CONSTRUCTIONS OF STRONGLY REGULAR CAYLEY GRAPHS AND SKEW HADAMARD DIFFERENCE SETS FROM CYCLOTOMIC CLASSES

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1 COMBINATORICA Bolyai Society Sringer-Verlag Combinatorica 35 (4) (05) DOI: 0.007/s CONSTRUCTIONS OF STRONGLY REGULAR CAYLEY GRAPHS AND SKEW HADAMARD DIFFERENCE SETS FROM CYCLOTOMIC CLASSES TAO FENG, KOJI MOMIHARA, QING XIANG Received October 3, 0 Online First Setember 0, 04 In this aer, we give a construction of strongly regular Cayley grahs and a construction of skew Hadamard difference sets. Both constructions are based on choosing cyclotomic classes of finite fields, and they generalize the constructions given by Feng and Xiang [0,]. Three infinite families of strongly regular grahs with new arameters are obtained. The main tools that we emloyed are index Gauss sums, instead of cyclotomic numbers.. Introduction We assume that the reader is familiar with the basic theory of strongly regular grahs and difference sets. For the theory of strongly regular grahs, our main references are [5] and [3]. For difference sets, we refer the reader to [7] and Chater 6 of []. We remark that strongly regular grahs are closely related to other combinatorial objects, such as two-weight codes, two-intersection sets in finite geometry, and artial difference sets. For these connections, we refer the reader to [5,. 3] and [7,]. Let Γ be a (simle, undirected) grah. The adjacency matrix of Γ is a (0,)-matrix A with both rows and columns indexed by the vertex set of Γ, where A xy = if there is an edge between x and y in Γ and A xy =0 otherwise. A useful way to check whether a grah is strongly regular is by using the eigenvalues of its adjacency matrix. For convenience we call an eigenvalue restricted if it has an eigenvector which is not a multile of the all-ones vector. (For a k-regular connected grah, the restricted eigenvalues are the eigenvalues different from k.) Mathematics Subject Classification (000): 05B0, 05E /5/$6.00 c 05 János Bolyai Mathematical Society and Sringer-Verlag Berlin Heidelberg

2 44 TAO FENG, KOJI MOMIHARA, QING XIANG Theorem.. For a simle grah Γ of order v, not comlete or edgeless, with adjacency matrix A, the following are equivalent:. Γ is strongly regular with arameters (v, k, λ, µ) for certain integers k,λ,µ,. A = (λ µ)a+(k µ)i +µj for certain real numbers k,λ,µ, where I,J are the identity matrix and the all-ones matrix, resectively, 3. A has recisely two distinct restricted eigenvalues. One of the most effective methods for constructing strongly regular grahs is by the Cayley grah construction. For examle, the Paley grah P(q) and the Clebsch grah are both Cayley grahs (moreover they are cyclotomic). Let G be an additively written grou of order v, and let D be a subset of G such that 0 D and D = D, where D = { d d D}. The Cayley grah on G with connection set D, denoted Cay(G,D), is the grah with the elements of G as vertices; two vertices are adjacent if and only if their difference belongs to D. In the case when Cay(G,D) is a strongly regular grah, the connection set D is called a (regular) artial difference set. The survey of Ma [] contains much of what is known about artial difference sets and about connections with strongly regular grahs. A difference set D in an additively written finite grou G is called skew Hadamard if G is the disjoint union of D, D, and {0}. The rimary examle (and for many years, the only known examle in abelian grous) of skew Hadamard difference sets is the classical Paley difference set in (F q,+) consisting of the nonzero squares of F q, where F q is the finite field of order q, and q is a rime ower congruent to 3 modulo 4. This situation changed dramatically in recent years. Skew Hadamard difference sets are currently under intensive study; see the introduction of [] for a short survey of known constructions of skew Hadamard difference sets and related roblems. As we have seen above, in order to obtain strongly regular Cayley grahs, we need to construct (regular) artial difference sets. A classical method for constructing both artial difference sets and difference sets in the additive grous of finite fields is to use cyclotomic classes of finite fields. Let be a rime, f a ositive integer, and let q = f. Let N > be an integer such that N (q ), and γ be a rimitive element of F q. Then the cosets C i =γ i γ N, 0 i N, are called the cyclotomic classes of order N of F q ; the numbers (C i +) C j are called cyclotomic numbers. Many authors have studied the roblem of determining when a union of some cyclotomic classes forms a (artial) difference set. A summary of results in this direction obtained u to 967 aeared in [5]. However, all the results in [5] are based on cyclotomic classes of small orders and this method has had only very limited success. In fact, known infinite series of difference sets were obtained only in the case

