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Certain strongly regular Cayley graphs on F22(2s+1) from cyclotomy
Finite Fields and Their Applications 25 (2014) 280–292
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Finite Fields and Their Applications www.elsevier.com/locate/ffa
Certain strongly regular Cayley graphs on F22(2s+1) from cyclotomy Koji Momihara 1 Faculty of Education, Kumamoto University, 2-40-1 Kurokami, Kumamoto 860-8555, Japan
a r t i c l e
i n f o
Article history: Received 9 February 2013 Received in revised form 10 August 2013 Accepted 5 October 2013 Available online 26 October 2013 Communicated by Dieter Jungnickel
a b s t r a c t We give a construction of negative Latin square type strongly regular Cayley graphs on F22(2s+1) with parameters (22(2s+1) , 2(22s − 1)(22s+1 + 1)/3, 4(22s − 1)2 /9 − 2, 2(22s+1 + 1)(22s − 1)/9) for every s 1 based on choosing suitable cyclotomic classes on F22(2s+1) and the computation of certain Gauss sums. © 2013 Elsevier Inc. All rights reserved.
MSC: 05E30 05B10 11T22 Keywords: Strongly regular graphs Cyclotomic cosets Gauss sums
1. Introduction In this paper, we will assume that the reader is familiar with the theory of strongly regular graphs. Our main reference is the book by Brouwer and Haemers [5]. We remark that strongly regular graphs are closely related to other combinatorial objects, such as two-weight codes, two-intersection sets in finite geometry, and partial difference sets [5, p. 132], [7,15]. Let Γ be a simple and undirected graph and A be its adjacency matrix. A useful way to check whether a graph 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 multiple of the all-ones vector 1. (For a k-regular connected graph, the restricted eigenvalues are the eigenvalues different from k.)
1
E-mail address: [email protected]. The work of K. Momihara was supported by JSPS under Grant-in-Aid for Research Activity Start-up 23840032.
1071-5797/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ffa.2013.10.006
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Theorem 1. For a simple graph Γ of order v, not complete or edgeless, with adjacency matrix A, the following are equivalent: (1) Γ is strongly regular with parameters ( v , k, λ, μ) for certain integers k, λ, μ, (2) A 2 = (λ − μ) A + (k − μ) I + μ J for certain real numbers k, λ, μ, where I , J are the identity matrix and the all-ones matrix, respectively, (3) A has precisely two distinct restricted eigenvalues. A strongly regular graph is said to be of Latin square type (respectively, negative Latin square type) if
( v , k, λ, μ) = (n2 , r (n − ), n + r 2 − 3 r , r 2 − r ) and = 1 (respectively, = −1). Typical examples of strongly regular graphs of Latin square type or negative Latin square type come from nonsingular quadrics in the projective space PG(m − 1, q), where m is even [15]. One of the most effective methods for constructing strongly regular graphs is by the Cayley graph construction. Let G be an additively written group 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 graph on G with connection set D, denoted by Cay (G , D ), is the graph with the elements of G as vertices; two vertices x, y ∈ G are adjacent if and only if x − y ∈ D. In the case when Cay (G , D ) is strongly regular, the connection set D is called a (regular) partial difference set. The survey of Ma [15] contains much of what is known about partial difference sets and about connections with strongly regular Cayley graphs. A classical method for constructing strongly regular Cayley graphs (i.e., partial difference sets) on elementary abelian groups is to use cyclotomic classes of finite fields. Let p be a prime, f a positive integer, and let q = p f . Let k > 1 be an integer such that k | (q − 1), and γ be a primitive (k,q)
= γ i γ k , 0 i k − 1, are called the cyclotomic classes of order element of Fq . Then the cosets C i k of Fq . Quite recently, new “cyclotomic” constructions of strongly regular graphs were given, see [8,9,11,16]. In [19], the authors discussed the problem when a Cayley graph on a finite field with a single cyclotomic class as its connection set is strongly regular. Such a strongly regular graph is called cyclotomic. They made the following conjecture: let F p f be the finite field of order p f , k | (k, p f )
k > 1, and C 0 := C 0 holds:
p f −1 p −1
with
with −C 0 = C 0 . If Cay (F p f , C 0 ) is strongly regular, then one of the following
(1) (subfield case) C 0 is the multiplicative group of a subfield of F p f , (2) (semi-primitive case) −1 ∈ p (Z/kZ)∗ , (3) (exceptional case) it is either of eleven sporadic examples of cyclotomic strongly regular graphs (see [19, Table 1]). This conjecture is still open. On the other hand, in [8,9,11,16,17], several of these sporadic examples have been generalized into infinite families by taking a union of cyclotomic classes. In particular, in [16], the author gave a recursive construction of strongly regular Cayley graphs using cyclotomic classes of finite fields. In the last section of this paper, we will apply the recursive construction to one example of strongly regular graphs obtained from our main theorem. For other constructions of strongly regular graphs from cyclotomy, we refer the reader to the references in [9,12,14,17]. In [18, Theorems 5.1 and 5.2], the author gave constructions of strongly regular Cayley graphs with the parameters (q6 , r1 (q3 + 1), −q3 + r12 + 3r1 , r12 + r1 ) on Fq2 × Fq4 and (q10 , r2 (q5 + 1), −q5 + r22 + 3r2 ,
r22 + r2 ) on Fq4 × Fq6 , where r1 = q(q − 1) and r2 = q(q2 + 1)(q − 1). One can easily generalize his constructions as follows: Proposition 2. Let p be a prime and let and t be positive integers. Assume that the following hold: (1) Let D i ⊆ F p 2(s+1) , 1 i p + 1, be a partition of F∗p 2(s+1) such that D i = − D i and | D i | =
( p 2(s+1) − 1)/( p + 1). The graphs Cay (F p 2(s+1) , D i ), 1 i p + 1, are all negative Latin square type or all Latin square type strongly regular graphs according as s is odd or even.
