Circulant modular Hadamard matrices

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(1)Article. Circulant modular Hadamard matrices. ELIAHOU, Shalom, KERVAIRE, Michel. Reference ELIAHOU, Shalom, KERVAIRE, Michel. Circulant modular Hadamard matrices. Enseignement mathématique, 2001, vol. 47, no. 1/2, p. 103-114. Available at: http://archive-ouverte.unige.ch/unige:12419 Disclaimer: layout of this document may differ from the published version..

(2) L'Enseignement Mathématique. Eliahou, Shalom / Kervaire, Michel. CIRCULANT MODULAR HADAMARD MATRICES. Persistenter Link: http://dx.doi.org/10.5169/seals-65430 L'Enseignement Mathématique, Vol.47 (2001). PDF erstellt am: 8 nov. 2010. Nutzungsbedingungen Mit dem Zugriff auf den vorliegenden Inhalt gelten die Nutzungsbedingungen als akzeptiert. Die angebotenen Dokumente stehen für nicht-kommerzielle Zwecke in Lehre, Forschung und für die private Nutzung frei zur Verfügung. Einzelne Dateien oder Ausdrucke aus diesem Angebot können zusammen mit diesen Nutzungsbedingungen und unter deren Einhaltung weitergegeben werden. Die Speicherung von Teilen des elektronischen Angebots auf anderen Servern ist nur mit vorheriger schriftlicher Genehmigung des Konsortiums der Schweizer Hochschulbibliotheken möglich. Die Rechte für diese und andere Nutzungsarten der Inhalte liegen beim Herausgeber bzw. beim Verlag.. SEALS Ein Dienst des Konsortiums der Schweizer Hochschulbibliotheken c/o ETH-Bibliothek, Rämistrasse 101, 8092 Zürich, Schweiz retro@seals.ch http://retro.seals.ch.

(3) CIRCULANT MODULAR HADAMARD MATRICES. Eliahou*). by Shalom. and. Michel Kervaire. Introduction. 1.. Besides Hadamard's conjecture proper, one of the major open problems concerning Hadamard matrices is Ryser's conjecture, according to which there cf. [R]. are no circulant Hadamard matrices of size greater than 4 ;. conjecture, it might prove interesting to consider the weaker notion of a circulant modular Hadamard matrix, and ask for what moduli and sizes such matrices happen to exist. In the light of Ryser's. On the one hand, one hopes that non-existence results may shed some light on Ryser's conjecture. On the other hand, computer expérimentation reveals. existence of families of modular circulant Hadamard matrices of given sizes and moduli with intriguing patterns. In the présent note we exhibit one. the. such infinité family with. Notation. xi. =±1. for ail. a. X=. Let. i=. 0,. (x o .. .. .. large modulus. m.. -i). be. ,x ,n. u. .. .. .. ,x n. -I.We. a. séquence of even size n, with. dénote the. £-th periodic corrélation. coefficient of X by. where the subscripts are read modulo n, for every. (0 <. k. k. <. n —. 1).. (X) is the dot product of X with its fc-th left shift a k (X), where a k (X) = (x k ,x k+u ,;fy +w _i), again with indices read modulo n. Observe that 7 o(X) =n and that -y n k (X) = -y k (X) for k= 1, -. Clearly,. j. k. .. .. .. .. .. .. ,. n-1.. *) During the préparation of this paper, the first author partially benefited from contract with the Fonds National Suisse pour la Recherche Scientifique.. a. research.