3 CAYLEY GRAPHS AND HADAMARD DIFFERENCE SETS 45 when N =. The situation for artial difference sets is slightly better (see [6], [5, ]). One of the reasons why very few difference sets have been discovered by this method is the difficulty of comuting cyclotomic numbers of order N when N is large. So far cyclotomic numbers have been evaluated for N 4 [] (but note that some of these evaluations are not exlicit). For large N, robably Van Lint and Schrijver [] are the first to use cyclotomic classes of order N of finite fields to construct strongly regular grahs, and Baumert, Mills and Ward [4] are the first to use cyclotomic classes of order N of finite fields to construct difference sets. We comment that the difference sets constructed in [4] are also artial difference sets since the finite fields involved have characteristic. Both constructions are based on the so-called uniform cyclotomy, which will be defined in Section. On the other hand, many soradic examles of strongly regular Cayley grahs have been found using unions of cyclotomic classes of F q through comuter search. For examle, the following are known: (i) (De Lange [0]) Let q = and N =45. Then, Cay(F q,c 0 C 5 C 0 ) is a strongly regular grah. (ii) (Ikuta and Munemasa [5]) Let q = 0 and N =75. Then, Cay(F q,c 0 C 3 C 6 C 9 C ) is a strongly regular grah. (iii) (Ikuta and Munemasa [5]) Let q = and N =49. Then, Cay(F q,c 0 C C C 3 C 4 C 5 C 6 ) is a strongly regular grah. Recently, in [0], the first and the third authors extended the above examles to infinite families by using index Gauss sums over F q. Below is the main theorem from [0]. Theorem.. (i) Let 3 (mod 4) be a rime, 3, N = m, and let be a rime such that f :=ord N ()=φ(n)/, where φ is the Euler totient function. Let q = f and D = m C i F q. Assume that + = 4 h, where h is the class number of Q( ). Then Cay(F q,d) is a strongly regular grah. (ii) Let and be rimes such that { (mod 4), (mod 4)} = {,3}, N = m, and let be a rime such that ord m () = φ( m ), ord () = φ( ), and f :=ord N ()=φ(n)/. Let q = f and D = m C i F q. Assume that = h/ +( ) b, = h/ ( ) b, h is even, and + =4 h, where b {, } and h is the class number of Q( ). Then Cay(F q,d) is a strongly regular grah. Furthermore, in [], the following two constructions of skew Hadamard difference sets and Paley tye artial difference sets were given. (A artial

4 46 TAO FENG, KOJI MOMIHARA, QING XIANG difference set D in a grou G is said to be of Paley tye if the arameters of the corresonding strongly regular Cayley grah are (v, v, v 5 4, v 4 ).) Theorem.3. (i) Let 7 (mod 8) be a rime, N = m, and let be a rime such that f := ord N () = φ(n)/. Let s be an odd integer, I any subset of Z N such that {i (mod m ) i I}=Z m, and let D = i I C i F fs. Then, D is a skew Hadamard difference set if 3 (mod 4) and D is a Paley tye artial difference set if (mod 4). (ii) Let 3 (mod 8) be a rime, 3, N =, and let 3 (mod 4) be a rime such that f :=ord N ()=φ(n)/. Let q = f, I = {0}, and let D = i I C i F q. Assume that + = 4 h, where h is the class number of Q( ). Then, D is a skew Hadamard difference set in the additive grou of F q. Note that in Theorem.3 (ii), we need to choose a suitable rimitive element γ of F q. For details, see []. To extend the construction of Theorem.3 (ii) to the general case where N = m was left as an oen roblem in []. The urose of this aer is to generalize the construction of strongly regular Cayley grahs in Theorem. (ii) to the case where N = m n and the construction of skew Hadamard difference sets in Theorem.3 (ii) to the case where N = m. Three infinite families of strongly regular grahs with new arameters are obtained (see Table in Section 3). An infinite series of skew Hadamard difference sets in (F q,+), where q = m, is also obtained. Imlications of these results on association schemes will be discussed in Section 4.. Index Gauss sums Let be a rime, f a ositive integer, and q = f. The canonical additive character ψ of F q is defined by ψ : F q C, ψ(x) = ζ Tr q/(x), where ζ =ex( πi ) and Tr q/ is the trace from F q to F. For a multilicative character χ of F q, we define the Gauss sum G(χ) = x F q χ(x)ψ(x). Below are a few basic roerties of Gauss sums [8]: (i) G(χ)G(χ)=q if χ is nontrivial;