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(2) Let E i ⊆ F p 2s , 1 i p + 1, be a partition of F∗p 2s such that E i = − E i and | E i | = ( p 2s − 1)/( p + 1). The graphs Cay (F p 2s , D i ), 1 i p + 1, are all negative Latin square type or all Latin square type strongly regular graphs according as s is even or odd. Then, Cay (F p 2(s+1) × F p 2s , D ) is a negative Latin square type strongly regular graph, where p +1
D=
D i ∪ { 0} × E i .
i =1
As an immediate corollary, we can show that there exists a negative Latin square type strongly regular graph with parameters
( v , k, λ, μ) = p 2(2s+1) , r p (2s+1) + 1 , − p (2s+1) + r 2 + 3r , r 2 + r ,
(1.1)
where r = p ( p 2s − 1)/( p + 1), as follows: each of the cyclotomic classes of order p + 1 in F p 2t becomes a connection set of Latin square type or negative Latin square type strongly regular Cayley graph on F p 2t according as t is odd or even (cf. [6, Theorem 2]). By applying Proposition 2 to these strongly regular graphs with t = s + 1 in (1) and with t = s in (2), we obtain a strongly regular graph with the parameters of (1.1). This existence result was also shown in [10, Proposition 4.3]. In this paper, we are interested in strongly regular Cayley graphs with parameters
2
22(2s+1) , 2 22s − 1 22s+1 + 1 /3, 4 22s − 1 /9 − 2, 2 22s+1 + 1 22s − 1 /9 ,
which is a special case of (1.1). The main objective of this paper is to present a construction of strongly regular Cayley graphs with these parameters using cyclotomy of order 3(22s+1 − 1) of F22(2s+1) , which is different from the product construction given in Proposition 2. The approach to proving the result is much more theoretical and it represents progress on the theory of cyclotomy. Furthermore, as described in Remark 13, the strongly regular graphs with parameters (210 , 330, 98, 110) obtained from Proposition 2 and our main theorem are nonisomorphic. The following is our main theorem.
Theorem 3. Let p = 2, m = 2 f − 1, q = 2 f , and f = 2s + 1. Let S = {logω (x)(mod m) | Trq/2 (x) = 0, x ∈ Fq∗ } and I = {3x + my (mod 3m) | x ∈ S , y = 1, 2}, where ω is a fixed primitive element of Fq and Trq/2 is the trace from Fq to F2 . Furthermore, let
D=
(3m,q2 )
Ci
,
i∈I
(3m,q2 )
where C i = γ i γ 3m and strongly regular.
γ is a primitive element of Fq2 such that γ q+1 = ω. Then, Cay (Fq2 , D ) is
2. Preliminaries 2.1. Preliminaries on characters Let p be a prime, f a positive integer, and q = p f . The canonical additive character ψ of Fq is defined by
ψ : Fq → C∗ ,
Trq/ p (x)
ψ(x) = ζ p
,
K. Momihara / Finite Fields and Their Applications 25 (2014) 280–292 i where ζ p = exp( 2π ) and Trq/ p is the trace from Fq to F p . For a multiplicative character p define the Gauss sum
G f (χ ) =
283
χ of Fq , we
χ (x)ψ(x),
x∈Fq∗
which belongs to Z[ζkp ] of integers in the cyclotomic field Q(ζkp ), where k is the order of be the automorphism of Q(ζkp ) determined by
σa,b (ζk ) = ζka ,
χ . Let σa,b
σa,b (ζ p ) = ζ pb
for gcd (a, k) = gcd (b, p ) = 1. Below are several basic properties of Gauss sums [13]: (i) (ii) (iii) (iv) (v)
G f (χ )G f (χ ) = q if χ is nontrivial; G f (χ p ) = G f (χ ), where p is the characteristic of Fq ; G f (χ −1 ) = χ (−1)G f (χ ); G f (χ ) = −1 if χ is trivial; σa,b (G f (χ )) = χ −a (b)G f (χ a ).
In general, explicit evaluations of Gauss sums are very difficult. There are only a few cases where the Gauss sums have been evaluated. The most well-known case is the quadratic case, i.e., the order of χ is two. The next simple case is the so-called semi-primitive case (also known as uniform cyclotomy or pure Gauss sum), where there exists an integer j such that p j ≡ −1 (mod k), where k is the order of the multiplicative character involved. The explicit evaluation of Gauss sums in this case is given as follows: Theorem 4. (See [2].) Suppose that k > 2 and p is semi-primitive modulo k, i.e., there exists an s such that p s ≡ −1 (mod k). Choose s minimal and write f = 2st. Let χ be a multiplicative character of order k. Then,
p − f /2 G f (χ ) =
if p = 2; (−1)t −1 s (−1)t −1+( p +1)t /k if p > 2.
This theorem was used to find strongly regular graphs on Fq (cf. [1,6]) and we will use this theorem in the next section. The next interesting case is the index 2 case where the subgroup p generated by p ∈ (Z/kZ)∗ is of index 2 in (Z/kZ)∗ and −1 ∈ / p . Many authors have investigated this case, see [20] for a complete solution to the problem of evaluating index 2 Gauss sums. Now we recall the following well-known lemma in Algebraic graph theory (cf. [5]). Lemma 5. Let (G , +) be an abelian group and D a subset of G such that 0 ∈ / D and D = − D. Then, the eigenvalues of Cay (G , D ) are given by ψ( D ), ψ ∈ G, where G is the character group of G. (k,q)
Let q be a prime power and let C i = γ i γ k , 0 i k − 1, be the cyclotomic classes of order k of Fq , where γ is a fixed primitive element of Fq . In order to check whether a candidate subset
(k,q)
is a connection set of a strongly D = i∈ I C i regular Cayley graph, by Theorem 1 and Lemma 5, it is enough to show that the sums ψ(γ a D ) = x∈ D ψ(γ a x), a = 0, 1, . . . , q − 2, take exactly two values, where ψ is the canonical additive character of Fq . Note that the sum ψ(γ a D ) can be expressed as a linear combination of Gauss sums (cf. [8]) by using the orthogonality of characters:
1 a i ψ γaD = G f χ −1 χ γ γ , k
χ ∈C 0⊥
i∈I
(2.1)
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K. Momihara / Finite Fields and Their Applications 25 (2014) 280–292
where C 0⊥ is the subgroup of Fq∗ consisting of all
χ which are trivial on C 0(k,q) . Thus, the computations
(k,q)
is a connection set of a strongly reguneeded to show whether a candidate subset D = i ∈ I C i lar Cayley graph are essentially reduced to evaluating Gauss sums. However, evaluating Gauss sums explicitly is very difficult. Therefore, we will need a technique in order to avoid explicitly evaluating Gauss sums. In particular, we use the Davenport–Hasse lifting formula and product formula stated below. Theorem 6. (See [2].) Let χ be a nontrivial multiplicative character of Fq = F p f and let χ be the lift of χ to Fq = F p f s , i.e., χ (α ) = χ (Normq /q (α )) for α ∈ Fq , where s 2 is an integer. Then
Gfs
s
χ = (−1)s−1 G f (χ ) .
Theorem 7. (See [2].) Let η be a multiplicative character of order > 1 of Fq = F p f . For every nontrivial multiplicative character χ of Fq ,
G f (χ ) =
−1 G f (χ ) G f (ηi )
χ ()
i =1
G f (χηi )
.
Now, we prove the following theorem, which will be used in the next section.