(4) A. circulant matrix circ(X). the shifts. a- k (X),. k — 0,. .. .. .. ,. associated to X. The rows of circ(X) of the séquence X.. is 1. n —. =0. Of course, the conjunction of the conditions 7^(X) is. équivalent. 7it(X). =. to. mod. 0. H. =. m. for. circ(X) ail. being. k. circulant. a. 1,...,|. =. means. for ail k—. Hadamard. 1,. .. .. .. ,. |. while. matrix,. // = circ(Z). that. are. is. an. m-modular circulant Hadamard matrix.. There are two trivial examples of circulant modular Hadamard matrices with a large modulus. One is the constant all-one matrix J = circ(l, 1). The matrix J is an n -modular circulant Hadamard matrix (CHM for short) .. .. .. ,. of size n.. other example is /' = circ(— 1, identity matrix of size n. Hère, /' (J'y an {n — 4) -modular CHM of size n. The. •. 1,...,1). = J. =ni+(n—. —. where. 21,. 4)(/. —. /). ,. /. is. whence J'. the is. order to exhibit interesting examples of circulant modular Hadamard matrices, we shall require that some of the corrélations jkQQ be actually zéro, In. not only zéro modulo. m.. Using the notation H{z) corrélations 7^ = 7^(X) arise. = as. Y!iô x i^ coefficients. n. £. in the. -. 1). as. usual,. the. identity. (1). where Z[z]/(z n —1) may be viewed of order n generated by z. The. as. Let. DEFINITION.. H=. n. is. if 7*(X) =. 2. .. 7». 0. in. for ail. this. be. treated. m-modular circulant Hadamard matrix if n / 2 00 —0• We say that H of type. be an. circ(X). We say that H. of even size. of type. the group ring ZC n of the cyclic group. formula suggests that it in the following (tentative) définition. position of. spécial. separately,. as. is. 1. k. + 0, |. 7n7. .. Ryser's conjecture amounts to saying that, in size greater than 4, there no circulant modular Hadamard matrix which is simultaneously of types and 2.. is 1.

(5) Even though the constraints for type 2 seem to be much stronger than the one for type 1, this may not necessarily be so. Consider, for example, the case of size. = 20 and modulus m = 16. Let. n. Then, quite surprisingly perhaps, circ(X) is X satisfles the equalities -y k (X) —0 for ail. 16-modular CHM of type. a. kÔ,. is z. 16-modular. no. =. as. -16.. 10, and 710 (X) =. However, it follows from formula (1) above that there CHM of type 1 in size 20. Indeed, for n = 20, substituting (1) with 710 = 0 yields H(\f = 20 + lk-. 2,. 1. in. formula. 2£Li. would imply (#(l)/) =5 is not a square modulo 8. Hence, the condition 7io(X) = 0 alone forbids the other corrélation coefficients of X, at positive indices k, to vanish simultaneously modulo 16. The same argument shows that for q odd with q mod 8 there is no 16-modular CHM of length Aq satisfying 72^ = 0 mod 32.. The condition 7* =0 mod 16 for mod 8, contradicting the fact that 5. =. fc. 1,. .. .. .. 2. 9. ,. 1. ,. exhibit (in the next section) a 4-parameter family of and of size 4p for (p — circulant Hadamard matrices of type mod 4 every prime number p such that this. In. note,. we. l)-. 1. p=l. .. circulant modular Hadamard matrices of type 2, it turns out that they can be obtained from a well known paper of Delsarte, Goethals and Seidel [DGS]. This is explained in Section 3. As to. - 1) -modular circulant Hadamard matrices of size 2.. A FAMILY OF (p. -. Let p be a prime satisfying p = 1 existence of (p - 1) circulant. mod. 4. We. 4p.. going to prove the Hadamard matrices of type 1 and size 4p. We give explicitly below the first row (x o ,x u ,x 4/? _i) of such a matrix as a polynomial H(z) = E^ô* x^ ZC V = Z[z]/(z 4p - 1), where ail .. are. .. coefficients x equal ±1 and H(z)H(z~ l ) =4p modulo to write down H(z) we need some notation. t. Let. -. U[p+. 1,2p which are prime to p. Note that if s square mod 2p. Indeed, if there exists then. So. C. (c + p). [1,/?. 2=2. =c. 2. 1]. + 2cp +. 2. pp. =s + 2cp. the. .. .. (p- l)ZC 4p. .. In order. of squares modulo 2p, is a square mod p, then s is also a 2 c such that =s +kp and k is odd, 1]. be. set. c2c. +(k+. p)p. =s. mod 2p..