5 CAYLEY GRAPHS AND HADAMARD DIFFERENCE SETS 47 (ii) G(χ )=G(χ), where is the characteristic of F q ; (iii) G(χ )=χ( )G(χ); (iv) G(χ)= if χ is trivial. In general, the exlicit evaluation of Gauss sums is a very difficult roblem. There are only a few cases where the Gauss sums have been evaluated. The simlest case is the so-called semi-rimitive case (also referred to as uniform cyclotomy or ure Gauss sum), where there exists an integer j such that j (mod N), here N is the order of the multilicative character χ involved. See [,4,7] for the exlicit evaluation in this case. The next interesting case is the index case where the subgrou generated by Z N has index in Z N and. In this case, it is known that N can have at most two odd rime divisors. Many authors have investigated this case, see, e.g., [3,9,3,4,7,8]. In articular, a comlete solution to the roblem of evaluating Gauss sums in this case was recently given in [7]. The following are the results on evaluation of Gauss sums which we will need in the next section. Theorem.. ([7], Case B; Theorem 4.0) Let N = m n, where m and n are ositive integers, and are rimes such that (mod 4) and 3 (mod 4). Assume that is a rime such that ord m () = φ( m ), ord n () = φ( n ), and [Z N : ] =. Let f = φ(n)/, q = f, and χ be a multilicative character of order N of F q. Then, for 0 s m and 0 t n, we have G(χ s t f h ) = G(χ m t ) = f ; G(χ s n ) = f, s t ( ) b + c s t ; where h is the class number of Q( ), and b and c are integers satisfying b,c 0 (mod ), 4 h =b + c, and b f h (mod ). Note that the sign of c in the above theorem deends on the character χ and the exlicit evaluation of c will not be needed in our argument in later sections. Furthermore, note that by the roerties (ii) and (iii) of Gauss sums stated above, we have G(χ l ) = f h (b + c )/ and G(χ l ) = f h (b c )/ for any l if we have G(χ)= f h (b+c )/. Theorem.. ([7], Case D; Theorem 4.) Let N = m, where > 3 is a rime such that 3 (mod 4) and m is a ositive integer. Assume that

6 48 TAO FENG, KOJI MOMIHARA, QING XIANG is a rime such that [Z N : ] =. Let f = φ(n)/, q = f, and χ be a multilicative character of order N of F q. Then, for 0 t m, we have ( ( ) G(χ t (m ) f ) t ht b+c ) =, if 3 (mod 8), ( ) m f, if 7 (mod 8); ( G(χ t f t ) ) = h b + c t ; G(χ m f f ) = ( ), where = ( ), h is the class number of Q( ), and b and c are integers satisfying 4 h =b + c and b f h (mod ). Note that Theorem. above is Theorem 4. in [7], whose statement contains several misrints. We corrected those misrints in the above statement. 3. Constructions of strongly regular Cayley grahs and skew Hadamard difference sets We first recall the following well-known lemma in the theory of difference sets (see e.g., [,6]). Lemma 3.. Let (G,+) be an abelian grou of odd order v, D a subset of G of size v. Assume that D D = and 0 D. Then, D is a skew Hadamard difference set in G if and only if χ(d) = ± v for all nontrivial characters χ of G. On the other hand, assume that 0 D and D =D. Then D is a Paley tye artial difference set in G if and only if χ(d) = ± v for all nontrivial characters χ of G. Let q = f, where is a rime and f a ositive integer, and let C i =γ i γ N, 0 i N, be the cyclotomic classes of order N of F q, where γ is a fixed rimitive element of F q. From now on, we will assume that D is a union of cyclotomic classes of order N of F q. In order to check whether a

7 CAYLEY GRAPHS AND HADAMARD DIFFERENCE SETS 49 candidate subset, D = i I C i, is a artial difference set or a skew Hadamard difference set in (F q,+), we will comute the sums ψ(ad):= x D ψ(ax) for all a F q, where ψ is the canonical additive character of F q. Note that the sum ψ(ad) can be exressed as a linear combination of Gauss sums using the orthogonality of characters: ψ(ad) = ψ(aγ i x N ) N = N = = i I x F q i I x F q (q )N (q )N = N χ C 0 q i I x F q χ F q i I χ F q G(χ ) i I ψ(y) y F q χ F q χ(aγ i x N )χ(y) G(χ )χ(aγ i x N ) G(χ )χ(aγ i ) x F q χ(aγ i ), χ(x N ) is the sub- where F q is the grou of multilicative characters of F q and C0 grou of F q consisting of all χ which are trivial on C Strongly regular grahs from unions of cyclotomic classes of order N = m n In this subsection, we assume that N = m n, where m,n are ositive integers, and are rimes such that (mod 4) and 3 (mod 4). Furthermore, we assume that is a rime such that ord m () = φ( m ), ord n () = φ( n ), and ord N() = φ(n)/. Let q = f and C i = γ i γ N, 0 i N, where f = ord N () and γ is a fixed rimitive element of F q. Define It is clear that D = D. D = m n C n i+ m j. Theorem 3.. The size of the set {ψ(γ a D) a = 0,,...,q } is at most five.