Theorem 8. Let p = 2, f = 2 f , and m = 2 f − 1, where f > 0 is any odd integer. Let χ3m be a multiplicative character of order 3m = 3(2 f − 1) of F2 f and write χ3m = χ3 · χm for multiplicative characters χ3 and χm of order 3 and m, respectively, of F2 f . Then, it holds that
G f (χ3m ) = G f
χm 3 G f χm −1 ,
is a multiplicative character of F such that χ is the lift of χ to F . where χm m m 2f 2f
Proof. By applying the Davenport–Hasse product formula (Theorem 7) as = 3, we have
1=
=
χ = χ3m , and η = χ3 ,
3 χ3m (3)G f (χ3m ) G f (χ3m χ3 )G f (χ3m χ32 ) · 3 G f (χ3 )G f (χ32 ) G f (χ3m )
G f (χ3m ) 3 G f (χm )
f
·
2 G f (χ3m )G f (χm )
2f
=
G f (χ3m )2 3 G f (χm )G f (χm−1 )
,
−1 ) = 2 f . Furthermore, by the Davenport–Hasse lifting where we used G f (χ3 )G f (χ32 ) = G f (χm )G f (χm formula (Theorem 6), we have
Gf
2
2
χm3 G f χm−1 = G f χm 3 G f χm −1 .
Hence, we have
(τ :=)
G f (χ3m ) 3 )G (χ −1 ) G (χm f m f
= 1 or −1.
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Now we consider the reduction of G f (χ3m )3 modulo 3. By Theorems 6 and 4, we have
G f (χ3m )3 ≡
z∈F∗ f
3 χ3m ( z)ψ(3z) (mod 3)
p
=
z∈F∗ f
3 χ3m ( z)ψ( z)
p
2 3 = −G f χm 3 . = G f χ3m On the other hand, we have
τ G f χm 3 G f χm −1
3
3 3 ≡ τ 3 G f χm 3 G f χm −1 (mod 3) 3 ≡ τ G f χm 3 G f χm −3 (mod 3) 2 = τ 2 f G f χm 3 .
3 )2 ≡ 0 (mod 3). By noting that 2 f + 1 ≡ 0 (mod 3) and 3 G (χ 3 ), Hence, we have (τ 2 f + 1)G f (χm f m we obtain τ = 1. This completes the proof of the theorem. 2
3. Construction of strongly regular Cayley graphs on F 22(2s+1) In this section, we give a proof of our main theorem. We fix the notation as follows: let p = 2, m = 2 f − 1, q = 2 f , and f = 2s + 1. Then, by Theorem 8, for any t such that gcd (t , 3) = 1 and m t we have
G2 f
t = G f χm 3t G f χm −t . χ3m
(3.1)
χm3 (γ ) = χm 3 (ω), by Theorem 6 we have
Furthermore, since
G2 f
2
3t = −G f χm 3t . χ3m
(3.2)
Moreover, we will use the following:
G f
χm t =
χm t (x) −
Tr
(x)=0 2 f /2
Tr
χm t (x) = 2
(x)=1 2 f /2
χm t (x).
Tr
(x)=0 2 f /2
3.1. Lemmas In this subsection, we give a few lemmas as preparation for proving our main theorem. Let χ1 and χ2 be multiplicative characters of Fq . We define the sum
J (χ1 , χ2 ) =
χ1 (x)χ2 (1 − x),
x∈Fq , x =0,1
the so-called Jacobi sum of Fq . We will use the following propositions.
(3.3)
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Proposition 9. (See Theorem 5.21 [13].) Let χ1 and χ2 be nontrivial multiplicative characters of F p f . Then, if χ1 χ2 is nontrivial, it holds that
J (χ1 , χ2 ) =
G f (χ1 )G f (χ2 ) G f (χ1 χ2 )
.
Proposition 10. (See Theorem 2.51 [2].) For i , j = 0, 1, . . . , k − 1, it holds that k −1 k −1 (k, p f ) 1 u (k, p f ) C ∩ + 1 C χ (−1)ζk−iu− jv J χ u , χ v , = i j 2
k
(k, p f )
where C i = ωi ωk with F p f such that χ (ω) = ζk .
u =0 v =0
ω a primitive element of F p f and χ is a multiplicative character of order k of
Throughout the rest of this paper, we use the notation of Theorem 3. Lemma 11. Let f a (x) := x3 + x2 + ωa ∈ Fq [x]. Then, f a (x) = 0 has exactly one solution in Fq if and only if a ∈ − S. Proof. Since x3 + x2 + ωa = 0 does not have any solution with multiplicity three in Fq , it is enough to consider the equations of the form (x + c )(x2 + dx + e ) = 0, where c = d + 1 and cd + e = 0, i.e., (x + d + 1)(x2 + dx + d(d + 1)) = 0, and x2 + dx + d(d + 1) is irreducible in Fq [x]. It is well known that the polynomial x2 + ux + v ∈ Fq [x] with u = 0 and q an even prime power is irreducible in Fq [x] if and only if Trq/2 (u −2 v ) = 1 (cf. [13, Corollary 5.35]). Hence, x2 + dx + d(d + 1) is irreducible in Fq [x] if and only if Trq/2 (d−1 ) = 0 by noting that Trq/2 (1) = 1. Furthermore, we have
Trq/2 d−1 (d + 1)−2 = Trq/2
1 d
+
1 d+1
+
1 d+1
2
= Trq/2 d−1 .
Thus, x2 + dx + d(d + 1) = 0 is irreducible in Fq [x] if and only if Trq/2 (d−1 (d + 1)−2 ) = 0, i.e., f a (x) = 0 has exactly one solution in Fq if and only if ωa = d(d + 1)2 and it satisfies Trq/2 (d−1 (d + 1)−2 ) = 0. For d, d ∈ Fq∗ with d = d such that Trq/2 (d−1 ) = Trq/2 (d −1 ) = 0, if ωa = d(d + 1)2 = d (d + 1)2 , then we have (x + d + 1)(x2 + dx + d(d + 1)) = (x + d + 1)(x2 + d x + d (d + 1)) = x3 + x2 + ωa . This means that f a (x) has two different factorizations in Fq [x], which is impossible. Hence, we have {d(d + 1)2 ∈ Fq∗ | Trq/2 (d−1 ) = 0} = {d ∈ Fq∗ | Trq/2 (d−1 ) = 0}. This shows the assertion of the lemma. 2 Lemma 12. There are subsets S 1 , S 2 ⊆ Zm such that S 1 ∪ S 2 = Zm \ (− S ), S 1 ∩ S 2 = ∅, and m −1
G f
−t 2
χm
G f
3t
t
χm χm ω
a
t =1
⎧ f ⎪ if a (mod m) ∈ − S ; ⎨22 f = −2 f + 2 2 +1 if a (mod m) ∈ S 1 ; ⎪ ⎩ f +1 f 2 −22 if a (mod m) ∈ S 2 .
