New Partial Geometric Difference Sets and Partial Geometric Difference Families
aa r X i v : . [ m a t h . C O ] D ec New Partial Geometric Difference Sets and Partial GeometricDifference Families
Jerod Michel ∗† Abstract
Olmez, in “Symmetric 1 -Designs and 1 -Difference Sets” (2014), introduced the concept of apartial geometric difference set (also referred to as a 1 -design), and showed that partial geometricdifference sets give partial geometric designs. Nowak et al., in “Partial Geometric Difference Fam-ilies” (2014), introduced the concept of a partial difference family, and showed that these also givepartial geometric designs. It was shown by Brouwer et al. in “Directed strongly regular graphsfrom 1 -designs” (2012) that directed strongly regular graphs can be obtained from partial geo-metric designs. In this correspondence we construct several families of partial geometric differencesets and partial difference families with new parameters, thereby giving directed strongly regulargraphs with new parameters. We also discuss some of the links between partially balanced designs,2-adesigns (which were recently coined by Cunsheng Ding in “Codes from Difference Sets” (2015)),and partial geometric designs, and make an investigation into when a 2-adesign is partial geometric. Key words and phrases : partial geometric design, cyclotomic class, directed strongly regular graph,partial geometric difference set, partial geometric difference family
Mathematics subject classifications : 05B10, 05B05, 05E30.
In this correspondence we discuss partial geometric difference sets and partial geometric differencefamilies, which were introduced by Olmez in [12] and Nowak et al. [10]. Here it was also shown thatpartial geometric difference sets and partial geometric difference families give partial geometric designs.It is clear that partial geometric designs have several applications in graph theory, coding theory andcryptography [3], [4], [11], [13]. It was shown by Brouwer et al. in [4] that directed strongly regulargraphs can be obtained from partial geometric designs. In [13] Olmez showed that certain partialgeometric difference sets can be used to construct plateaued functions. Recently, a generalization ofcombinatorial designs related to almost difference sets, namely the t-adesign , was introduced and theirapplications to constructing linear codes discussed [5], [6], [7]. In this correspondence we construct ∗ Corresponding author. Email Address: [email protected]. † J. Michel is with the School of Mathematics, Zhejiang University, Hangzhou 310027, China. An incidence structure is a pair ( V, B ) where V is a finite set of points and B is a finite set blockscomposed of points of V . For a given point u ∈ V , its replication number is the number of blocks of B in which it occurs, and is denoted by r u . Given two distinct points u, w ∈ V , their index is the numberof blocks in which they occur together, and is denoted r uw . A tactical configuration is an incidencestructure ( V, B ) where the cardinalities of blocks in B and the replication numbers of points in V areboth constant.Let ( V, B ) be a tactical configuration where | V | = v , each block has cardinality k , and each pointhas replication number r . For each point u ∈ V and each block b ∈ B , let s ( u, b ) denote the numberof flags ( w, c ) ∈ V × B such that w ∈ b \ { u } , u ∈ c and c = b . If there are integers α ′ and β ′ such that s ( u, b ) = ( α ′ , if u ∈ b,β ′ , otherwise , as ( u, b ) runs over V × B , then we say that ( V, B ) is a partial geometric design with parameters( v, k, r ; α ′ , β ′ ). Let G be a finite (additive) Abelian group and S ⊂ G . We denote the multiset [ x − y | x, y ∈ S ] by∆( S ). For a family S = { S , ..., S n } of subsets of G , we denote the multiset union F ni =1 ∆( S i ) by ∆( S ).For a subset S ⊂ G we denote |{ ( x, y ) ∈ S × S | z = x − y }| by δ S ( z ). For a family S = { S , ..., S n } of subsets of G we denote |{ ( x, y ) ∈ S i × S i | x = x − y }| by δ i ( z ).Let v, k and n be integers with v > k >
2. Let G be a group of order v . Let S = { S , ..., S n } bea collection of distinct k -subsets of G . If there are constants α, β such that for each x ∈ G and each i ∈ { , ..., n } , n X i =1 X y ∈ S i δ S i ( x − y ) = ( α, if x ∈ S i ,β, otherwise , then we say S is a partial geometric difference family with parameters ( v, k, n ; α, β ). When n = 1 and S = { S } , we simply say that S is a partial geometric difference set with parameters are ( v, k ; α, β ).2gain let G be a group and S = { S , ..., S n } a collection of distinct k -subsets of G . We call theset of translates S ni =1 { S i + g | g ∈ G } the development of S , and denote it by Dev ( S ). The followingtheorem was proved in [10]. Theorem 2.1.
Let S = { S , ..., S n } be a collection of distinct k -subsets of a group G of order v . If S is a partial geometric difference family with parameters ( v, k, n ; α, β ) , then ( G, Dev ( S )) is a partialgeometric design with parameters ( v, k, kn ; α ′ , β ′ ) where α ′ = P ni =1 P y ∈ S i \{ x } ( δ S i ( x − y ) − for x / ∈ S ,and β ′ = P ni =1 P y ∈ S i δ S i ( x − y ) for x ∈ S (see Remark 2.1). Remark 2.1.