(6) Let. Si. -I]U[p+. = ([l,/7. 1,2/7. - I])\S. of non-squares mod n[/7 + 1,2/?- I]| = Çp, so. be the. O. prime to p. We hâve |S o n[l,/7- I]| = |S o that 0 Similarly, |Si H [1,/7 - I]| - |Si. 2/7,. |So|S. and. |Si|. |. =P-1.. =p. —. 1. Mz) = Esest ?e 2 fi(—z ) — S.seSfC— l) is,. 1.. n[p+. 1,2/7. - I]|=. also.. Let/o(z) and/i(z). THEOREM. set. be the. Hall polynomials of. zc *p for «= 0, 5^- O ur objective. We shall need. 1. Let /o and f\. be as. and Si respectively. That. So. .. is. the. f(z. 2. ). independent parameters with values H(z) G ZC 4p = Z[z]/(z 4p -1) given by. has ail its coefficients of the monomials. £o5o. 5. £ jes/ &. and. proof of the following theorem.. defined above and let. 4. =. = =tl-. £i? £i-> £3. l,z,z. 2 ,. .. .. .. ,z. 1. 4/?. £\, £2, £3 be The polynomial. eo,. egwa/ to. and. ±1. satisfies the identity. for some polynomial. R{z). which the coefficient of. z. Z[z]/(z 4p —1). G. 2p. is. gi'vera. below in formula. (11). m. zéro.. exponents of z in H and 7? are to be read modulo 4p. We use 4p The — 1). (abusively) the terni "polynomial" for the éléments of Z[z]/(z assertion on the coefficients of H is easy to verify by direct observation and The. is. left to the reader.. parameter £0 is clearly the coefficient of the constant term in the displayed expression for H(z). The coefficient of z in H(z) is £\ on the condition that p = 1 mod 8. Indeed, in this case 2 is a square mod p. G So. Thus, the term Also 3/7+l is a square mod 2/7 and therefore. The. z= £*P+p. appears in. =5 coefficient (—l)^-. etfotf)?. .. If p. mod 8, then. G. Si. and. z. e?> = -j-£ 3 The first appearance appears in H(z) with the of £2 in H(z) dépends on the minimum of Si a number for which there is .. ,. no. known formula.. For the proof of the theorem we separate a preliminary part, which only dépends on symmetry properties of the set So, from the final calculation,. which properly dépends on the hypothesis that of quadratic residues mod p.. So. is. constructed from the. set.

(7) We first dérive the properties of. of the set. So. and its complément. dénote by. <p:. [1,/?. -I]U. hence. well. set. So. is. =. Si. —. x. l. ). coming from the symmetries. -I]U\p+. ([l,p. -I]->. [p+ 1,2p. defined by the formula (p(x) = 2p. Whenever the. H(z)H(z~ [lsjpl. p. 5j. - l])\5 0. 1,2/?. -I]U[p+. -1]. 1, 2/7. as. Si. the flip. .. stable under. tp. ,. the existence of. ip. So — > So. :. and. ,. Y^seSt z2s. implies the following properties of the sums YsesS-^^ for the sets Si with = °' l. :si—>. cp. We. .. ,. as. :. (2). This follows simply by applying the invorution. cp. For instance,. since. z. 4p. =1.. This means that /o(— 2. 2. z. ). and. /i(—. 2. z. ). 2. both self-reciprocal z~ ). The proof for the are. polynomials: /o(-z ) =/o(-~ ) and /i(— z ) =/i(— other formula (without the sign) is essentially the same. We also hâve. a. 2. "baker's flip" p, mapping [1,/?. —. 2. l]U[/?+. 2p —1]. 1,. onto. itself, defined by. If for. So. and S\ are stable under. /=. 0,. 1. p. ,. the existence of the. implies the following formulas. automorphisms p:. Si — > S;. :. (3). Hère we apply. ponSifl. [1,/?. —. I]. ,. and on. S. -. z. fl [p +. 1,. 2p. —. I]. .. We hâve.