2 ) = G (χ ). Then, by Proposition 9, we have Proof. First note that G f (χm f m m −1 t =1
G f
2
χm −t G f χm 3t χm t ωa = 2 f
m −1 t =1
J
χm −t , χm −2t χm t ωa .
(3.4)
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287
On the other hand, by Proposition 10, we have m −1 m −1 a (m,q) (m,q) 1 j (2u − v ) −u a u v ω C ∩ C +1 = ζm χ ω J χ ,χ ,
−2 j
j
where we can assume that
m
m2
m
m
u =0 v =0
χm (ω) = ζm . Hence, we further have
m −1
m −1 m −1 m −1 a (m,q) (m,q) 1 j (2u − v ) −u a u v ω C ∩ C +1 = ζm χ ω J χ ,χ
j =0
−2 j
j
m2
m
m
m
j =0 u =0 v =0
m −1 1 −u a u 2u χm ω J χm , χm m
=
u =0
1
=
m
m −1
χm −u ωa J χm u , χm 2u +
m−1
u =1
m
.
Then, (3.4) is reformulated as follows:
(3.4) = 2
f
−1 a (m,q) (m,q) m f 2 −1 ω C −2 j ∩ C j +1 − 2 −2 . f
j =0
Now, we must show that m −1
a (m,q) (m,q) ω C ∩ C +1 = −2 j
j =0
m−1
j
1 if a(mod m) ∈ − S ; 0 if a (mod m) ∈ S 1 ; 3 if a (mod m) ∈ S 2 .
+ 1)| is the number of solutions of ωa x−2 = x + 1, i.e., x3 + x2 + ω = 0 in Fq . Furthermore, it is impossible that x3 + x2 + ωa = 0 has exactly two distinct solutions. As shown in Lemma 11, x3 + x2 + ωa = 0 has exactly one solution in Fq if and only if a ∈ − S. This completes the proof. 2 It is clear that a
j =0
(m,q)
(m,q)
|ωa C −2 j ∩ (C j
3.2. Proof of Theorem 3 In this subsection, we give a proof of Theorem 3. Now we must compute the sum ψ(γ a D ) for 0 a q2 − 2, where ψ is the canonical additive character of F2 f . By the orthogonality of characters, we have
3m · ψ
γaD =
3m −1 j =0
G2 f
−j χ3m
j χ3m γ a +i .
i∈I
We compute this sum by partitioning the range of the outer sum in the right-hand side of the above equation into four parts: (i) j = 0; (ii) j = ms for s = 1, 2; (iii) j = 3t + ms for t = 1, . . . , m − 1 and s = 1, 2; and (iv) j = 3t for t = 1, . . . , m − 1.
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0 (i) By G 2 f (χ3m ) = −1, we have
G2 f
0 χ3m
0 χ3m γ a+i = −| I | = −2 f + 2.
i∈I
(ii) We denote
m η = χ3m , which is a multiplicative character of order 3 of Fq2 . Then, we have
G2 f
−ms χ3m
s=1,2
ms χ3m γ a +i =
G2 f
s=1,2
i∈I
η s γ a η s γ 3x+my
x∈ S y =1,2
= −| S |
η −s
G2 f
η −s η s γ a .
(3.5)
s=1,2
Then, by Theorem 4, we obtain
(3.5) = −(−1)
f −1
f
2 |S|
s
η γ
a
s=1,2
f +1 f −1 − 1) if 3 | a; = −f2 f −(21 2 (2 − 1) if 3 a.
(iii) By (3.1), we have −1 m
G2 f
−3t −ms χ3m
s=1,2 t =1
=
3t +ms χ3m γ a +i
i∈I
−1 m
G f
χm 3(−3t −ms) G f χm 3t +ms
s=1,2 t =1
=
m −1
G f
m −1
χm −9t G f χm 3t
m −1
3t +ms χ3m γ a+3x+my
s=1,2 x∈ S y =1,2
G f
χm −9t G f χm 3t χm 3t ωa
t =1
=−
i∈I
t =1
=−
3t +ms χ3m γ a +i
ηs γ a
χm −3t G f χm t χm t ωa
t =1
x∈ S
s=1,2
G f
χm 3t ω3x
ηs γ a
χm t ω3x
.
(3.6)
x∈ S
s=1,2
) = G (χ 2 ), we have Furthermore, by (3.3) and G f (χm f m
(3.6) = −
m −1 1
2
= −2 f
G f
χm −3t G f χm t G f χm 3t χm t ωa
t =1 −1
ηs γ a
s=1,2
−1 m t =1
G f
χm t χm t ωa ·
2 if 3 | a; −1 if 3 a.
(3.7)
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289
t ), we have Let ψ denote the canonical additive character of Fq . By the definition of G f (χm m −1
G f
χm t χm t ωa =
t =1
m −1 m −1
ψ ω s χm t ω s χm t ωa
t =1 s =0
=
m −1
−1 −1 m m ψ ωs χm t ω s+a − ψ ωs
s =0
t =0
s =0
if a (mod m) ∈ − S ; = m · ψ ω−a + 1 = 2 f / −S. −2 + 2 if a (mod m) ∈ f
Therefore, we obtain
⎧ −2 f ⎪ ⎪ ⎨ f 2 − 2 f +1 (3.7) = f −1 2 ⎪ ⎪ ⎩ −2 f −1 + 2 f
if 3 | a and a (mod m) ∈ − S ; if 3 | a and a (mod m) ∈ / −S; if 3 a and a (mod m) ∈ − S ; if 3 a and a (mod m) ∈ / −S.
(iv) By (3.2) and (3.3), we have m −1
G2 f
−3t χ3m
t =1
3t χ3m γ a +i = −
m −1
G f
χm −3t
2
t =1
i∈I
=−
m −1
χm 3t ωa+i
i∈I
G f
2
χm −3t χm 3t ωa
m −1
G f
2
χm −3t χm 3t ωa
m −1
G f
χm 9t ωx
x∈ S
t =1
=−
χm 3t ω3x+my
x∈ S y =1,2
t =1
= −2
2
χm −t G f χm 3t χm t ωa .
(3.8)
t =1
Furthermore, by Lemma 12, there exist subsets S 1 , S 2 ⊆ Z2 f −1 such that S 1 ∪ S 2 = Z2 f −1 \ (− S ), S 1 ∩ S 2 = ∅, and
⎧ f ⎪ ⎨ −2 2 f (3.8) = 2 f − 2 2 +1 ⎪ ⎩ f −2 f +1 + 2 2
if a (mod m) ∈ − S ; if a (mod m) ∈ S 1 ; if a (mod m) ∈ S 2 .