8 40 TAO FENG, KOJI MOMIHARA, QING XIANG Proof. Let χ e denote the multilicative character of order e of F q such that χ e (γ) = ζ e, where ζ e := ex( πi e ). Then χd e = χ e for any divisor d of d e. Note that since D is a union of cyclotomic classes of order N, we have {ψ(γ a D) a=0,,...,q }={ψ(γ a D) a=0,,...,n }. To rove the theorem, it is sufficient to evaluate the sums T a := N ψ(γ a D) = m n l=0 m G(χ l m ) n n where a=0,,...,n. For l=0, by noting that G(χ 0 m )=, we have n m G(χ 0 m ) n n For l= l but l 0 (mod m ), we have m χ l m n (γn i ) = χ l m n (γa+n i+m j ), χ 0 m n (γa+n i+m j ) = m n. m For l= l but l 0 (mod n ), we have n χ l m n (γm j ) = n χ l m χ l n (γ i ) = 0. (γ j ) = 0. Note that for each a {0,,..., N }, there is a unique i {0,,..., m } such that m (a + n i); we write a + n i = m i a. Define δ ia = or 0 according as i a 0 (mod ) or not. Similarly, for each a {0,,...,N }, there is a unique j {0,,..., n } such that n (a+ m j); we write a+m j = n j a. Define δ ja = or 0 according as j a 0 (mod ) or not. For l = m l but l 0 (mod ), since G(χ l n ) = f by Theorem., we have l : gcd (l, )= G(χ l n m ) = m f x Z n y=0 χ l n (γa+n i+m j ) n n χ x+ y n (γ a+m j )

9 CAYLEY GRAPHS AND HADAMARD DIFFERENCE SETS 4 = m n f x Z χ x n (γja ) = m n f ( δ ja ). Similarly, for l= n l but l 0 (mod ), since G(χ l m )=f by Theorem., we have m G(χ l m ) l : gcd (l, )= n For the remaining cases, we consider the sum χ l m (γa+n i+m j ) = m n f ( δ ia ). (3.) l: gcd (l, )= m G(χ l m ) n n χ l m n (γa+n i+m j ). Note that any multilicative character of order m n can be written as χ u mχv n for some u Z m and v Z n. Let σ u,v, be the Galois automorhism of Q(ζ m,ζ n,ζ ) defined by σ u,v, (ζ m ) = ζ u m, σ u,v,(ζ n ) = ζ v n, σ u,v,(ζ ) = ζ. A well known roerty of Gauss sums tells us that (3.) G(χ u m χ v n ) = σ u,v,(g(χ m χ n )). By Theorem. there are integers b and c such that G(χ mχ n ) = f h b + c, where b,c 0 (mod ), b + c =4 h, b f h (mod ). Note that Q( ) Q(ζ, ζ ) Q(ζ m, ζ n, ζ ), and [Q( ):Q]=. We see that the restriction of σ u,v, to Q( ) mas to η (u)η (v), where η and η are the quadratic characters of F and F, resectively. It follows from (3.) that G(χ u mχ v n ) = f h b + cη (u)η (v).

10 4 TAO FENG, KOJI MOMIHARA, QING XIANG Now the sum (3.) can be rewritten as f h u Z m v Z n b+cη (u)η (v) (3.3) b = f h u Z m m n m χ u m (γa+n i+m j )χ v n (γa+n i+m j ) χ u m (γa+n i ) v Z n n χ v n (γa+m j ) (3.4) + f h c η (u) u Z m m χ u m (γa+n i ) η (v) v Z n n χ v n (γa+m j ) For (3.3), we have f h b m χ u u Z m (γa+n i ) n χ v n v Z n (γa+m j ) ( ) = f h b m m χ x+ y (γ a+n i ) x Z y=0 ( ) n n χ x + y n (γ a+m j ) x Z y =0 ( ) ( ) = f h b m χ x (γ ia ) n (γ ja ) x Z x Z χ x = f h b m n ( δ ia )( δ ja ). Let G(η i ), i=,, be the quadratic Gauss sums of F i, resectively. It is well known that G(η i )= ( ) (i )/ i (c.f. [8]). Then, for (3.4), we have f h c m m η (x) χ x+ y m (γ a+n i ) x Z y=0 ) ( x Z n y =0 η (x ) n χ x + y n (γ a+m j )

