J = (xl + x 2)J ~2J li3J - (x3 + x 4)J , ~J !, .~ - 0 • , • j • r X. is a (l,-l)-matrix 1=1 -l. = similar equations (x, x 2)J (x3 - x 4)J , We consider the excess of the different orthogonal designs 01 , O2, 0 3 , D4 and find for 0i YIY1i + Y2Y2i + Y3Y3i + Y4'l4i - 240 -Using' the two methods for form.inq Ai we obtain Hence we obtain To see both are needed we consider t _ 5 .. 22 + 12 + 0 -+ 0 , a(20) '" 20 max{4,3) .. 80, a(28) '" 28 max{4,5) = 140. (4' > and when the Ai are oooBtrueted using' U and G or V and H we ba~ - 241 -LEMMA 1. SWpp08i1 the matri"E!8 Xl' X2 ' x,, X, dilscl'ibed abovfI (2) e:x:is-t foro order lJith , , , , Then t t .. Xl + x, + x3 + K, , thlll'e i8 an Ol'thouona1. design 0 of order 4t aM type (t,t,t,t) with a (4tl ;a 0(0) 4t max(2x1 , x1+x2+x3+x4), ",here x, > x, > x, :. x," Fut'ther if t .. 8 2 and x, = -, 0(0) '" 8s3 and a (Yi ) .. 2s3. Aho o(Bt) :. 16t(x1 + x 2}. A similar process, using 4 circulant (l,-l)-matrices A, B, C, D of order t with row S\lIQ8 Xl' x 2' x 3' x, allows us to say; LEMMA 2. Let n=4t""12+X22+X32+X,2;;:4(DlOdB) be the O1'd8Z> of an H<%damard lIIat:ri:r, H, co.ult:ruc-ted !<Ising lola' circu'/:ant matrices &lith 1'0101 8Wi'!e Xl' Xl' x3 ' x, respective1.y, Xl ~ x2 ~ x3 :. x," ThBn Q' (HI and a(28) • and in same suitable first rows to use in Lemmas 1 and 2 are giVen in Table 1, others can be found in J. Wallis [3J. SO we have 0(12) ~ 36, 0(16);t 64. o(20);t 80 and 0(24) l! 96. - 242 -t , • 5 , W(H) " 160 240 '" o(Hl " 64 BO " A 111 -111 l-ll-111-1-Table 1 B 1---111 11--1 1-1111 In particular, using a l .. a 2 = a] .. a 4 .. 2 n = • we obtain { o(22r J Jr-l > 2 , (5) O{22r+l) > 2Jr • We also ••• j 0"(22X'.J} > 2 3r- 1 .9. (6) o (22r+l. 3) > 2 3r+2 .9 • { o(22r .S} > 23r+1 .S , (7) O(22r+l.S) > 2 3r+1 .15 . - 243 -c D 1-- 1---Ill -111 -1111 -1111 11---- 1--1-1 =d H of order If P = l(mod 4) is a ·prime power, then p ~ x2 + y2. Moreover, a suitable Hadamard matrix can be found [3, p.3lS} of order 2 (P+l) with rOW' sums x + 1 , " ", .. ~. 0' (2 (p+l)) l; 2 (p+l) (x+y) , which gives ·0'(12) l; 36, g(28);J: 140, 0'(52);J: 364 and 0'(60. "" 420. 2. An Upper Bound. y • Let &i l; 0 be the column sum of the ith column of a weiqhill9' matrix, W, of order n and weiqht k. '!'hen wwr = kIn Let e be the l><n IQatrix of ones. Then eww'teT ekI e'r n nk ("1 ... anlJ n 2 ! a .'-1 • - 244 -Now to find the maximum excess a (W) we wish to Il\aximize , , 1 " 1=1 subject ta 1 , a i • nk. We use Schwarz' inequality i=l to see that the maximum is for " " so the maximum excess is a(w) '1 " ,-, " ,'" . Since this is the maximum, we have for any weighing matrix and Haddmard matrix (" a(W) " a{nl " nln • This impraves Schmidt and Wang's bound (1) and is that given by Best. In particular, we have for Hadamard matrices of order 452 Compare this with (of). NOW one Qf us (SeberryJ has shown: THEOREM 3. Given an!! natural numbsl' q > 3 thel'lII elf:ist/l a l'Bfj'IoIZal' lI!!mmet-l'i" Hadamal'd mat-l'ilf: ",ith constant diafj'onal. of Ol'dBl' 22sq 2 fol' everl/ 0$ a [? 