Thus, we finally obtain
⎧ −2 f +1 + 2 f ⎪ ⎪ ⎪ f f +1 ⎪ 2 −2 ⎪ ⎪ ⎨ a −2 f +1 + 2 f f 3m · ψ γ D + 2 − 2 = ⎪ 2 f − 2 f +1 ⎪ ⎪ ⎪ ⎪ 2 f − 2 f +1 ⎪ ⎩ −2 f +1 + 2 f
if 3 | a and a (mod m) ∈ − S ; if 3 | a and a (mod m) ∈ S 1 ; if 3 | a and a (mod m) ∈ S 2 ; if 3 a and a (mod m) ∈ − S ; if 3 a and a (mod m) ∈ S 1 ; if 3 a and a (mod m) ∈ S 2 ,
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K. Momihara / Finite Fields and Their Applications 25 (2014) 280–292
which says that the sums ψ(γ a D ), a = 0, 1, . . . , 3m − 1, take exactly two values. This completes the proof of the theorem. Remark 13. In this remark, we discuss the “isomorphism” problem of strongly regular graphs obtained by Proposition 2 and Theorem 3. Consider the case where ( p , , s) = (2, 1, 2) in Proposition 2. For any partition { S 0 , S 1 , S 2 } of Z/9Z such that | S 0 | = | S 1 | = | S 2 | = 3, the graph Cay (F26 × F24 , D ) with
D =
2 i =0
(9,26 )
Ch
∪ {0}
h∈ S i
3,24 ) × C i(+ j
or
D =
2 i =0
(9,26 )
Ch
h∈ S i
4
(3,2 ) ∪ { 0} × C − , i+ j
is a strongly regular graph with parameters (1024, 330, 98, 110) since every Cay (F26 ,
0 j 2,
h∈ S j
(9,26 )
Ch
)
forms a Latin square type strongly regular graph (cf. [6, Theorem 2]). On the other hand, by Theo-
(93,210 )
rem 3, the graph Cay (F210 , D ) with D = x∈ S y =1,2 C 3x+31 y is a strongly regular graph with the same parameters, where S = {logω (x)(mod 31) | Tr210 /2 (x) = 0, x ∈ F∗210 }. Now, we are interested in the graph isomorphism of Cay (F210 , D ) and Cay (F26 × F24 , D ). (It seems that only one construction of strongly regular graphs with these parameters was known in [18] as listed in Brouwer’s table [4].) By using Magma [3], the author checked that Cay (F210 , D ) and Cay (F26 × F24 , D ) for any { S 0 , S 1 , S 2 } are nonisomorphic. The author believes that the strongly regular graphs obtained by Proposition 2 and Theorem 3 are nonisomorphic for every s 2. 4. Concluding remarks In this paper, we constructed negative Latin square type strongly regular Cayley graphs on F22(2s+1) with parameters (22(2s+1) , 2(22s − 1)(22s+1 + 1)/3, 4(22s − 1)2 /9 − 2, 2(22s+1 + 1)(22s − 1)/9) based on choosing suitable cyclotomic classes of order 3(22s+1 − 1) of F22(2s+1) . In this section, we point out an advantage in obtaining strongly regular graphs with these parameters from cyclotomy of F22(2s+1) . Let h = p 1 p 2 · · · p be a positive integer, where the p i ’s are distinct odd primes. Let p be a prime satisfying the following: for any divisor d = p i 1 · · · p im of h, if p is of index u modulo d, then so is p modulo d = p ix1 · · · p ixm for all xi 1, 1 i m. Let e denote the index of p modulo h. Let p 1 1
m
be a prime factor of k = p 11 p 22 · · · p and set k = kp 1 . Then, by the assumption, p is again of index e
e
e
e in both of (Z/kZ)∗ and (Z/k Z)∗ . Let q = p f and q = p f , where f = φ(k)/e and f = φ(k )/e. In [16], the following recursive construction of strongly regular graphs was given. Theorem 14. Let q = p f and q = p f , where f = φ(k)/e and f = φ(k )/e. Let
J = x x divides k and x is not divisible by p 11 ⊆ N. e
Let I ⊆ {0, 1, . . . , k − 1}. Assume that there is a subset J 1 ⊆ J satisfying the following conditions: (i)
ij i ∈ I ζk
= 0 for all j ∈ J 1 . φ(k )−φ(k)
(ii) G f (χ j ) = p 2e G f (χ j ) for all j ∈ J \ J 1 (=: J 2 ), where = 1 or −1 not depending on j. Here, χ is a multiplicative character of order k of Fq and χ is its restriction to Fq . e 1 +1
Furthermore, assume that (iii) if 2, it holds that G f (χ p 1 e k/ p 11 − 1. Let
v
) = p
φ(k )−φ(k) 2e
e1
G f (χ p 1 v ) for all 1 v
K. Momihara / Finite Fields and Their Applications 25 (2014) 280–292
D=
(k,q)
Ci
and
D =
i∈I
1 −1 p
i ∈ I j =0
C
291
(kp 1 ,q ) e . ip 1 + jk/ p 11
∗
∗
q } contains exactly two values, then so does {ψ ( D ) | ψ ∈ F If {ψ( D ) | ψ ∈ F q }. The following theorem was also given in [16]. Theorem 15. Assume that k is odd and gcd (k , p − 1) = 1. Let χ be a multiplicative character of order k of Fq and χ its restriction to Fq . Then, it holds that
G f
χ j = p
φ(k )−φ(k) 2e
Gf
χj
e
for any j such that p 11 j, where e 1 is the highest power of p 1 dividing k. We now consider the strongly regular graph Cay (F210 , D ) with the parameters (210 , 330, 98, 110) obtained from Theorem 3. Apply Theorem 14 to this strongly regular graph as (, p 1 , p 2 , k) = (2, 31, 3, 93) and J = {1, 3} and J 1 = ∅. Note that 2 is of order 10 × 31i −1 modulo 3 · 31i for every i 1. In order to apply Theorem 14, we need to check that (1) G 10·31 (χ3 ·312 ) = 2(10·31−10)/2 G 10 (χ3·31 ),
) = 2(10·31−10)/2 G (χ ), (2) G 10·31 (χ31 10 31 2
(3) G 10·31 (χ3 ) = 2(10·31−10)/2 G 10 (χ3 ),
where χn is a multiplicative character of order n of F210·31 and χm appeared in the right-hand side of each equation is the restriction of χn to F210 , which is of order m. By Theorem 15, it is clear that (1) and (2) hold. Furthermore, by Theorem 4 we have the equality of (3). Thus, the graph Cay (F210·31 , D ) with D =
i∈ S
30 j =0
h=1,2
(3·312 ,210·31 )
C 31(3i +31h)+3 j becomes a strongly regular graph. By applying Theorem 14
recursively, it follows that the graph Cay (F210·31r −1 , D ) with r −1
D =
31−1 i∈ S
j =0
h=1,2
r −1
(3·31r ,210·31
)
C 3( j +31r −1 i )+31r h
is a strongly regular graph for all r 1. Thus, we obtain an infinite series of strongly regular Cayley graphs on F210·31r −1 . This example points out an advantage in obtaining strongly regular graphs using cyclotomy of F22(2s+1) . Interesting problems which are worth studying in future work are (1) Determine whether the strongly regular graphs obtained from Proposition 2 and Theorem 3 are nonisomorphic for every s 2. (2) Generalize the construction of Theorem 3 to general p and for obtaining strongly regular graphs with the parameters of (1.1). Acknowledgments The author would like to thank the referees and the editor for their helpful and constructive comments which improved the readability of this paper. The author is also very grateful to Professor Qing Xiang, University of Delaware, for his helpful comments and improvements.