11 = f h c CAYLEY GRAPHS AND HADAMARD DIFFERENCE SETS 43 m η (x)χ x (γ ia ) x Z n = f h c m n η (i a )η (j a )G(η )G(η ) = f h c m n η (i a )η (j a ) ( ) + = f h c m n η (i a )η (j a ). Thus, we obtain T a + m n = m n f ( δ ja + δ ia ) b + f h f h c Now, we comute S a :=(T a + m following four cases: η (x )χ x (γ ja ) x Z m n ( δ ia )( δ ja ) m n η (i a )η (j a ). n )/ m n ± h c f by considering the (i) If δ ja =δ ia =0, we have S a = h b. (ii) If δ ja =,δ ia =0, we have S a = h b ( ). (iii) if δ ja =0,δ ia =, we have S a = h b ( ). (iv) if δ ja =δ ia =, we have S a = + + h b ( )( ). As can be seen above, T a take at most 5 ossible values. The roof is now comlete. Corollary 3.. If b,c {, }, h is even, = h/ +b, = h/ b, and b f h (mod ), then Cay(F q,d) is a strongly regular grah. Proof. Since D = D and 0 D, the Cayley grah Cay(F q,d) is undirected and without loos. It is also regular of valency D. The restricted eigenvalues of this Cayley grah, as exlained in [5,. 34], are ψ(γ a D), where a = 0,,...,q. By Theorem., it suffices to show that the set {ψ(γ a D) a = 0,,...,q } has recisely two elements. We substitute = h/ + b, = h/ b, and b,c {, } into the exressions for S a in the roof of Theorem 3., and find that S a indeed take only two distinct

12 44 TAO FENG, KOJI MOMIHARA, QING XIANG values. This roves that Cay(F q,d) is a strongly regular grah. In articular, the two restricted eigenvalues r and s (r > s) are given by r = f+h + f h f+h f+h f h f+h and s =, or s = and r = deending on whether b = or b =. Furthermore, the arameters k,λ, and µ of the strongly regular grah are given by k =( f )/, λ=s+r +k +sr, and µ=k +sr. Remark 3.3. One can show that the assumtions that b,c {, }, h is even, = h/ + b, = h/ b, and b f h (mod ) are also necessary for Cay(F q,d) to be strongly regular by a roof similar to that of Corollary 5. in [0]. The construction of strongly regular Cayley grahs given in this subsection is a generalization of Theorem. (ii) [0]. In [0], the six infinite series of strongly regular grahs in Table below were obtained. Note that the case when m= of the st series of Table is the examle found by De Lange [0] and the case when m = of the nd series of Table is the examle found by Ikuta and Munemasa [5]. These six infinite series are combined and generalized to three infinite families of strongly regular grahs in Table. No. N h b v k r s 3 m 5 4 3m 4 3m 5 5 m 3 4 5m 4 5m m 7 3 5m 3 5m m 5 3 7m 3 7m m m m m m m m 5 8 5m m m m m m 5 7 5m m m m m 33 Table. Some strongly regular grahs obtained in [0]. The arameters r, s are the two nontrivial eigenvalues of Cay(G,D), i.e., the two values in {ψ(γ a D) a=0,,...,q }. The arameters λ and µ of the strongly regular grahs can be comuted by λ=s+r +sr +k and µ=k +sr Examle 3.4. Table gives generalizations of strongly regular grahs in Table. Here, the arameters,n,h,b satisfy the conditions of Corollary 3., i.e., is a rime such that [Z N : ] =, b,c {, }, = h/ + b, = h/ b, h 0 (mod ), and b f h (mod ), where h is the class

13 CAYLEY GRAPHS AND HADAMARD DIFFERENCE SETS 45 number of Q( ). It is easy to see by induction that ord N ()=φ(n)/ for all airs (,N) in Table. Furthermore, since ( m n ) (mod ), the condition b f h as b (mod ) can be rewritten h (mod ), which is indeendent of m and n. There are only these three series satisfying the conditions of Corollary 3. when 0 7. No. N h b v k r, s 7 3 m 5 n 4 3m 5 n 4 3m 5 n r = 8 3 m 5 n 5 5 s = 7 3m 5 n m 7 n 3 5m 7 n 3 5m 7 n 35 r = m 9 n m 9 n m 9 n 33 r = m 7 n 35 s = m 7 n 35 m 9 n 33 s = m 9 n 33 Table. Generalizations of the strongly regular grahs in Table. The arameters λ and µ of the strongly regular grahs can be comuted by λ=s+r +sr +k and µ=k +sr 3.. Skew Hadamard difference sets from unions of cyclotomic classes of order N = m In this subsection, we assume that. 3 (mod 8), ( 3),. N = m, 3. + =4 h, where h is the class number of Q( ), 4. is a rime such that ord N ()=φ( m )/. Let q = f, where f = ord N (). Let ζ q = ex( πi q ) and P be a rime ideal in Z[ζ q ] lying over. Then, Z[ζ q ]/P is the finite field of order q and written as Z[ζ q ]/P={ζ i q 0 i q } {0}, where ζ q =ζ q +P. Hence, γ := ζ q is a rimitive element of F q = Z[ζ q ]/P. Let ω P be the Teichmüller character of F q. Then, ω P (γ)=ζ q. Put χ N :=ω q N P. Then χ N is a multilicative character of order N of F q. For this χ N, by the results of

