Signings of group divisible designs and projective planes

"We investigateof the all-ones matrix .]V for1) 16 over va,rious admissible groups. For 1) 16 allhave been PIlumera,ted. For the case 7) 16 only some classes have been completely enumerated. Of particularare signings over the group since these can alsobe considHed as GDD(16 X 2,16,these overexpanding,GDD(16 4,16,4)'s. We continne in this way until we obtainG DD(16 16,16, 1)'s, which can be extended to projective planes of order 16.\Ve discuss the properties of classes of G DD's obtained this well identifying the projective planes obtained at the final stage. 1. Definitions and Background A balanced incomplete block design, BIBD(v, b, 1', k, A), is an arrangement of v elementsinto b blocks such that (i) each element appears in exactly r blocks; (ii) each blockcontains exactly k elements; and (iii) each pair of distinct elem,ents appear together inA blocks. Well known necessary conditions for a BlBD(v, b, r, k, A) to existare vr bk and A( vI) = r( k 1). Because of this dependence we shall usethe abbreviated notation BIBD(v,k,A) to denote a BIBD(v,b,r,k,A). A BIBD isnontrivial if :3 k < v. A symmetric BIBD (SBIBD) is a BIBD with v b. Aprojective plane of order v is an SBIBD( v 2+ v + 1, v + 1, 1).Two BIBD( v, k, A)'S, with element sets VI and 112 respectively, are said to be iso-morphic if there is a bijection () : VI --t 112 which preserves blocks. An automorphismof a BlBD is an isomorphism of the BIBD with itself. The set of all automorphisms,under the usual composition of mappings, forms the a71tom07'phism group of theBlBD. Australasian Journal of Combinatorics 11( 1995), pp.79-104 A gf 1I.cralizcd Bhaskar Rao (G BRD)follows. Let VV be av x b matrix with dements from 0 u {a}, where Gfinite group of order y, withelement f. '['hen Hl canT¥ hI A I + +. -\-.,., Ag are u x hIladamardx a for any i j. (That is, for i j, no 1 of occurs in the saHW position 1 ofAj ). Denote }V+ the transpose ofhl 1 Al -\... + h.;lAgand kt.V A1 + .. AI' Then vr isBha.-;kar Rao denoted byOBRD(v. h. r, k,"\: if (i) }V W + r (1 + ,,\ / 9 ( hI + ... + h9 ) ( J 1 L and (ii)( )' ,\ ) I-\-,,\ J . The second conditionthat N be the incidence matrix ofBIBD(v,b,T,k,"\). Because of the parameter dependencies for BIBD's mentionedabove we shall use the shorter notation GBRD( v, k, A: G) for a generalized Bhaskar RaoAk, A; with v b is aGBRD orweighingmatri:r. If VV has no a entries then the GERD is also known a generalized Hadamardmatri:r or GHAt. A GHAl the group is a Hadamard matrix provided themotivation forOHMs). Producing a GBRD(v, k, from aBTBD(v, k, A)referred to as signing the BlBD over the group G.A group-divi8ibh design, ODD(v x g,k,A) is an incidence structure B) con-sisting of a sd X,vg, partitioned into v disjoint g-8ubsets (groups), X =Xl U ... U and coIled ion B of k-subsets of X (blocks) such that: (i) Each point .r E X is incident with r blocks. (ii) 1 L nXi 1 for every block L E Band i 1, .. " v. A blocks incident with x and y. If bg then bk = rv and >..g( v-I) r( k 1). Note that the groups in the above definition correspond to subsets rather than algebraic groups.An incidence structure (X, B) (which can be a BIBD or a. ODD) is said to bert8oh'(lblr if there exists a partition R of the set of blocks B into subsets R l , ... , R'11'ca.lled JHlralld da88fs, such that each Ri is a partition of X. It is well known that athe dua.l of a. syrnmetric ODD (where v = band r k) is again a ODD, which meansthat the blocks of the origina.l CDD partition into v sets of 9 disjoint blocks each (corresponding to groups in the dua.l) constituting a resolution. Hence symmetricODD and its dual are both resolvable. In this paper the property of resolvability isused in the final of obtaining a projective plane of order 16 from GDD(16 x 16,16,1).A (iDD(v x g,k,A) with 9 1 is a BIBD(v,k,"\). ODD isomorphism is definedin the same way as BIBD isomorphism.F'rom a. GBRD(v, k, A; 0), IGI= g, we can form a ODD(vxg, k, A/g) as follows. For(tllY h E G let 1\ denote the corresponding 9 x 9 permutation matrix, PhI + ... + Ph g =