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
L.D. Baumert, W.H. Mills, R.L. Ward, Uniform cyclotomy, J. Number Theory 14 (1982) 67–82. B. Berndt, R. Evans, K.S. Williams, Gauss and Jacobi Sums, Wiley, 1997. W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user language, J. Symb. Comput. 24 (1997) 235–265. A.E. Brouwer, Parameters of strongly regular graphs, http://win.tue.nl/~aeb/graphs/srg/srgtab.html. A.E. Brouwer, W.H. Haemers, Spectra of Graphs, Springer, Universitext, 2012. A.E. Brouwer, R.M. Wilson, Q. Xiang, Cyclotomy and strongly regular graphs, J. Algebr. Comb. 10 (1999) 25–28. R. Calderbank, W.M. Kantor, The geometry of two-weight codes, Bull. Lond. Math. Soc. 18 (1986) 97–122. T. Feng, Q. Xiang, Strongly regular graphs from union of cyclotomic classes, J. Comb. Theory, Ser. B 102 (2012) 982–995. T. Feng, K. Momihara, Q. Xiang, Constructions of strongly regular Cayley graphs and skew Hadamard difference sets from cyclotomic classes, arXiv:1201.0701. G.A. Fernández-Alcober, R. Kwahira, L. Martínez, Cyclotomy over products of finite fields and combinatorial applications, Eur. J. Comb. 31 (2010) 1520–1538. G. Ge, Q. Xiang, T. Yuan, Construction of strongly regular Cayley graphs using index four Gauss sums, J. Algebr. Comb. 37 (2013) 313–329. C.L.M. de Lange, Some new cyclotomic strongly regular graphs, J. Algebr. Comb. 4 (1995) 329–330. R. Lidl, H. Niederreiter, Finite Fields, Cambridge Univ. Press, 1997. J.H. van Lint, A. Schrijver, Construction of strongly regular graphs, two-weight codes and partial geometries by finite fields, Combinatorica 1 (1981) 63–73. S.L. Ma, A survey of partial difference sets, Des. Codes Cryptogr. 4 (1994) 221–261. K. Momihara, Cyclotomic strongly regular graphs, skew Hadamard difference sets, and rationality of relative Gauss sums, Eur. J. Comb. 34 (2013) 706–723. K. Momihara, Q. Xiang, Lifting constructions of strongly regular Cayley graphs, arXiv:1212.5752. J.B. Polhill, Generalizations of partial difference sets from cyclotomy to nonelementary abelian p-groups, Electron. J. Comb. 15 (2008) R125. B. Schmidt, C. White, All two-weight irreducible cyclic codes?, Finite Fields Appl. 8 (2002) 1–17. J. Yang, L. Xia, Complete solving of explicit evaluation of Gauss sums in the index 2 case, Sci. China Ser. A 53 (2010) 2525–2542.
Contents lists available at ScienceDirect
Finite Fields and Their Applications www.elsevier.com/locate/ffa
Certain strongly regular Cayley graphs on F22(2s+1) from cyclotomy Koji Momihara 1 Faculty of Education, Kumamoto University, 2-40-1 Kurokami, Kumamoto 860-8555, Japan
a r t i c l e
i n f o
Article history: Received 9 February 2013 Received in revised form 10 August 2013 Accepted 5 October 2013 Available online 26 October 2013 Communicated by Dieter Jungnickel
a b s t r a c t We give a construction of negative Latin square type strongly regular Cayley graphs on F22(2s+1) with parameters (22(2s+1) , 2(22s − 1)(22s+1 + 1)/3, 4(22s − 1)2 /9 − 2, 2(22s+1 + 1)(22s − 1)/9) for every s 1 based on choosing suitable cyclotomic classes on F22(2s+1) and the computation of certain Gauss sums. © 2013 Elsevier Inc. All rights reserved.
MSC: 05E30 05B10 11T22 Keywords: Strongly regular graphs Cyclotomic cosets Gauss sums
1. Introduction In this paper, we will assume that the reader is familiar with the theory of strongly regular graphs. Our main reference is the book by Brouwer and Haemers [5]. We remark that strongly regular graphs are closely related to other combinatorial objects, such as two-weight codes, two-intersection sets in finite geometry, and partial difference sets [5, p. 132], [7,15]. Let Γ be a simple and undirected graph and A be its adjacency matrix. A useful way to check whether a graph 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 multiple of the all-ones vector 1. (For a k-regular connected graph, the restricted eigenvalues are the eigenvalues different from k.)
1
E-mail address: [email protected]. The work of K. Momihara was supported by JSPS under Grant-in-Aid for Research Activity Start-up 23840032.