The parameters for the corresponding partial geometric designs seem to disagree inLemma . of [12] and Theorem of [10]. To see this, the reader should compare Definition . of[12] to Definition of [10]. In this correspondence all graphs are assumed to be loopless and simple. Let Γ be an undirectedgraph with v vertices. Let A denote the adjacency matrix of Γ. Then Γ is called strongly regular withparameters ( v, k, λ, µ ) if A = kI + λA + µ ( J − I − A ) and AJ = J A = kJ. A directed graph Γ with adjacency matrix A is said to be directed strongly regular with parameters( v, k, t, λ, µ ) if A = tI + λA + µ ( J − I − A ) and AJ = J A = kJ. The following theorems were proved in [4].
Theorem 2.2.
Let ( V, B ) be a tactical configuration, and let Γ be the directed graphs with vertex set V = { ( u, b ) ∈ V × B | u / ∈ V } and adjacency given by ( u, b ) → ( w, c ) if and only if u ∈ c. Then Γ is directed strongly regular if and only if ( V, B ) is partial geometric. Theorem 2.3.
Let ( V, B ) be a tactical configuration, and let Γ be the directed graphs with vertex set V = { ( u, b ) ∈ V × B | u ∈ V } and adjacency given by ( u, b ) → ( w, c ) if and only if ( u, b ) = ( w, c ) and u ∈ c. Then Γ is directed strongly regular if and only if ( V, B ) is partial geometric. .4 Group Ring Notation For any finite group G the group ring Z [ G ] is defined as the set of all formal sums of elements of G ,with coefficients in Z . The operations “+” and “ · ” on Z [ G ] are given by X g ∈ G a g g + X g ∈ G b g g = X g ∈ G ( a g + b g ) g and X g ∈ G a g g X h ∈ G b h h ! = X g,h ∈ G a g b h ( g + h ) . where are a g , b g ∈ Z .The group ring Z [ G ] is a ring with multiplicative identity = Id , and for any subset X ⊂ G , wedenote by X the sum P x ∈ X x , and we denote by X − the sum P x ∈ X ( − x ). Let q be a prime power, and γ ∈ F q primitive. The cyclotomic classes of order e are given by C ei = γ i h γ e i for i = 0 , , ..., e −
1. Define ( i, j ) = | C ei ∩ ( C ej + 1) | . It is easy to see there are at most e different cyclotomic numbers of order e . When it is clear from the context, we simply denote ( i, j ) e by ( i, j ). We will need the following lemma. Lemma 2.4. [9]
Let q = em + 1 be a prime power for some positive integers e and f . In the groupring Z [ F q ] we have C ei C ej = a ij + e − X k =0 ( j − i, k − i ) e C ek where a ij = f, if m is even and j = i,f, if m is odd and j = i + e , , otherwise. We will need the following lemmas.
Lemma 3.1. [2]
Let q be a prime power and let C i , for i = 0 , , ..., q denote the cyclotomic classes oforder q + 1 in F q . Then the cyclotomic numbers are given by (0 ,
0) = q − , ( i, i ) = ( i,
0) = (0 , i ) = 0 , ( i, j ) = 1 , ( i = j ) . emma 3.2. [10] Let p be a prime and let C i , for i = 0 , , ..., p denote the cyclotomic classes oforder p + 1 in F p . Let S i = C i ∪ { } for i = 0 , , ..., p . If x / ∈ S j then | ( x − S j ) ∩ C i | = 1 for each i ∈ { , , ..., p } \ { j } . The following is our first construction.
Theorem 3.3.