14 46 TAO FENG, KOJI MOMIHARA, QING XIANG [9], we have (3.5) G(χ N) = G(χ m ) = f h ( b + c where b,c 0 (mod ), b + c = 4 h, and b f h (mod ). By our assumtion that + = 4 h, we have b,c {,}, where the sign of c deends on the choice of P. In articular, in [], it was shown that bc (mod P). On the other hand, since + = 4 h, we have ( + )( ) P, from which it follows that + P or P for any rime ideal P in Q(ζ q ) lying over. We may choose a rime ideal P such that + P. Then, bc (mod P) with b,c {,} imlies that bc=. From now on, we fix this choice of P. Let C i =γ i γ N, where 0 i N, and γ is the fixed rimitive element of F q as above. It is clear that C 0 or C m deending on whether (mod 4) or 3 (mod 4) since and f are odd. Let J = {0} (mod ) and define D = m C i+ m j. From the facts that is a nonsquare of F and that the reduction of Z N modulo is the subgrou of index of Z we deduce that J = Z (mod ), and D = D or D D = deending on whether (mod 4) or 3 (mod 4). Theorem 3.. The size of the set {ψ(γ a D) a = 0,,...,q } is recisely two. Proof. Set A := ( ) (m ) f h and B := ( ) f f, where = ( ). First of all, we note that ( ) f = ( ) m since 3 (mod 8). It follows that h A=B. Secondly, since D is a union of cyclotomic classes of order N, we have {ψ(γ a D) a=0,,...,q }={ψ(γ a D) a=0,,...,n }. It is sufficient to evaluate the sums T a := N ψ(γ a D) = m l=0 m G(χ l m) ), χ l m (γa+i+m j ),

15 CAYLEY GRAPHS AND HADAMARD DIFFERENCE SETS 47 where a=0,,...,n. For l=0, by noting that G(χ 0 m )=, we have m G(χ 0 m) χ 0 m (γa+i+m j ) = m. For l=l but l 0 (mod ), since J =Z (mod ), we have χ l m (γm j ) = χ l (γ j ) = 0. For l= l but l 0 (mod m ), we have m χ l m (γ i ) = m χ l m (γ i ) = 0. For l= m, since G(χm m )=B by Theorem., we have m G(χ m m) χ m m (γa+i+m j ) = B m ( ) a. For the remaining cases, we evaluate the sum (3.6) l m G(χ l m) + χ l m (γa+i+m j ) l m G(χ l m) χ l m (γa+i+m j ). By Theorem., we have G(χ l m ) = A ( b + c for l, where b,c are the same as in the evaluation (3.5) of G(χ m ). By the choice of P, it is exanded as ( G(χ l m) = A + ). 4 )

16 48 TAO FENG, KOJI MOMIHARA, QING XIANG Since χ l m ( ) = for any odd l by the assumtion 3 (mod 8), i.e., f is odd, the above sum is reformulated as ( + ) A 4 l ( ) + A 4 m l m χ l m (γa+i+m j ) χ l m (γa+i+m j ). Note that can be written as {x + y x (mod ),y {0,,..., m }}. Furthermore, there is a unique i {0,,..., m } such that a+i 0 (mod m ); we write a+i= m i a. Then, the above sum is rewritten as ( + ) A 4 ( ) + A 4 x (mod ) ( = m + ) A 4 ( + m ) A 4 x (mod ) ( = m + ) A 4 ( + m ) A 4 m m y=0 m m y=0 x (mod ) x (mod ) x (mod ) x (mod ) χ x m (γa+i+m j ) χ y m (γ a+i+m j ) χ x (γ ia+j ) χ x χ x m (γa+i+m j ) χ y m (γ a+i+m j ) χ x (γ ia+j ) (γ ia ) χ j (γ) χ x (γ ia ) χ j (γ). Put X a := χ j (γ) and Y a := x (mod ) χ x (γ ia ). Let η be the quadratic character of F and ψ be the canonical additive character of F.

17 CAYLEY GRAPHS AND HADAMARD DIFFERENCE SETS 49 Noting that is a nonsquare in F. For i Z \{0, } it holds that x (mod ) Hence, we have and X a = χ x Thus, we obtain T a + m (γ i ) = x (mod ) χ ix (γ)χ ix (γ) = ( ) i ( + η(x))ψ ( ix) x F = ( ) i + η( i)g(η) j (mod ) = χ j (γ) + + j (mod ) + = Y a = ( ) + η(i a) ia, i a 0,. = ( ) i + η(i). χ j (γ) + = B m ( ) a + m A ( ( + )( )Y a 4 +( )( + )Y a ). We comute T a + m by considering the following six cases: (i) i a = 0: In this case, we have a 0 (mod ), Y a =, and T a + m = m ( A 4 ( )+B). (ii) i a = : In this case, we have a (mod ), Y a =, and T a + m = m ( A 4 ( +) B). (iii) i a : In this case, we have a (mod ), Y a =, and T a + m = m ( A 4 ( + +) B). (iv) i a : In this case, we have a (mod ), Y a = +, and T a + m =m ( A 4 ( ) B). (v) i a : In this case, we have a 0 (mod ), Y a = +, and T a + m =m ( A 4 ( )+B). (vi) i a : In this case, we have a 0 (mod ), Y a = +, and T a + m =m ( A 4 ( )+B).