1071-5797/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ffa.2013.10.006
K. Momihara / Finite Fields and Their Applications 25 (2014) 280–292
281
Theorem 1. For a simple graph Γ of order v, not complete or edgeless, with adjacency matrix A, the following are equivalent: (1) Γ is strongly regular with parameters ( v , k, λ, μ) for certain integers k, λ, μ, (2) A 2 = (λ − μ) A + (k − μ) I + μ J for certain real numbers k, λ, μ, where I , J are the identity matrix and the all-ones matrix, respectively, (3) A has precisely two distinct restricted eigenvalues. A strongly regular graph is said to be of Latin square type (respectively, negative Latin square type) if
( v , k, λ, μ) = (n2 , r (n − ), n + r 2 − 3 r , r 2 − r ) and = 1 (respectively, = −1). Typical examples of strongly regular graphs of Latin square type or negative Latin square type come from nonsingular quadrics in the projective space PG(m − 1, q), where m is even [15]. One of the most effective methods for constructing strongly regular graphs is by the Cayley graph construction. Let G be an additively written group 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 graph on G with connection set D, denoted by Cay (G , D ), is the graph with the elements of G as vertices; two vertices x, y ∈ G are adjacent if and only if x − y ∈ D. In the case when Cay (G , D ) is strongly regular, the connection set D is called a (regular) partial difference set. The survey of Ma [15] contains much of what is known about partial difference sets and about connections with strongly regular Cayley graphs. A classical method for constructing strongly regular Cayley graphs (i.e., partial difference sets) on elementary abelian groups is to use cyclotomic classes of finite fields. Let p be a prime, f a positive integer, and let q = p f . Let k > 1 be an integer such that k | (q − 1), and γ be a primitive (k,q)
= γ i γ k , 0 i k − 1, are called the cyclotomic classes of order element of Fq . Then the cosets C i k of Fq . Quite recently, new “cyclotomic” constructions of strongly regular graphs were given, see [8,9,11,16]. In [19], the authors discussed the problem when a Cayley graph on a finite field with a single cyclotomic class as its connection set is strongly regular. Such a strongly regular graph is called cyclotomic. They made the following conjecture: let F p f be the finite field of order p f , k | (k, p f )
k > 1, and C 0 := C 0 holds:
p f −1 p −1
with
with −C 0 = C 0 . If Cay (F p f , C 0 ) is strongly regular, then one of the following
(1) (subfield case) C 0 is the multiplicative group of a subfield of F p f , (2) (semi-primitive case) −1 ∈ p (Z/kZ)∗ , (3) (exceptional case) it is either of eleven sporadic examples of cyclotomic strongly regular graphs (see [19, Table 1]). This conjecture is still open. On the other hand, in [8,9,11,16,17], several of these sporadic examples have been generalized into infinite families by taking a union of cyclotomic classes. In particular, in [16], the author gave a recursive construction of strongly regular Cayley graphs using cyclotomic classes of finite fields. In the last section of this paper, we will apply the recursive construction to one example of strongly regular graphs obtained from our main theorem. For other constructions of strongly regular graphs from cyclotomy, we refer the reader to the references in [9,12,14,17]. In [18, Theorems 5.1 and 5.2], the author gave constructions of strongly regular Cayley graphs with the parameters (q6 , r1 (q3 + 1), −q3 + r12 + 3r1 , r12 + r1 ) on Fq2 × Fq4 and (q10 , r2 (q5 + 1), −q5 + r22 + 3r2 ,
r22 + r2 ) on Fq4 × Fq6 , where r1 = q(q − 1) and r2 = q(q2 + 1)(q − 1). One can easily generalize his constructions as follows: Proposition 2. Let p be a prime and let and t be positive integers. Assume that the following hold: (1) Let D i ⊆ F p 2(s+1) , 1 i p + 1, be a partition of F∗p 2(s+1) such that D i = − D i and | D i | =
( p 2(s+1) − 1)/( p + 1). The graphs Cay (F p 2(s+1) , D i ), 1 i p + 1, are all negative Latin square type or all Latin square type strongly regular graphs according as s is odd or even.
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K. Momihara / Finite Fields and Their Applications 25 (2014) 280–292
(2) Let E i ⊆ F p 2s , 1 i p + 1, be a partition of F∗p 2s such that E i = − E i and | E i | = ( p 2s − 1)/( p + 1). The graphs Cay (F p 2s , D i ), 1 i p + 1, are all negative Latin square type or all Latin square type strongly regular graphs according as s is even or odd. Then, Cay (F p 2(s+1) × F p 2s , D ) is a negative Latin square type strongly regular graph, where p +1
D=
D i ∪ { 0} × E i .
i =1
As an immediate corollary, we can show that there exists a negative Latin square type strongly regular graph with parameters
( v , k, λ, μ) = p 2(2s+1) , r p (2s+1) + 1 , − p (2s+1) + r 2 + 3r , r 2 + r ,
(1.1)
where r = p ( p 2s − 1)/( p + 1), as follows: each of the cyclotomic classes of order p + 1 in F p 2t becomes a connection set of Latin square type or negative Latin square type strongly regular Cayley graph on F p 2t according as t is odd or even (cf. [6, Theorem 2]). By applying Proposition 2 to these strongly regular graphs with t = s + 1 in (1) and with t = s in (2), we obtain a strongly regular graph with the parameters of (1.1). This existence result was also shown in [10, Proposition 4.3]. In this paper, we are interested in strongly regular Cayley graphs with parameters
2
22(2s+1) , 2 22s − 1 22s+1 + 1 /3, 4 22s − 1 /9 − 2, 2 22s+1 + 1 22s − 1 /9 ,
which is a special case of (1.1). The main objective of this paper is to present a construction of strongly regular Cayley graphs with these parameters using cyclotomy of order 3(22s+1 − 1) of F22(2s+1) , which is different from the product construction given in Proposition 2. The approach to proving the result is much more theoretical and it represents progress on the theory of cyclotomy. Furthermore, as described in Remark 13, the strongly regular graphs with parameters (210 , 330, 98, 110) obtained from Proposition 2 and our main theorem are nonisomorphic. The following is our main theorem.
Theorem 3. Let p = 2, m = 2 f − 1, q = 2 f , and f = 2s + 1. Let S = {logω (x)(mod m) | Trq/2 (x) = 0, x ∈ Fq∗ } and I = {3x + my (mod 3m) | x ∈ S , y = 1, 2}, where ω is a fixed primitive element of Fq and Trq/2 is the trace from Fq to F2 . Furthermore, let
D=
(3m,q2 )
Ci
,
i∈I
(3m,q2 )
where C i = γ i γ 3m and strongly regular.
γ is a primitive element of Fq2 such that γ q+1 = ω. Then, Cay (Fq2 , D ) is
2. Preliminaries 2.1. Preliminaries on characters Let p be a prime, f a positive integer, and q = p f . The canonical additive character ψ of Fq is defined by
ψ : Fq → C∗ ,
Trq/ p (x)
ψ(x) = ζ p
,
K. Momihara / Finite Fields and Their Applications 25 (2014) 280–292 i where ζ p = exp( 2π ) and Trq/ p is the trace from Fq to F p . For a multiplicative character p define the Gauss sum
G f (χ ) =
283
χ of Fq , we
χ (x)ψ(x),
x∈Fq∗
which belongs to Z[ζkp ] of integers in the cyclotomic field Q(ζkp ), where k is the order of be the automorphism of Q(ζkp ) determined by
σa,b (ζk ) = ζka ,
χ . Let σa,b
σa,b (ζ p ) = ζ pb
for gcd (a, k) = gcd (b, p ) = 1. Below are several basic properties of Gauss sums [13]: (i) (ii) (iii) (iv) (v)
G f (χ )G f (χ ) = q if χ is nontrivial; G f (χ p ) = G f (χ ), where p is the characteristic of Fq ; G f (χ −1 ) = χ (−1)G f (χ ); G f (χ ) = −1 if χ is trivial; σa,b (G f (χ )) = χ −a (b)G f (χ a ).