Let p be a prime and let C i , for i = 0 , , ..., p denote the cyclotomic classes of order p + 1 in F p . Let S i = C i ∪ { } for i = 0 , , ..., p . Let i ′ , j ′ ∈ { , , ..., p } be fixed with i ′ = j ′ . Let m ≡ mod be a positive integer, and define Σ l = l + { , , ..., m − } ⊂ Z m for l = 0 , . Then S i ′ j ′ = Σ × S i ′ ∪ Σ × S j ′ is a partial geometric difference set in ( Z m × F p , +) with parameters ( mp , mp ; ( m ) p ( p + 3) , m p ) .Proof. First note that S ∼ = F p , and S i ′ , S j ′ are both subgroups of ( F p , +). For each z ∈ F p and each i ∈ { , , .., p } we have δ S i = ( | S i | , if z ∈ S i , , otherwise , = ( p, if z ∈ S i , , otherwise . (1)Now suppose that ( h, z ) ∈ S i ′ j ′ . Then we have( h, z ) − S i ′ j ′ = ( Σ × S i ′ ∪ Σ × ( z − S j ′ ) , if h ∈ Σ , z ∈ S i ′ , Σ × ( z − S i ′ ) ∪ Σ × S j ′ , if h ∈ Σ , z ∈ S j ′ . (2)Denote the number of occurrences of u in ∆( S i ′ j ′ ) by n u . Then P ( h ′ ,z ′ ) ∈ S i ′ j ′ δ S i ′ j ′ (( h, z ) − ( h ′ , z ′ )) canbe written (P v ∈ P ×{ } n v + P v ∈ Σ × ( S i ′ \{ } ) n v + P v ∈ Σ × (( z − S j ′ ) \{ z } ) n v + P v ∈ Σ ×{ z } n v , if h ∈ Σ , z ∈ S i ′ , P v ∈ Σ ×{ } n v + P v ∈ Σ × ( S j ′ \{ } ) n v + P v ∈ Σ × (( z − S i ′ ) \{ z } ) n v + P v ∈ Σ ×{ z } n v , if h ∈ Σ , z ∈ S j ′ . which, by (2), in both cases gives α = m p + ( m ) ( p − p + ( m ) p + ( m ) p = ( m ) p ( p + 3).Now suppose that ( h, z ) / ∈ S i ′ j ′ . We have( h, z ) − S i ′ j ′ = Σ × ( z − S i ′ ) ∪ Σ × S j ′ , if h ∈ Σ , z ∈ S j ′ , Σ × S i ′ ∪ Σ × ( z − S j ′ ) , if h ∈ Σ , z ∈ S i ′ , Σ × ( z − S i ′ ) ∪ Σ × ( z − S j ′ ) , if h ∈ Σ , z / ∈ S i ′ ∪ S j ′ , Σ × ( z − S i ′ ) ∪ Σ × ( z − S j ′ ) if h ∈ Σ , z / ∈ S i ′ ∪ S j ′ . (3)Using Lemma 3.2, it is easy to see that ( h, w ), where h ∈ Σ , w ∈ ( z − S i ′ ) ∩ S j ′ and z / ∈ S i ′ ,appears m p times in ∆( S i ′ j ′ ), and each member of Σ × S j ′ appears m times. Similarly, (0 , w ), where5 ∈ S i ′ ∩ ( z − S j ′ ) and z / ∈ S j ′ , appears ( m ) p times in ∆( S i ′ j ′ ), and each member of Σ × S i ′ appears m times.We need to consider the two cases h ∈ Σ , z / ∈ S i ′ ∪ S j ′ and h ∈ Σ , z / ∈ S i ′ ∪ S j ′ . Notice∆( S i ′ j ′ ) = S i ′ j ′ S − i ′ j ′ = (Σ × S i ′ ∪ Σ × S j ′ )(Σ × S i ′ ∪ Σ × S j ′ ) − = m , S i ′ S − i ′ ) + m , S i ′ S − j ′ ) + m , S j ′ S − i ′ ) + m , S j ′ S − j ′ )= m , p ( S i ′ ∪ S j ′ )) + (Σ , S i ′ S − j ′ + S j ′ S − i ′ )Since C i ∩ C j = ∅ for i = j and f = p − p +1 is even, we have by Lemma 2.4 that C i C − j = C i C j = p X l =0 ( j − i, l − i ) C l = X l = i,j ( j − i, l − i ) C l . Thus, by using (3) and Lemma 3.1, we can see that in the case where h ∈ Σ and z / ∈ S i ′ ∪ S j ′ ,each member of Σ × ( z − S j ′ ) appears m times in ∆( S i ′ j ′ ), and a member ( h, w ), where h ∈ Σ and w ∈ ( z − S i ′ ) ∩ S j ′ , appears m p times. Similarly, in the case where h ∈ Σ and w / ∈ S i ′ ∪ S j ′ , each memberof Σ × ( z − S i ′ ) appears m times in ∆( S i ′ j ′ ), and a member ( h, w ), where h ∈ Σ and w ∈ S i ′ ∩ ( z − S j ′ ),appears m p times. Thus we have β = P ( h ′ ,z ′ ) ∈ S i ′ j ′ δ S i ′ j ′ (( h, z ) − ( h ′ , z ′ )) = ( m ) p + m ( m ) p = m p .We give another construction of partial geometric difference sets in products of Abelian groups. Theorem 3.4.