18 430 TAO FENG, KOJI MOMIHARA, QING XIANG By the assumtion that + = 4 h and the fact that h A = B, it is easily checked that T a + m, a=0,,...,n, take recisely two values. The roof is now comlete. Corollary 3.5. The set D is a skew Hadamard difference set or a Paley tye artial difference set according as 3 (mod 4) or (mod 4). Proof. By Theorem 3., the set {ψ(γ a D) a = 0,,...,q } has recisely two elements, which are ( m N = ( A ± m ( )) 4 ( ) + B ± ( ) ( )(m ) ( ) f ). By Lemma 3., the assertion of the corollary follows immediately. The construction of skew Hadamard difference sets and Paley tye artial difference sets given in this subsection is a generalization of Theorem.3 (ii) []. In articular, in [], one examle of skew Hadamard difference sets with arameters (,N,h,b,v) = (3,,,,3 5 ) was given. Unfortunately, we can not generalize this examle to N = m because = 3 does not satisfy the condition ord N () = φ(n)/ for m >. Below are some infinite series of skew Hadamard difference sets and Paley tye artial difference sets obtained by Corollary 3.5. Examle 3.6. Table 3 shows all ossible skew Hadamard difference sets and Paley tye artial difference sets obtained by alying Corollary 3.5 to all 0 6 excet for the case when = and m =. In articular, the 3rd case of Table 3 gives skew Hadamard difference sets and the other cases give Paley tye artial difference sets. Here, the arameters, N, h, b satisfy the conditions of Corollary 3.5, i.e., is a rime such that [Z N : ] =, + = 4 h, and b f h (mod ), where h is the class number of Q( ). Note that it is easy to rove by induction that ord N ()=φ(n)/ for all airs (,N) in Table 3. Furthermore, since m ( )/ h m ( +) m h+ 4 (mod ), the condition b f h (mod ) can be rewritten as b h 4 (mod ), which is indeendent of m.

19 CAYLEY GRAPHS AND HADAMARD DIFFERENCE SETS 43 No. N h b v 5 9 m 5 9 9m 7 67 m m m m m m m m Table 3. Some Paley tye artial difference sets and skew Hadamard difference sets obtained by Corollary Concluding remarks In this aer, we have given two constructions of strongly regular grahs and skew Hadamard difference sets, which are generalizations of those given by the first and third authors [0,]. As a consequence, we obtain three infinite series of strongly regular grahs with new arameters and a family of skew Hadamard difference sets in (F q,+), where q = m. The results on strongly regular grahs have imlications on association schemes. Given a d-class (symmetric) association scheme (X,{R l } 0 l d ), we can take union of classes to form grahs with larger edge sets (this rocess is called a fusion); it is not necessarily guaranteed that the fused collection of grahs will form an association scheme on X. If an association scheme has the roerty that any of its fusions is also an association scheme, then we call the association scheme amorhic. A well-known and imortant examle of amorhic association schemes is given by the cyclotomic association schemes on F q when the cyclotomy is uniform [4]. In [6], A.V. Ivanov conjectured that if each nontrivial relation in an association scheme is strongly regular, then the association scheme must be amorhic. This conjecture turned out to be false. A first counterexamle was found by Van Dam [8] in the case when the association scheme is imrimitive. Afterwards, Van Dam [9] and Ikuta and Munemasa [5] gave more counterexamles in the case when the association scheme is rimitive. However, there had been known only a few counterexamles in the rimitive case. Recently, in [], the authors generalized the counterexamles of Van Dam and Ikuta-Munemasa into infinite series using strongly regular Cayley grahs based on index Gauss sums of tye N = m and tye N = m. Our generalization (Corollary 3.) of the second construction in [0] roduces further new counterexamles to Ivanov s conjecture and association schemes with very interesting roerties. More recisely, under the same