(8) Remembering that. z. Ap. =1,. using the automorphism. ip. we obtain. as. above. Again, the proof for the formula without. the sign is the same.. As. corollary, we get. a. (4). obtained by observing that (1 +z 2p ) and (1 —z 2p ) both kill the above product. The first factor is killed by 1+ z 2p The second one by 1- z 2p It follows that 2-= (1 -f z 2p ) +(l -z 2p ) annihilâtes the left-hand side of (4), which must be 0 since 2 is not a zero-divisor in ZC 4j? .. .. .. We. can. hypothesis. begin the calculation of some terms in H(z)H(z *). Under the mod 4 of the theorem, -1 is a square mod p and —1 is. p=l. square mod 2p Therefore, p— 1 G Sq and it follows that So S\ stable by both involutions p, (p. The formulas (2), (3) and (4) apply.. also. a. .. ,. are. conséquence, we obtain that the coefficients of £o£2, £i£2, £o^3 l in H(z)H(z~ ) ail vanish. For instance, in the coefficient of £ o £3 and £i£ l in H(z)H(z~ ), which is As. a. 3. the products of. 1+. z. 2p. with. the products of Y^seSo***. 1-. witn. z. 2p. 1~. 2*2. P. *. S^C— l^z are 0. and with Sjgî^" l^^. and. Furthermore, also vanisn. -. The coefficients of the other terms 0. by the same arguments based on. of £2^3. e§ei, e\£ 2 and e\ £3 are seen to be formulas (2), (3) and (4). The coefficient. is. Although of a somewhat différent nature, 2p observing that z p + z~ p = z p (l + z ).. it. also. vanishes by formula (3),.

(9) x. The only remaining terms in H(z)H(z. are. ). H(z)H(z ) =C+ 0 ,i sq £\ calculation using formula (3) and the simple remarks l. We end up with an expression. An easy 2(1 +. 2p. z. ),. -. (1. f. 2p. z. -. = 2(1. Co,iC. .. z. 2p. +. z. 2p. 2. ). =. yields. ),. similarly. and. which require the computation of the two squares 2. and(/ (-z. )). 1. =(E^. 2. (-). 1. (/o(z. 2. (M~z. /. 2. )). =. 2. 0,. (f\(-z. ,. U-{\-. )). 2. 2. (ZêSo z^). =. -. 2. (/o(z. )). 2 ,. (fi(z. 2. )). 2 ,. For brevity, we use the notation. .. 1 .. Note first that hâve set. 2. )). 2. We shall actually need to calculate ail four quantities. for. (1. Xo+X. 0. T=E^ z. 2p. ), where 2. Observe that. T. z. z. 2u. -(1+ 2v Similarly, Y +Yl= U= E'Co^-l)^ ". +X. x. =. Y^Zq. z. z. =T-(1+. ). z. Y,l P^{~Wz 2v. o. .. 2p. 2p. ). ,. where we. -(1-. z. 2p. ). =. 2. .. =T. and. 2. z. -U.. U =. It. follows that. (5). We also hâve. -. (X o. -. X )T = \S O \T X. \S. {. \T. =0,. and thus. (6). remembering formula (3). 2. The main point is the calculation of (Ko —X\ the. ). ,. which. is. reminiscent of. familiar calculation with Gauss sums. Let. (— ). :. Z. —. {±1}. >•. be the. quadratic character if. at the. prime. p. extended. divisible by p, (-) = +1 if x, prime to p, is a quadratic residue modulo p (Le., x=y2 modulo p for some y) and (-) = —1 if x is prime to p and not a quadratic residue modulo p. We to the. are. integers. assuming. as. usual:. p=\. mod. (^). 4,. =. 0. x. is. and hence. (— ) =. 1 ..