Let p be a prime and let C i , for i = 0 , , ..., p , denote the cyclotomic classes of order p + 1 in F p . Let S i = C i ∪ { } for i = 0 , , ..., p . Let i ′ , j ′ ∈ { , , ..., p } be fixed with i ′ = j ′ . Let { , } , { , } ⊂ Z . Then S i ′ j ′ = { , } × S i ′ ∪ { , } × S j ′ is a partial geometric difference set in ( Z × F p , +) with parameters (6 p , p ; 20 p, p ) .Proof. We have already established in (1) that δ S i ( z ) = ( p, if z ∈ S i , , otherwise . Now suppose that ( h, z ) ∈ S i ′ j ′ . Then we have( h, z ) − S i ′ j ′ = ( { , } × S i ′ ∪ { , } × ( z − S j ′ ) , if h ∈ { , } , z ∈ S i ′ , { , } × ( z − S i ′ ) ∪ { , } × S j ′ , if h ∈ { , } , z ∈ S j ′ . (4)If we denote the number of occurrences of u in ∆( S i ′ j ′ ) by n u then, using (4), we have α = X ( h ′ ,z ′ ) ∈ S i ′ j ′ δ S i ′ j ′ (( h, z ) − ( h ′ , z ′ )) = X v ∈{ , }×{ } n v + X v ∈{ , }× ( S i ′ \{ } ) n v + X v ∈{ , }× ( z − S j ′ ) n v = 8 p + 8 p + 4 p = 20 p. h, z ) − S i ′ j ′ depending on whether h is contained in { , } , { , } or { , } , and whether z is contained in S i ′ or S j ′ , or contained in neither. These are simple tocompute and we do not list them. Using Lemma 3.2 it is easy to see that ( h, w ), where h ∈ { , } and w ∈ ( z − S j ′ ) for z / ∈ S i ′ , appears 2 p times in ∆( S i ′ j ′ ), and each member of { , } × S j ′ appears 2 p times. The cases where h ∈ { , } , z ∈ S i ′ , where h ∈ { , } , z ∈ S j ′ , and where h ∈ { , } , z ∈ S i ′ , aresimilar to the previous case. We need to consider the cases where z / ∈ S i ′ ∪ S j ′ . Notice∆( S i ′ j ′ ) = S i ′ j ′ S − i ′ j ′ = ( { , } × S i ′ ∪ { , } × S j ′ )( { , } × S i ′ ∪ { , } × S j ′ ) − = 2( { , } , S i ′ S − i ′ ) + 2( { , } , S i ′ S − j ′ ) + 2( { , } , S j ′ S − i ′ ) + 2( { , } , S j ′ S − j ′ )= 2(( { , } , p ( S i ′ ∪ S j ′ )) + ( { , } , S i ′ S − j ′ ) + ( { , } , S j ′ S − i ′ ))Since C i ∩ C j = ∅ for i = j and f = p − p +1 is even, we have by Lemma 2.4 that C i C − j = C i C j = p X l =0 ( j − i, l − i ) C l = X l = i,j ( j − i, l − i ) C l . Thus, by using the expressions for ( h, z ) − S i ′ j ′ and Lemma 3.1, we can see that, in the case where h ∈ { , } and z / ∈ S i ′ ∪ S j ′ , each member of { , } × ( z − S j ′ ) appears twice in ∆( S i ′ j ′ ), and amember ( h, w ), where h ∈ { , } and w ∈ ( z − S i ′ ) ∩ S j ′ , appears 2 p times. The cases where h ∈{ , } , z / ∈ S i ′ ∪ S j ′ and where h ∈ { , } , z / ∈ S i ′ ∪ S j ′ are similar. Thus we can conclude that β = P ( h ′ ,z ′ ) ∈ S i ′ j ′ δ S i ′ j ′ (( h, z ) − ( h ′ , z ′ )) = 8 p .We next construct partial geometric designs from planar functions. For a more detailed introduc-tion to planar functions the reader is referred to [1] and [5].Let ( A, +) and ( B, +) be Abelian groups of order n and m respectively. Let f : A → B be afunction. One measure of the nonlinearity of f is given by P f = max = a ∈ A max b ∈ B P r ( f ( x + a ) − f ( x ) = b ),where P r ( E ) denotes the probability of the event E . The function f is said to have perfect nonlinearity if P f = m . The following lemma gives many examples of perfect nonlinear functions in finite fields. Lemma 3.5. [1]
The power function x s from F p m to F p m , where p is an odd prime, has perfectnonlinearity P f = p m for the following values of s :1. s = 2 ,2. s = p k + 1 , where m/gcd ( m, k ) is odd,3. s = (3 k + 1) / , where p = 3 , k is odd, and gcd ( m, k ) = 1 . We will use the following lemma. 7 emma 3.6. [1]
Let f be a function from an Abelian group ( A, +) of order n to another Abeliangroup ( B, +) of order n with perfect nonlinearity P f = n . Define C b = { x ∈ A | f ( x ) = b } and C = S b ∈ B { b } × C b ⊂ B × A . Then | C ∩ ( C + ( w , w )) | = n, if ( w , w ) = (0 , , , if w = 0 , w = 0 , , otherwise . The following is a construction.
Theorem 3.7.
Let f be a function from an Abelian group ( A, +) of order n to another Abeliangroup ( B, +) of order n with perfect nonlinearity P f = n . Define C b = { x ∈ A | f ( x ) = b } and C = S b ∈ B { b } × C b ⊂ B × A . Then C is a partial geometric difference set in A × B with parameters ( n , n ; 2 n − , n − .Proof. Suppose ( h, z ) ∈ C . Then we have( h, z ) − C = [ b ∈ B { h − b } × ( z − C b )= [ b ∈ B { h − b } × { z − x | z, x ∈ A, f ( x ) = b } . Denote the number of occurrences of u in ∆( C ) by n u . Define V = S b ∈ B \{ h } { h − b } × { } and V = S b ∈ B { h − b } × { z − x | x, z ∈ A, z = x, f ( x ) = b } . Then, using Lemma 3.6, we have X ( h ′ ,z ′ ) ∈ C δ (( h, z ) − ( h ′ , z ′ )) = n (0 , + X v ∈ V n v + X v ∈ V n v = n + 0 + ( n − n − . Now suppose ( h, z ) / ∈ C . Define U = S b ∈ B \{ f (0) } { h − b } × { z − x | x ∈ A, f ( x ) = b } . Then, usingLemma 3.6, we have X ( h ′ ,z ′ ) ∈ C δ (( h, z ) − ( h ′ , z ′ )) = n ( h − f (0) , + X v ∈ U n v = 0 + ( n − n − . Corollary 3.8.
