Cyclic difference sets with parameters (511, 255, 127)
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(2) 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 [email protected] http://retro.seals.ch.
(3) CYCLIC DIFFERENCE SETS WITH PARAMETERS (511, 255,. 127). Roland Bacher. by. 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.. différence. with the above heavy computer calculations and we. The technique used to produce ail cyclic. parameters is the same will only sketch it. Let. in [C]. It uses. as. positive integers. Dénote by C the cyclic group of order v identify with the additive group of the ring Z/uZ. A cyclic. v, k, X be. which. we. v. différence set with parameters (v, k, such that. ways. with. set. every élément. c. in. & 0. C. is. X) v. a. subset D of. can be. cardinality. written in exactly. X. k in C. v. différent. as. in D. We. identify D with. characteristic function. Hence D can be considered as an élément of the group ring ZC Dénote by o: CC -+ C the automorphism of C which sends ce C to its inverse -c. For an élément Xof ZC set X= o(X). An élément X eZC is a cyclic différence set if and only if ail coefficients of X belong to the set {0, } and X satisfies the équation d. d2d. x. ,. 2. its. V. u. .. v. v. v. V. u. 1. for appropriate X and k. Différence sets are preserved under translation in C and under automorphisms of C Two différence sets related by a translation and an automorphism are considered équivalent.. v. v. ..
(4) For. define an l-orbit X in C which sends ce C to le.. orbit of the auto The multiplier theorem morphismcp/ :Cv -» C in [EK]) shows that each cyclic différence set D with (see e.g. section parameters (511, 255, 127) is équivalent to a différence set D fixed under cp i.e. Disa union of 2-orbits in 5U .We can hence restrict the search to such différence sets. Réduction modulo 73 shows that ail 2-orbits of 5U hâve 9 éléments, with prime to. /. v. we. as. v. v. an. u. r. 1. 2. ,. CSUC. CSUC. exceptions:. 3. Since. that {0} cannot morphismcp of. be CSUC. 3. that A. D and B. C. of cardinality 255, we see in D and that exactly one of A, B is in D. The auto 5U exchanges the sets A and B, hence we may suppose. for. searching. are. we. <£. différence. a. D. set. D.. In order to reduce the search further we use the fact that 511 =. 7. 73 is. •. introduce the obvious surjective group homomorphisms CC For Me ZC 511 we set M7M = tt (M), and 7i 73 C sn -+ 7i 73 n -» M7M If the coefficients of and = ti (M), 73 = 7i 73 (M). 73 = 7i 73 (M) Me ZC 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 and satisfying ACD. Clearly D6ZC 5n is an élément with ail coefficients constant on 2-orbits and in the set {0, I}. not. prime. Let. a. C7C. :. 7. 5. 7. us. C73.C. :. 7. 7. M73M. M73M. 7. 7. .. D. 2. we hâve considering the 2-orbits of =Xo uXx uX2 where = Ix/X/. X = {o}, X! = {1,2,4}, X2X = {3,5,6}. Hence we can write 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 if its éléments are in the kernel of the projection ti or 3 to X\ or according to whether it. By. C7C. C7C. 7. 7. D7D. 2. o. 7. 0. x2x. 7. 2. consists of squares or non-squares (mod 7). Moreover A e D contributes and their sum is equal Ail coefficients of D are either 0 or to X\ to 255, hence xx ,Xi,x are positive integers not greater than 73 such 1. 1. .. 2. o. that +. in. =. Ko. 0. (mod. 9),. X\. =. 1. (mod. 3),. x2x. 2. =0. (mod. 3). and. 3x 2 = 255. Since the différence set D satisfies the équation. ZC. 5n5. n. the. projection. D7D. The only solutions in ZC. 7. 7. satisfies. satisfying ail the above requirements are. x0 +. 3x\.
(5) For. is. C73C. we hâve. 73. C73C. 73. =. =. décomposition of. the. C73C. y j where. o. into. 73. r. 0. =. 2-orbits.. {0} and. We. are. 13 = integers ...,7s not greater than 7 such that know that D contains a unique 2-orbit of cardinality Dl3D. yO,y. contain and. D73D. 0 ,. This implies that y = E^/7/ satisfies. 0.. 73. 0. =3. Moreover. for. positive We already. searching. E^o-K/*}3. we hâve y. and that D does not 0. +9. £. 8 y 8y. =. x. yj. = 255. A small computer program gives (up to multiplication by an invertible élé the following four solutions (51), ..,(54) with y 0 =3, and mentin 73 ) C73)C. (SI) (S 2). (53) (54). Hence. équivalent 2 solutions reduce the. différence. différence set with parameters (511,255,127) is to a différence set D which projects onto one of the above in and onto one of the above 4 solutions in One can 73 number of cases somewhat more by considering the orbit of a set D under multiplication by 74 which is an invertible square cyclic. every. C7C. C73.C. 7. .. (and hence keeps A fixed) and the identity (mod 73). There remain 10 unions of 2-orbits which hâve thèse projections. A computer more than 10. (mod. 7). program written in FORTRAN running about 50 hours on a SUN-workstation found (up to isomorphism) exactly five différence sets denoted by D\ .., D The notation D = (#i .., ai) means that the différence set D is the union of ,. t. ,. t. 5. ..
(6) the 2-orbits generated by. a. x. ,. ..,. a. t. .. The polynomial P (x) associated to D t. t. is. defined by. where D. t. dénotes the translate of D. + a. t. by a.. One easily checks that the usual triple intersection. polynomial. is. just. a. scalar multiple of the one above.. Remark. 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. For. a. symmetric design coming from. a. 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. We also give. -. 1). a. where. factorization of D-(y). in. not unique). The first such différence set. D[(y) factorizes. in. Z[y]/(y. 5U. The second such différence set. is. -I)as. is. Z[y]/(y. 5U. -1). (the. factorization. is.
(7) 5U. -I)as. D' {y) factorizes in. Z[y]/(y. Two différence. project onto (52):. 2. sets. and. No différence set projects onto (£3). One différence set projects onto (S 4):. 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.. Acknowledgements.. This note owes more to. S..
(8) REFERENCES [C]. [EK]. Cheng, U. Exhaustive Construction of (255, 127, 63)-Cyclic Différence Sets. /. Combin. Theory Ser. A 35 (1982), 115-125. 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. Université. Mathématiques de. Genève. C.P. 240 1211. Genève 24 (Switzerland).
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