(10) Notice (*±£). =. that. (|). -Xx. Xo. for ail. Y%£ (f)z 2x. = For. x.. ail. integers. (^X. (Eô. =. +. z. 2p. since. ). (^-) = (*)(*). hâve. we. jt,;y. 1. and. thus. Now, observe that. z. 2{t+p. The. F^. crucial. ,. 2p. For fixed xG F*. Summing over. Since. E yeF ;. coming back. This gives (7). us. for. x. the. to. a. the. side. right-hand. expression. J2 x. GF. *. is. well. (ïi^^x+y). defined, in. itself. without is. only. 1).. —. ,. that. is. as. y. runs over F*. y. =. =o,we. (P. 2t. Fp\{o}F. ambiguity even though (z. z. we hâve. point. defined modulo. +z 2p ) =. (l +z 2p ) for any integer t. It follows of integers [1,/?— 1] with F* = p \{0} by the natural. that, identifying the set. projection Z—». \l. 1. ,. and then for y. hâve. £y6. summation over. does. so. G. F*\{l},. (|) =. [l,p - I],. —yx; therefore. we get. -1. Using. (f). = +1, and.

(11) Combining this resuit with the équations (5) and (6), we. see that. from thèse équations the resuit. It is now easy to deduce. (8). Of course we would also like to hâve. analogue of équation (5). similar formula for. a. Yo,. Y\. The. is. 2 2 2s = 2pU It is observing that z U = -U, so that z U = (-1) U and and Y\ the dérivation of 0 easy, though somewhat boring, to imitate with 2t the formulas (5), (6) and (7). The needed assertion, that (l - z 2p ) only dépends on the class of t mod p is valid and the argument goes through. S. on. U2U. .. 707. (f)(-z. ,. The analogue of the above équation (8). is. (9). However, we can simply embed the ring ring of C 4p over the Gaussian integers Z[i],. ZC^ into Z[Ï\C4 P. i= (a/~Î),. ,. the. group. and then apply to. calculations of X o X\ the automorphism a of the ring Z[i][z]/(z 4/? -1) induced by a(z) = (\f-i)z. The substitution of (V^T)^ for z is compatible with z 4p =1 and a(X ) = Y i9 a(T) =U and a(z 2p ) = -z 2p The resuit is indeed formula (9) above.. the. ,. t. Using. r+£/. for H(z)H( z -. 1. ). =. 2. .. XX=o z^. =C+. Co,iC. 0. ,i. '. anc^ pl u gg m g. e. x. ,. mese expressions into the formula. we get. and. Finally, H(z)H(z~ (10). l. ). =Ap+(p-. l)R(z), where.

(12) Equivalently, this "remainder" R(z) can. written. be. (H) The (periodic) corrélations of H(z) in degrees = 2 mod 4 are strictly zéro. This includes in particular the corrélation of degree 2p. Hence, the modular. Hadamard matrix associated with the séquence (polynomial) of the Theorem is indeed of type 1 as asserted. The corrélations in degrees = 0 mod 4 are 2(p — 1). Note that the corrélation in degree p is 2(p— 1) sq £i because z p J rz~ p. Eîfe 2^". also appears in the sum. Remark. p—\ may be of type. + z". (2ï/. ". 1}. v=. for. ). .. It seems probable, from computer-assisted. maximum modulus for. the. and size. 1. 1. Ap. not always maximal. CHM of type. .. expérimentation, that modular circulant Hadamard matrix. a. However, the power of. as. the power of. 2. is. dividing p— dividing the modulus of 2. 1. certainly modular. a. There are many values of p (where p is prime and satisfies p = 9 mod 16) for which a variant of the formula for H(z) in the above Theorem yields a 16-modular CHM. The first few such values of p. and size Ap. 1. 233,. 89,. are p = 73,. .. On the other hand, it seems for example that. ..... indeed no 16-modular, type. CHM of size. 1. Ap. exists for p = 41. .. hope to corne back on the gênerai question of 16-modular circulant Hadamard matrices of type 1 in a future publication. We. 3.. Circulant modular Hadamard matrices of type. 2. In this section we produce circulant modular Hadamard matrices of type and size. 2(q-\- 1), where. n=. of such objects. is. a. 2. odd prime power. The existence theorem from the 1971 paper [DGS].. qisan arbitrary. corollary of. a. grateful to Roland Bâcher for valuable discussions about some unpublished work of his which helped in obtaining the following resuit. We. are. THEOREM. there (0. exists. </<n-. is a. 2. a. For every. n=. binary séquence. 2(q. X=. +1), (x 0,. where .. ... ,x n. l), such that k (X) =0 for ail k circulant modular Hadamard matrix of type. is. q. -i) 0, \ 2. with .. odd prime power,. an X[. —±1 for ail. In other words,. and size n.. circ(X). i.