Let f ( x ) = x s be a function from F p m to F p m , where p is an odd prime. Define C b = { x ∈ F p m | f ( x ) = b } and C = S b ∈ F pm { b } × C b ⊂ F p m × F p m . If:1. s = 2 , . s = p k + 1 , where m/gcd ( m, k ) is odd, or3. s = (3 k + 1) / , where p = 3 , k is odd, and gcd ( m, k ) = 1 .Then ( F p m × F p m , Dev ( C )) is a partial geometric difference set with parameters ( p m , p m ; 2 p m − , p m − . Remark 3.1.
Interestingly, the partial geometric difference sets constructed in Theorem 3.7 are almostdifference sets [1] and so correspond to planar -adesigns (see Section 5). Nowak et al., in [10], constructed partial geometric difference families in groups G = Z n where n = 4 l for some positive integer l . We close this section by further generalizing this idea. The proofis a simple counting exercise, and so is omitted. Theorem 3.9.
Let G = Z × Z n where n = 4 l for some positive integer l . Let H = h i be theunique subgroup of Z n of order l . Define H + i = { z + i | z ∈ H } = { x ∈ Z n | x ≡ i ( mod } for i = 0 , , , (i.e. the cosets of H in Z n ). Then both { } × ( H ∪ ( H + 1)) ∪ { } × ( H ∪ ( H + 3)) and { } × ( H ∪ ( H + 1)) ∪ { } × ( H ∪ ( H + 3)) are partial geometric difference sets in G with parameters (8 l, l ; 6 l , l ) . We next discuss some new partial geometric difference families.
We begin with the following construction.
Theorem 4.1.
Let n = p u where p is an odd prime and u ≥ is an integer. Let S = { , , ..., p u − − } and, for l = 0 , , ..., p u − , define S l = ( pl − S = { , pl − , pl − , ..., ( p u − − pl − ( p u − − } . Then S = { S l | l = 1 , , ..., p u − } is a partial geometric difference family with parameters ( p u , p u − , p u − ; ( p u − − p u − p u − , p u + ( p u − − p u − p u − ) .Proof. We will use the following property, which is easily seen to hold:
The members ± ( pl − , ± (2 pl − , ..., ± ( p u − − pl − ( p u − − each appearin the multiset ∆( S l ) with multiplicities p u − − , p u − − , ..., respectively . (5)Also note that for s ∈ S we have that ± s ( pl − ≡ ∓ ( p u − − s )( pl − p u − ) for each l , l = 0 , , ..., p u − . Claim:
For each s ∈ S , the equation s ( pl − ≡ α (mod p u ) has p u − solutions ( s, l ) for s, l ∈{ , , ..., p u − } . Proof of Claim:
Notice if pl − ≡ α (mod p u ) we have s ( p ( l + β ) − ≡ α (mod p u ) if and only if sα + spβ ≡ α (mod p u ) . (6)9e can see that (6) holds if and only if s ≡ p ). We know that ( s, β ) = (1 ,
0) is a solution. Nowset s = 1 + p and β = pl ′ + 1 for some l ′ ∈ { , ..., p u − } . Then we have( p + 1)( pl −
1) + ( p + 1)( pl ′ + 1) p = pl − ⇔ pα + ( p l ′ + p + pl ′ + 1) p = 0 ⇔ p ( pl −
1) + p (1 + l ′ ) + p = 0 ⇔ p ( l + l ′ + 1) = 0 ⇔ l + l ′ + 1 ≡ p (mod p u − ) . Thus we can choose β = pl ′ + 1 where l ′ ≡ − l − p u − ) and we have a solution. Since there are p u − such solutions, the claim is proved.Thus we have that each element of Z p u not congruent to 0(mod p u − ) appears in S l for p u − differentvalues of l . Since 0 appears p u times in the multiset F p u − l =1 ∆( S l ), and by (5), we must have that if x ∈ S l for some l then α = X l X y ∈ S l δ S l ( x − y ) = p u + X l X y ∈ S l ,x = y p u − p u − = p u + ( p u − − p u − p u − , (since each S l contains 0)Now notice that if we reduce the elements of S l modulo p u − we get the set S = S . It follows thenthat for any S l , and any x / ∈ S l , the set x − S l contains exactly one member congruent to 0(mod p u − ).Then we have that if x / ∈ S l for all l , then β = X l X y ∈ S l δ S l ( x − y ) = X l X y ∈ S l ,x y (mod p u − ) p u − p u − = ( p u − − p u − p u − . Our next two constructions further generalize Theorem 3.3.