10g2 {q-3Q • - 245 -So we have, using & result of Best: COROLLARY 4. Let q > 3 be an!l natural. number and S l': 2!:092(q-3l] be an inteu6r. Then If Ik is not an integer we use a lattice point nearest (aI' •.. , an) .. Ik(l, .•• , 1). For the next discussion we confine ourselves to k .. n. Then, with x the greatest even integer <.;n, " • ", • • " • t • • nd &i+l a i +2 a t + • , n where t • x if In-x'i < I (X+2)2_nl and t .. x-2 otherwise. - , , , t' ., " " < n and , , (t+4) 2 a i +1 • an > n . n , it2 + Cn-i) (t+4)2 , SO ! 'j n j=l that ,. i • n(t+4)2 _ nJ 8(t+2j This value of i may not be integral and must be an intege~ so we - 246 -choose the integer part of ~((t+4)2 - III 8{t+2) for i. increase I aj by choosing i smaller. Thus 0' (n) Hence '" WIn) n ! 'J' j:l ti + (t+4) (n-i) , ~((t+4)2 _ nlt+4) - 4L 8(t+2J \n{t+n+4) -We note we n~ • where x is the greatest integer ,,0 (mod 2) and s /ii, t .. x if In-x2 1 < I (x+2)2-nl and t .. x-2 otherwise. 3. A lower Bound. An examination of Best's lower bound shoWs that we can exclude the rows of the Hadamard matrix of order n and so • obtain a(n) While we were always able to improve on this bound constructively, it is, nevertheless, the best general lower bound available. - 247 -f. The Po",eFB of 2. Schillidt and Wang noted that if H and G are Hadamard matrices of orders h and 9, .respectively, and maxima.l weights, then (10) W (hq) '" w (HxG) THEOREM 5. i.e. Proof. Let Xl=[l], X2 .. xl .. X4 .. {OJ be circulant matrices of order 1. Then by repeated use of the doubling construction of section 1 we obtain four circulant matrices of order t. 22r-2 with row sums 2r-l, 0, 0, 0 respectively and by ()) we have By equation (8) we have So we have the result. - 248 -LEMMA 6. , 22«t + 4 + 22r+l) _ 2 2 (t+4)-~2r-2 ( 2 t + 2 or odd. Proof. The right-hand-side fOllows immediately from (9). Also we have Since we know W(22S) exactly, we use a" 2p±l when 2r+l '" 4pil in (10) to obtain _ 24r+l- [>, (r+l)j • Now we use (5) to obtain the quoted lower bound. COROLLARY 7. Bounds on W(22r+1) arB gil'''" by the tabte - 249 -r x < Wl22r+1) < y 1 " w(a) 2 '" < W(l2) < GOO J 8832 < 1'1'(128) < 8910 • 136192 < W(512) < 136854 5 2139112 < W(2048) < 2143466 Proof. The lower bounds for 1'1(32) and W(l2S) are found using W(S) and (111. while those for W(S12) and W(2048) are found using W{l2} and (11). S. The EZCBS8 of Orthogonal Delligna. From the discussion at the beginning of section 2 we see that II. weighing matrix of order n with k non-zero entries per rOll' has excess nIX. 'l'hus an orthogonal desi9'n D'" Y1Y1 + YZYZ + •.• + YtY t of type {51' 52' •.