(13) Set xn. Proof.. =x0 =1. séquence X== (jci,jc 2 ,. Y = (xi,x 2 ,...,X|_i).. .. .. and xa+. ,* n. -i). 7o(X) = n, jn_(X) =. We hâve. -x. =. t. i=. for ail. t. 1, 2,. .. .. .. ,. \-1.. The. therefore determined by its subsequence. is. n, and. -. 4. easily checked, where a* is the feth aperiodic 1 corrélation coefficient. Of course, j n - k QO = 7*PO for ail A; = 1, 2, In order to prove the theorem, it therefore sufflces to exhibit a binary ,**-i) of length § - 1 = q, satisfying the équation séquence Y = (x u x 2. fc=1,2,...,§-l. for ail. ,. .. .. as. ...,§-. .. .. |-1.. for every k= 1, 2, For this purpose, we recall the notion of a negacyclic matrix, introduced by Delsarte, Goethals and Seidel in their paper [DGS]. By définition it is simply a matrix of the form a k (Y). -. an__. k. (Y). =0. .. which we will dénote by NC(uq, u\,. Explicitly,. the entries. cc t. j. .. .. .. ,. .. ur. of the matrix. .. ,. ).. NC(u 0. u. ,. u. very easy to see that the binary séquence satisfies a k (Y) - a>±- k {Y) =0 for every k= 1, 2, It. C=. negacyclic matrix. CC. = (f. jVC(O,jci,. .. .. .. ,jc»_i). is. a. .. .. .. .. ,. Y=. is. .. .. ,. ur ). are. (xi,X2,. \-lif. .. .. .. >*§-i). and only if the. conférence matrix, that. is. if. - 1)/. Now, Delsarte, Goethals and Seidel hâve explicitly constructed negacyclic conférence matrices of every size of the form q+l, where q=pf with p an odd prime and. a. positive integer, in Section. 7. of [DGS]. Thèse negacyclic. équivalent to the usual Paley conférence matrices the quadratic character %: F* -» {±1} of the finite field F^. D. conférence based on. /. matrices. are. After having submitted the présent paper for publication, we came across the Thèse d'Habilitation of Philippe Langevin (Toulon). There, a concept which is closely related to our type 2 séquences is studied. P. Langevin uses the terminology "almost perfect séquences" and his treatment also relies on [DGS].. Note..

(14) Thus, we now find it préférable to drop the type 1 / type 2 terminology and rather call enhanced modular the modular matrices of type 1. We intend to use this new désignation in future publications on the subject.. BIBLIOGRAPHY J.M. Goethals and J.J. Seidel. Orthogonal matrices with zéro diagonal, IL Canad J. Math. 23 (1971), 816-832. RYSER, H. J. Combinatorial Mathematics. Carus Monograph 14. Math. Assoc. of America, 1963.. Delsarte,. [DGS] [R]. P.,. (Reçu. Shalom Eliahou. Département de Mathématiques LMPA Joseph Liouville Université du Littoral Côte d'Opale Bâtiment Poincaré 50, rue Ferdinand Buisson, B.P. 699 F-62228 Calais France. e-mail. :. eliahou@lmpa.univ-littoral.fr. Michel Kervaire Département de Mathématiques Université de Genève 2-4, rue du Lièvre B.P.. 240. CH-1211 Genève 24 Suisse. e-mail. :. Michel.Kervaire@math.unige.ch. le. 18. juillet 2000).

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