Theorem 4.2.
Let p be a prime, and for each i ∈ { , , ..., p } let S i = C i ∪ { } where C i is the i th cyclotomic class of order p + 1 in F p . Let I ⊂ { , , ..., p } such that | I | = 2 κ for some positiveinteger κ . Say I = { i , ..., i κ } , and define Θ to be the set of all pairs ( i, j ) ∈ I × I such that i = j and each member of I appears in exactly one ordered pair. Let m be a positive, even integer. Define Σ l = l + { , , ..., m − } ⊂ Z m for l = 0 , , and for each ( i ′ , j ′ ) ∈ Θ define S i ′ j ′ = Σ × S i ′ ∪ Σ × S j ′ ⊂ Z m × F p . Then S = { S i ′ j ′ | ( i ′ , j ′ ) ∈ Θ } is a partial geometric difference family with parameters ( mp , mp, κ ; ( m ) p ( p + 3) + ( κ − m p, κ m p ) .Proof. We have already established in Theorem 3.3 that if ( h, z ) ∈ S i ′ j ′ then X ( h ′ ,z ′ ) ∈ S i ′ j ′ δ S i ′ j ′ (( h, z ) − ( h ′ , z ′ )) = ( m p ( p + 3) , h, z ) / ∈ S i ′ j ′ then X ( h ′ ,z ′ ) ∈ S i ′ j ′ δ S i ′ j ′ (( h, z ) − ( h ′ , z ′ )) = 34 m p. Then if ( h, z ) ∈ S i ′ j ′ for some ( i ′ , j ′ ) ∈ Θ we have α = X ( i,j ) ∈ Θ X ( h ′ ,z ′ ) ∈ S ij δ S ij (( h, z ) − ( h ′ , z ′ )) = ( m p ( p + 3) + ( κ −
1) 34 m p, and if ( h, z ) / ∈ S i ′ j ′ for all ( i ′ j ′ ) ∈ Θ we have β = X ( i,j ) ∈ Θ X ( h ′ ,z ′ ) ∈ S ij δ S ij (( h, z ) − ( h ′ , z ′ )) = κ m p. The proofs of the following corollaries are omitted as they use simple counting principals similarto those of Theorem 4.2.
Corollary 4.3.
Let p be a prime, and for each i ∈ { , , ..., p } let S i = C i ∪ { } where C i is the i th cyclotomic class of order p + 1 in F p . Let I ⊂ { , , ..., p } such that | I | = 2 κ for some positiveinteger κ . Say I = { i , ..., i κ } , and define Θ = { ( i κ , i ) , ( i , i ) , ( i , i ) , ..., ( i κ − , i κ ) } . Let m be apositive integer. Define Σ l = l + { , , ..., m − } ⊂ Z m for l = 0 , , and for each ( i ′ , j ′ ) ∈ Θ define S i ′ j ′ = Σ × S i ′ ∪ Σ − × S j ′ ⊂ Z m × F p . Then S = { S i ′ j ′ | ( i ′ , j ′ ) ∈ Θ } is a partial geometricdifference family with parameters ( mp , mp, κ ; ( m ) p ( p + 3) + (2 κ − m p, κ m p ) . Corollary 4.4.
Let p be a prime, and for each i ∈ { , , ..., p } let S i = C i ∪ { } where C i is the i thcyclotomic class of order p + 1 in F p . Let the integer κ and Θ e , for e = 0 resp. , be defined as inTheorem 4.2 resp. Corollary 4.3. For each ( i ′ , j ′ ) ∈ Θ e define S ei ′ j ′ = { , }× S i ′ ∪{ , }× S j ′ ⊂ Z × F p for e = 0 , , and S e = { S ei ′ j ′ | ( i ′ , j ′ ) ∈ Θ e } . Then S resp. S is a partial geometric difference familywith parameters (6 p , p, κ ; 20 p + 8( κ − p, κp ) resp. (6 p , p, κ ; 20 p + 8(2 κ − p, κp ) . We close this section with the following construction.
Theorem 4.5.