•• Btl would have maximal excess while the orthogonal design ~ = y (Y1 + ••• + Yt) obtained by equating all the variables has maxiroal excess - 250 -Clearly, in general, a (E,J niSI + ••• + St :s; nIB"! + ••• + nrs;;- • So the question arises: QUESTION. Let D "" YIY1 + ••• + YtYt be an o1Othogonat. deeign of O1Oa61O n and type (sl' ••• , Stl on the <lommuting va1Oiabt.se Y1' ••• , Yt' Fo1O "'hat types ana opaere of orthouona~ deeign ie the e~asee the earns as the e~<leee of the "'6iuhing matri~ of oPde1O As a partial answer we see the following orthogonal deSigns of orders 4 and 8 and types (1,1,1,1) and (1,1,1,1,1,1,1,1), respectively, have the same excess as the underlying Hadamard matrix. [~ b c -~ and • d -d a c -b -. b c d b • d -c c -d • b d c -b a e -f g -h fe-h-g 9 -h -e f h g- f e - 251 -• f g h f -. h -g -g h • -f h g -f -. a b-c d -b a d c -d a b -d -c -b a Replacing the variables of this orthogonal design of order eight by circulant matrices with first rows 111, for a and band -11 for c, d, e, f, g, h gives an Hadamard matrix of order 24 with 0(24) = 108. Replacinq the variables by the circulant JDatrices with first rows 1111111 for a and 111-1-- for b, c, d, e, f, q, h gives an Hadamard matrix of order 56 with 0(56) _ 392). Also the orthogonal designs of order 4t, t odd and square free, and type (t,t,t,t) constructed via Lemma 1 bave the same excess as the underlying Hadamard matrix for t 3, 5, 7 and· 13 and qive the best known lower bound for t 11 and· 15. This suggests another possible conjecture:-014t) Where and squa.re-free. , x, • x ' 3 , + x4 ' , odd Previously wallis had found W(8) ~ 42, W(l2) .. 90 and W(20) '" 240. Theorem 5 of Wallis 12l gave W(16) ~ 154. The results of Wallis (4J and Schmidt and Wanq [2] qave - 252 -334 ,; W(24) ,; 360 756 ,; W(36) 866 ,; W(40) ,; 960 . Best found W(24) 344. We find Best's conjecture fo< n cr(n) Wen) cr (n) • , 8 20 20 12 36 " 16 64 64 20 80 80 24 108 342 112 28 140 462 140 32 176 600 592 36 216 756 216 40 240 '" 240 .. 264 1100 1100 .. 312 1308 1.2'6 52 364 1534 36. 56 392 1.764 392 1.764 60 420 2010 201.0 64 512 23-04 512 Best conjectured - 253 -Calculated WIn) 3 " " 160 240 344 '" , W(32) , 600 756 920 , W(44) , 1108 , W(48) , 1312 1.534 , W(56) , 1774 , W(60) , 2030 2304 I, (n~ + I,n [2m]) , n - O(mod 8), WIn) l,(n2 + o(n)) l,(n2 + n[rnJ) , n - 4(mod 8). comparison with the above results shows this is an excellent conjecture. References ell M.R. Best, The ezoess of a Hadama7'd ma:t;7'iz, Indag. Math., 39 (1977), 357-361. {2] R.W. Schmidt and Edward T.H. Wang, The weights of Hadama7'd matrices. J. combinatorial Theory, Ser. A (to appear). [3] Jennifer Seberry Wallis, Hadama7'd matriass, Room squares. Sum-free sets, Hadamard matrices, by W.D. Wallis, Anne Penfold Street, Jennifer Seberry Walais, in Lecture Notes in Mathematics, Vol. 292, Springer-Verlag, Berlin-Heidelberg-New York, 1972, p.273-4B9. [4] W.D. Wallis, On the weights of Hadamard matrices, Ars Combinatoria, 3 (1977), 2B7-292. University of Sydney - 254 -

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A remark on the excess of Hadamard matrices and orthogonal designs

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