20 43 TAO FENG, KOJI MOMIHARA, QING XIANG assumtions as in Corollary 3., define D k = m n C n i+ m j+m n k for each 0 k. Let R 0 ={(x,x) x F q } and R k := {(x, y) x, y F q, x y D k }. Then, one can similarly rove that (F q,{r k } 0 k ) is a seudocyclic and non-amorhic association scheme in which every nontrivial relation is a strongly regular grah. Table yields three new infinite series of seudocyclic and non-amorhic association schemes, where each of the nontrivial relations is strongly regular. Moreover, further fusion schemes of these association schemes are ossible by alying Corollary 3. and Theorem 4. of [5]. In articular, Examles and of [5] are generalized into an infinite series by using the above association scheme with =, b=, (, )=(5,3), and h=. Acknowledgements The work of Tao Feng was suorted in art by the Fundamental Research Funds for the central universities, Zhejiang Provincial Natural Science Foundation under Grant LQA009, the National Natural Science Foundation of China under Grant 048, and the Research Fund for Doctoral Programs from the Ministry of Education of China under Grant The work of K. Momihara was suorted by JSPS under Grant-in-Aid for Research Activity Start-u The work of Qing Xiang was suorted in art by NSF Grant DMS and by the Y. C. Tang discilinary develoment fund of Zhejiang University. References [] B. Berndt, R. Evans and K. S. Williams: Gauss and Jacobi Sums, Wiley, 997. [] T. Beth, D. Jungnickel and H. Lenz: Design Theory. Vol. I. Second edition, Encycloedia of Mathematics and its Alications, 78. Cambridge University Press, Cambridge, 999. [3] L. D. Baumert, J. Mykkeltveit: Weight distributions of some irreducible cyclic codes, DSN Progr. Re., 6 (973), 8 3. [4] L. D. Baumert, W. H. Mills, R. L. Ward: Uniform cyclotomy, J. Number Theory 4 (98), 67 8.

21 CAYLEY GRAPHS AND HADAMARD DIFFERENCE SETS 433 [5] A. E. Brouwer, W. H. Haemers: Sectra of Grahs, Sringer, Universitext, 0. [6] A. E. Brouwer, R. M. Wilson and Q. Xiang: Cyclotomy and strongly regular grahs, J. Alg. Combin. 0 (999), 5 8. [7] R. Calderbank and W. M. Kantor: The geometry of two-weight codes, Bull. London Math. Soc. 8 (986), 97. [8] E. R. van Dam: A characterization of association schemes from affine saces, Des. Codes Crytogr. (000), [9] E. R. van Dam: Strongly regular decomositions of the comlete grahs, J. Alg. Combin. 7 (003), 8 0. [0] T. Feng and Q. Xiang: Strongly regular grahs from union of cyclotomic classes, J. Combin. Theory (B) 0 (0), [] T. Feng, F. Wu and Q. Xiang: Pseudocyclic and non-amorhic fusion schemes of the cyclotomic association schemes, Designs, Codes and Crytograhy 65 (0), [] T. Feng and Q. Xiang: Cyclotomic constructions of skew Hadamard difference sets, J. Combin. Theory (A) 9 (0), [3] C. Godsil and G. Royle: Algebraic Grah Theory GTM 07, Sringer-Verlag, 00. [4] M. Hall, Jr.: A survey of difference sets, Proc. Amer. Math. Soc. 7 (956), [5] T. Ikuta and A. Munemasa: Pseudocyclic association schemes and strongly regular grahs, Euro. J. Combin. 3 (00), [6] A. A. Ivanov, C. E. Praeger: Problem session at ALCOM-9, Euro. J. Combin. 5 (994), 05. [7] E. S. Lander: Symmetric designs: an algebraic aroach, Cambridge University Press, 983. [8] R. Lidl and H. Niederreiter: Finite Fields, Cambridge University Press, 997. [9] P. Langevin: Calcus de certaines sommes de Gauss, J. Number Theory 63 (997), [0] C. L. M. de Lange: Some new cyclotomic strongly regular grahs, J. Alg. Combin. 4 (995), [] J. H. van Lint and A. Schrijver: Construction of strongly regular grahs, twoweight codes and artial geometries by finite fields, Combinatorica (98), [] S. L. Ma: A survey of artial difference sets, Des. Codes Crytogr. 4 (994), 6. [3] O. D. Mbodj: Quadratic Gauss sums, Finite Fields Al. 4 (998), [4] P. Meijer, M. van der Vlugt: The evaluation of Gauss sums for characters of -ower order, J. Number Theory 00 (003), [5] T. Storer: Cyclotomy and Difference Sets, Lectures in Advanced Mathematics, Markham Publishing Comany, 967. [6] R. J. Turyn: Character sums and difference sets, Pacific J. Math. 5 (965), [7] J. Yang and L. Xia: Comlete solving of exlicit evaluation of Gauss sums in the index case, Sci. China Ser. A 53 (00), [8] J. Yang and L. Xia: A note on the sign (unit root) ambiguities of Gauss sums in the index and 4 case, rerint, arxiv:09.44v.

22 434 FENG, MOMIHARA, XIANG: CONSTRUCTIONS OF STRONGLY REGULAR... Tao Feng Deartment of Mathematics Zhejiang University Hangzhou 3007, Zhejiang, China Koji Momihara Faculty of Education Kumamoto University -40- Kurokami Kumamoto , Jaan Qing Xiang Deartment of Mathematical Science University of Delaware Newark, DE 976, USA