Let G be an Abelian group of odd composite order n . Let H be a proper, nontrivialsubgroup of G of order m , and set κ = n/m − . Suppose that g , ..., g κ ∈ G are such that { H ± g i | ≤ i ≤ κ } is a partition of G \ H . Then S = { H ∪ ( H + g i ) | ≤ i ≤ κ } is a partial geometric differencefamily with parameters ( n, m, κ ; n + m ( m − , m ) .Proof. If h ∈ H ∪ ( H + g i ′ ) for some fixed i ′ ∈ { , ..., κ } then g − ( H ∪ ( H + g i ′ )) = ( H + g ) ∪ ( H + ( g − g i ′ )) = ( ( H + g i ′ ) ∪ H, if h ∈ H + g i ′ ,H ∪ ( H − g i ′ ) otherwise . (7)11f g / ∈ H ∪ ( H + g i ) for all i ∈ { , ..., κ } then, for each i , g − ( H ∪ ( H + g i )) = ( H + g ) ∪ ( H + ( g − g i )) (8)where H + g and H + ( g − g i ) are distinct members of { H ± g i | ≤ i ≤ κ } . Also notice that, for fixed i ′ ∈ { , ..., κ } , we have∆( H ∪ ( H + g i ′ )) = ( H ∪ ( H + g i ′ ))( H ∪ ( H + g i ′ )) − = 2 HH − + H ( H + g i ′ ) − + ( H + g i ′ ) H − = 2 mH + m ( H − g i ′ ) + m ( H + g i ′ ) . (9)Let n u denote the number of occurrences of u in ∆( S ) = F κi =1 ∆( H ∪ ( H + g i )). Then, using (7) and(9), if g ∈ S we have α = κ X i =1 X g ′ ∈ H ∪ ( H + g i ) δ H ∪ ( H + g i ) ( g − g ′ ) = 2 κ | H | + m | H ± g + i | = ( nm − m + m = n + m ( m − , and, using (8) and (9), if g / ∈ S we have β = κ X i =1 X g ′ ∈ H ∪ ( H + g i ) δ H ∪ ( H + g i ) ( g − g ′ ) = 2 m . Table 1:
Parameters of partial geometric difference sets constructed in this paper.
Reference ( v, k ; α, β ) Group InformationTheorem 3.3 ( mp , mp ; ( m ) p ( p + 3) , m p ) Z m × F p m even, p an odd primeTheorem 3.4 (6 p , p ; 20 p, p ) Z × F p p an odd primeTheorem 3.7 ( n , n ; 2 n − , n − ∗ A × B (generic) A, B both Abelian groups of order n Theorem 3.9 (8 l, l ; 6 l , l ) Z × Z n n = 4 l for positive integer l *: This partial geometric difference set is an almost difference set and corresponds to a planar 2-adesign (see Section 5). Parameters of partial geometric difference families constructed in this paper.
Reference ( v, k ; α, β ) Group InformationTheorem 4.1 ( p u , p u − , p u − ; α, β ) α = p u + ( p u − − p u − p u − β = ( P u − − p u − p u − Z p u p an odd prime, u ≥ mp , mp, κ ; ( m ) p ( p + 3) + ( κ − m p, κ m p )1 ≤ κ ≤ p +12 Z m × F p m even, p an odd primeCorollary 4.3 ( mp , mp, κ ; ( m ) p ( p + 3) + (2 κ − m p, κ m p )1 ≤ κ ≤ p +12 Z m × F p m even, p an odd primeCorollary 4.4 (6 p , p, κ ; 20 p + 8( κ − p, κp )1 ≤ κ ≤ p +12 Z × F p p an odd primeCorollary 4.4 (6 p , p, κ ; 20 p + 8(2 κ − p, κp )1 ≤ κ ≤ p +12 Z × F p p an odd primeTheorem 4.5 ( n, m, κ ; n + m ( m − , m ) κ = n/m − G (generic) G Abelian of odd,composite order n with m | n We first discuss an important connection between partial geometric designs and tactical configurationsthat have exactly two indices, i.e., tactical configurations ( V, B ) where there are integers µ = µ suchthat for any pair of distinct points x, y ∈ V , r xy ∈ { µ , µ } . If A is the v × b incidence matrix of atactical configuration ( V, B ) with v points, b blocks, and the two indices µ = µ , then we will denoteby A the symmetric matrix whose ( i, j )th entry is 1 if the points corresponding to the i th and j throws of A are contained in exactly µ blocks, and is 0 otherwise. We will need the following lemma. Lemma 5.1. [8]
An incidence structure ( V, B ) is a partial geometric design with parameters ( v, k, r ; α ′ , β ′ ) if and only if its incidence matrix A satisfies AJ = rJ, J A = kJ and AA T A = n ′ A + α ′ J, where n ′ = r + k + β ′ − α ′ − . Suppose ( V, B ) is a partial geometric design with parameters ( v, k, r ; α ′ , β ′ ) and the two indices µ = µ . Let A be the incidence matrix of ( V, B ). It is easy to see that A satisfies AA T = ( r − µ ) I + ( µ − µ ) A + µ ( J − A − I ) . (10)13ince ( V, B ) is partial geometric, by Lemma 5.1 we have that A also satisfies n ′ A + α ′ J = AA T A = ( r − µ ) A + ( µ − µ ) A A + µ kJ. (11)Then, using (10) and (11), we must have that A A = νA + ζ ( J − A ) for some integers ν and ζ .Moreover we must have n ′ + α ′ = r − µ + ν and α ′ = ζ + µ k . This means that, for each pair( x, b ) ∈ V × B , we have |{ y ∈ b | y = x, r xy = µ }| = ( ν (= n ′ + α ′ − r + µ ) , if x ∈ b,ζ (= α ′ − µ k ) , otherwise . (12)Note that condition (12) is necessary and sufficient.Now set σ = r − µ , φ = µ − µ and ψ = ν − ζ . Then we can write AA T = σI + φA + µ J and A A = ψA + ζJ . By Lemma 5.1, and since A is symmetric, we have krJ = AA T J = ψA J + σJ + µ kJ = J AA T . Then, after some simple arithmetic, we can get A J = J A = κJ (13)where κ = ( k − r + µ (1 − v ) µ − µ . Now set ǫ = ζr − µ ( κ − ψ ). Then we have( ψA + ζJ ) A T = A AA T = φA + σA + µ κJ ⇔ φA + σA − ψφA + ψσI = ǫJ ⇔ A = k ′ I + aA + b ( J − I − A ) , (14)where k ′ = κ = ǫ − ψσφ , a = ǫ + ψφ − σφ and b = ǫφ are integers (note that k ′ = κ follows from (13)). From(13) and (14) it is clear that A is the adjacency matrix of a strongly regular graph with parameters( v, k ′ , a, b ) (see Subsection 2.3). We have thus shown the following. Lemma 5.2.
