Article Cyclic difference sets with parameters (511, 255, 127) BACHER, Roland Reference BACHER, Roland. Cyclic difference sets with parameters (511, 255, 127). Enseignement mathématique, 1994, vol. 40, no. 1/2, p. 187-192 DOI : 10.5169/seals-61110 Available at: http://archive-ouverte.unige.ch/unige:12831 Disclaimer: layout of this document may differ from the published version.
L'Enseignement Mathématique Bacher, Roland CYCLIC DIFFERENCE SETS WITH PARAMETERS (511, 255, 127) Persistenter Link: http://dx.doi.org/10.5169/seals-61110 L'Enseignement Mathématique, Vol.40 (1994) PDF erstellt am: 7 déc. 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
CYCLIC DIFFERENCE SETS WITH PARAMETERS (511, 255, 127) by Roland Bacher This note describes the (équivalence classes of) cyclic différence sets with parameters (511, 255, 127). There are five non-isomorphic such classes which are listed at the end of this note. We give also a version of their triple intersection polynomial (our version differs by a numerical factor from the polynomial used in [EK]). The first and second belong to known families and are obtained by a géométrie construction and by the GMW-construction described (for example) in [EK]. The last three appear to be exotic. The technique used to produce ail cyclic différence set with the above parameters is the same as in [C]. It uses heavy computer calculations and we will only sketch it. Let v, k, X be positive integers. Dénote by C v the cyclic group of order v which we identify with the additive group of the ring Z/uZ. A cyclic différence set with parameters (v, k, X) is a subset D of cardinality k in C v such that every élément c & 0 in C v can be written in exactly X différent ways as with d x, d2d 2 in D. We identify D with its characteristic function. Hence D can be considered as an élément of the group ring ZC V. Dénote by o: CC v -+ C v the automorphism of C u which sends ce C v to its inverse -c. For an élément Xof ZC V set X= o(x). An élément X ezc u is a cyclic différence set if and only if ail coefficients of X belong to the set {0, 1 } and X satisfies the équation for appropriate X and k. Différence sets are preserved under translation in C v and under automorphisms of C v. Two différence sets related by a translation and an automorphism are considered équivalent.
For / prime to v we define an l-orbit X in C v as an orbit of the auto morphismcp/ :Cv -» C v which sends ce C u to le. The multiplier theorem (see e.g. section 1 in [EK]) shows that each cyclic différence set D r with parameters (511, 255, 127) is équivalent to a différence set D fixed under cp 2, i.e. Disa union of 2-orbits in CSUC 5U.We can hence restrict the search to such différence sets. Réduction modulo 73 shows that ail 2-orbits of CSUC 5U hâve 9 éléments, with 3 exceptions: Since we are searching for a différence set D of cardinality 255, we see that {0} cannot be in D and that exactly one of A, B is in D. The auto CSUC morphismcp 3 of 5U exchanges the sets A and B, hence we may suppose that A C D and B < D. In order to reduce the search further we use the fact that 511 = 7 73 is not a prime. Let us introduce the obvious surjective group homomorphisms 7i 7 : CC 5 n -» C7C 7 and 7i 73 : C sn -+ C73.C 73. For Me ZC 511 we set M7M 7 = tt 7 (M), M7M 7 = ti 7 (M), M73M 73 = 7i 73 (M) and M73M 73 = 7i 73 (M). If the coefficients of Me ZC D are constant on 2-orbits then so are the coefficients of M/, M/ for / = 7, 73. We take now for M a cyclic différence set D preserved by cp 2 and satisfying ACD. Clearly D6ZC 5n is an élément with ail coefficients constant on 2-orbits and in the set {0, I}. By considering the 2-orbits of C7C 7 we hâve C7C 7 =Xo uxx ux2 where X o = {o}, X! = {1,2,4}, X2X 2 = {3,5,6}. Hence we can write D7D 7 = Ix/X/. Ail 2-orbits of D except A contain 9 éléments. Since 2 is a square (mod 7), any 2-orbit of D distinct from A contributes 9 to x 0 if its éléments are in the kernel of the projection ti 7 or 3 to X\ or x2x 2 according to whether it consists of squares or non-squares (mod 7). Moreover A e D contributes 1 to X\. Ail coefficients of D are either 0 or 1 and their sum is equal to 255, hence xx o,xi,x 2 are positive integers not greater than 73 such that Ko = 0 (mod 9), X\ = 1 (mod 3), x2x 2 =0 (mod 3) and x 0 + 3x\ + 3x 2 = 255. Since the différence set D satisfies the équation in ZC 5n5 n the projection D7D 7 satisfies The only solutions in ZC 7 satisfying ail the above requirements are
For C73C 73 we hâve C73C 73 = = o y j where r 0 = {0} and is the décomposition of C73C 73 into 2-orbits. We are searching for positive integers yo,y 0,...,7s not greater than 7 such that Dl3D 13 = E^o-K/*}- We already know that D contains a unique 2-orbit of cardinality 3 and that D does not contain 0. This implies that y 0 =3. Moreover we hâve y 0 +9 y and D73D 73 = E^/7/ satisfies 8 8y = yj = 255 x A small computer program gives (up to multiplication by an invertible élé C73)C mentin 73 ) the following four solutions (51),..,(54) with y 0 =3, and (SI) (S 2) (53) (54) Hence every cyclic différence set with parameters (511,255,127) is équivalent to a différence set D which projects onto one of the above 2 solutions in C7C 7 and onto one of the above 4 solutions in C73.C 73. One can reduce the number of cases somewhat more by considering the orbit of a différence set D under multiplication by 74 which is an invertible square (mod 7) (and hence keeps A fixed) and the identity (mod 73). There remain more than 10 10 unions of 2-orbits which hâve thèse projections. A computer program written in FORTRAN running about 50 hours on a SUN-workstation found (up to isomorphism) exactly five différence sets denoted by D\,.., D 5. The notation D t = (#i,.., ai) means that the différence set D t is the union of
the 2-orbits generated by a x,.., a t. The polynomial P t (x) associated to D t is defined by where D t + a dénotes the translate of D t by a. One easily checks that the usual triple intersection polynomial is just a scalar multiple of the one above. Remark : For a différence set this polynomial does not dépend on the choice of the fixed first block. For a symmetric design whose automorphism group is not transitive on the blocks it does in gênerai. So this gives a condition fulfilled by a symmetric design coming from a différence set. Two différence sets project onto (SI): Thèse two correspond to known constructions and give factorizations of the éléments D-(y) in the ring Z[y]/(y 5U - 1) where We also give a factorization of D-(y) in Z[y]/(y 5U -1) (the factorization is not unique). The first such différence set is D[(y) factorizes in Z[y]/(y 5U -I)as The second such différence set is
D' 2 {y) factorizes in Z[y]/(y 5U -I)as Two différence sets project onto (52): and No différence set projects onto ( 3). One différence set projects onto (S 4): Acknowledgements. This note owes more to S. Eliahou and M. Kervaire than this short sentence can suggest. I also thank the Fonds National suisse pour la Recherche Scientifique for a grant during this work.
REFERENCES [C] Cheng, U. Exhaustive Construction of (255, 127, 63)-Cyclic Différence Sets. /. Combin. Theory Ser. A 35 (1982), 115-125. [EK] Eliahou, S. and M. Kervaire. Barker Séquences and Différence Sets. L'Ensei gnementmathématique 38 (1992), 345-382. (Reçu le 21 mars 1994) Roland Bâcher Section de Mathématiques Université de Genève C.P. 240 1211 Genève 24 (Switzerland)