A tactical configuration with the two indices µ = µ and incidence matrix A is partialgeometric with parameters ( v, k, r ; α ′ , β ′ ) if and only if there are integers ν and ζ such that for eachpair ( x, b ) ∈ V × B , |{ y ∈ b | y = x, r xy = µ }| = ( ν (= n ′ + α ′ − r + µ ) , if x ∈ b,ζ (= α ′ − µ k ) , otherwise , and A is the adjacency matrix of a strongly regular graph with parameters ( v, k ′ , a, b ) where k ′ = ǫ − ψσφ , a = ǫ + ψφ − σφ and b = ǫφ . Moreover, k ′ = ( k − r + µ (1 − v ) µ − µ . It is interesting that condition (12), when combined with (10), leads to the strongly regular graphdescribed by (14). We can see that Lemma 5.2 describes a special class of block designs that have twoindices. We now discuss a particular subclass of these block designs.A tactical configuration ( V, B ) with the two indices µ and µ such that µ − µ = 1 is called a2- adesign . Adesigns were were recently introduced in [5], and reported on in [6] and [7], where severalconstructions are given, and codes generated by the incidence matrices are computed.14 heorem 5.3. A - ( v, k, λ ) adesign with incidence matrix A is partial geometric with parameters ( v, k, r ; α ′ , β ′ ) if and only if there are integers ν and ζ such that for each pair ( x, b ) ∈ V × B , |{ y ∈ b | y = x, r xy = µ }| = ( ν (= n ′ + α ′ − r + λ ) , if x ∈ b,ζ (= α ′ − λk ) , otherwise , and A is the adjacency matrix of a strongly regular graph with parameters ( v, k ′ , a, b ) where k ′ = ǫ − ψσ, a = ǫ + ψ − σ and b = ǫ . Moreover, the following relations hold: σ = r − λ, ψ = n ′ − r + λ ( k +1) , ǫ = λ v + λ ( k + r − kr + β ′ − α ′ −
1) + α ′ r and k ′ = ( k − r + λ (1 − v ) . We can see that Theorem 5.3 describes a special class of 2-adesigns. There seem to be even fewerexamples of these, and the few examples we can find have long since been discovered.
Example 5.4.
Let ( V, B ) be a quasi-symmetric design with intersection numbers s and s such that s − s = 1 . Several families of such quasi-symmetric designs are known to exist [14] . It is well-knownthat the dual of any balanced incomplete block design is a partial geometric design [11] . Then the dual ( V, B ) ⊥ of ( V, B ) is a partial geometric -adesign. Example 5.5.
Let p be an odd prime. Let D p +1 i denote the i th cyclotomic class or order p + 1 in F p . It was shown in [10] that ( F p , Dev ( D p +1 i )) is a partial geometric design. It is easy to seethat ( F p , Dev ( D p +1 i )) has the two indices µ = 1 and µ = 0 (see [9] ). Then ( F p , Dev ( D p +1 i )) is asymmetric partial geometric -adesign. Example 5.6.
Let C be the partial geometric difference set from Theorem 3.7 in the Abelian group A × B of order n . Then ( A × B, Dev ( C )) is a partial geometric design, and it was shown in [1] that ( A × B, Dev ( C )) has the two indices µ = 1 and µ = 0 . Then ( A × B, Dev ( C )) is a symmetric partialgeometric -adesign. We have constructed several families of partial geometric difference sets and partial geometric differencefamilies which are recorded in Table 1 and Table 2 respectively. These families have new parametersand so give directed strongly regular graphs with new parameters. We discussed some links betweenpartially balanced designs, 2-adesigns, and partial geometric designs and made an investigation intowhen a 2-adesign is partial geometric. The condition noted in Lemma 5.2 seems surprisingly strong,and describes a special class of partial geometric designs that correspond (via (14)) to strongly regulargraphs. The condition noted in Theorem 5.3 is also strong and describes a special class of 2-adesigns.We wonder whether Lemma 5.2 gives an indirect but viable way of searching for strongly regulargraphs with new parameters.
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