In general, explicit evaluations of Gauss sums are very difficult. There are only a few cases where the Gauss sums have been evaluated. The most well-known case is the quadratic case, i.e., the order of χ is two. The next simple case is the so-called semi-primitive case (also known as uniform cyclotomy or pure Gauss sum), where there exists an integer j such that p j ≡ −1 (mod k), where k is the order of the multiplicative character involved. The explicit evaluation of Gauss sums in this case is given as follows: Theorem 4. (See [2].) Suppose that k > 2 and p is semi-primitive modulo k, i.e., there exists an s such that p s ≡ −1 (mod k). Choose s minimal and write f = 2st. Let χ be a multiplicative character of order k. Then,
p − f /2 G f (χ ) =
if p = 2; (−1)t −1 s (−1)t −1+( p +1)t /k if p > 2.
This theorem was used to find strongly regular graphs on Fq (cf. [1,6]) and we will use this theorem in the next section. The next interesting case is the index 2 case where the subgroup p generated by p ∈ (Z/kZ)∗ is of index 2 in (Z/kZ)∗ and −1 ∈ / p . Many authors have investigated this case, see [20] for a complete solution to the problem of evaluating index 2 Gauss sums. Now we recall the following well-known lemma in Algebraic graph theory (cf. [5]). Lemma 5. Let (G , +) be an abelian group and D a subset of G such that 0 ∈ / D and D = − D. Then, the eigenvalues of Cay (G , D ) are given by ψ( D ), ψ ∈ G, where G is the character group of G. (k,q)
Let q be a prime power and let C i = γ i γ k , 0 i k − 1, be the cyclotomic classes of order k of Fq , where γ is a fixed primitive element of Fq . In order to check whether a candidate subset
(k,q)
is a connection set of a strongly D = i∈ I C i regular Cayley graph, by Theorem 1 and Lemma 5, it is enough to show that the sums ψ(γ a D ) = x∈ D ψ(γ a x), a = 0, 1, . . . , q − 2, take exactly two values, where ψ is the canonical additive character of Fq . Note that the sum ψ(γ a D ) can be expressed as a linear combination of Gauss sums (cf. [8]) by using the orthogonality of characters:
1 a i ψ γaD = G f χ −1 χ γ γ , k
χ ∈C 0⊥
i∈I
(2.1)
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K. Momihara / Finite Fields and Their Applications 25 (2014) 280–292
where C 0⊥ is the subgroup of Fq∗ consisting of all
χ which are trivial on C 0(k,q) . Thus, the computations
(k,q)
is a connection set of a strongly reguneeded to show whether a candidate subset D = i ∈ I C i lar Cayley graph are essentially reduced to evaluating Gauss sums. However, evaluating Gauss sums explicitly is very difficult. Therefore, we will need a technique in order to avoid explicitly evaluating Gauss sums. In particular, we use the Davenport–Hasse lifting formula and product formula stated below. Theorem 6. (See [2].) Let χ be a nontrivial multiplicative character of Fq = F p f and let χ be the lift of χ to Fq = F p f s , i.e., χ (α ) = χ (Normq /q (α )) for α ∈ Fq , where s 2 is an integer. Then
Gfs
s
χ = (−1)s−1 G f (χ ) .
Theorem 7. (See [2].) Let η be a multiplicative character of order > 1 of Fq = F p f . For every nontrivial multiplicative character χ of Fq ,
G f (χ ) =
−1 G f (χ ) G f (ηi )
χ ()
i =1
G f (χηi )
.
Now, we prove the following theorem, which will be used in the next section.
Theorem 8. Let p = 2, f = 2 f , and m = 2 f − 1, where f > 0 is any odd integer. Let χ3m be a multiplicative character of order 3m = 3(2 f − 1) of F2 f and write χ3m = χ3 · χm for multiplicative characters χ3 and χm of order 3 and m, respectively, of F2 f . Then, it holds that
G f (χ3m ) = G f
χm 3 G f χm −1 ,
is a multiplicative character of F such that χ is the lift of χ to F . where χm m m 2f 2f
Proof. By applying the Davenport–Hasse product formula (Theorem 7) as = 3, we have
1=
=
χ = χ3m , and η = χ3 ,
3 χ3m (3)G f (χ3m ) G f (χ3m χ3 )G f (χ3m χ32 ) · 3 G f (χ3 )G f (χ32 ) G f (χ3m )
G f (χ3m ) 3 G f (χm )
f
·
2 G f (χ3m )G f (χm )
2f
=
G f (χ3m )2 3 G f (χm )G f (χm−1 )
,
−1 ) = 2 f . Furthermore, by the Davenport–Hasse lifting where we used G f (χ3 )G f (χ32 ) = G f (χm )G f (χm formula (Theorem 6), we have
Gf
2
2
χm3 G f χm−1 = G f χm 3 G f χm −1 .
Hence, we have
(τ :=)
G f (χ3m ) 3 )G (χ −1 ) G (χm f m f
= 1 or −1.
K. Momihara / Finite Fields and Their Applications 25 (2014) 280–292
285
Now we consider the reduction of G f (χ3m )3 modulo 3. By Theorems 6 and 4, we have
G f (χ3m )3 ≡
z∈F∗ f
3 χ3m ( z)ψ(3z) (mod 3)
p
=
z∈F∗ f
3 χ3m ( z)ψ( z)
p
2 3 = −G f χm 3 . = G f χ3m On the other hand, we have
τ G f χm 3 G f χm −1
3
3 3 ≡ τ 3 G f χm 3 G f χm −1 (mod 3) 3 ≡ τ G f χm 3 G f χm −3 (mod 3) 2 = τ 2 f G f χm 3 .
3 )2 ≡ 0 (mod 3). By noting that 2 f + 1 ≡ 0 (mod 3) and 3 G (χ 3 ), Hence, we have (τ 2 f + 1)G f (χm f m we obtain τ = 1. This completes the proof of the theorem. 2
3. Construction of strongly regular Cayley graphs on F 22(2s+1) In this section, we give a proof of our main theorem. We fix the notation as follows: let p = 2, m = 2 f − 1, q = 2 f , and f = 2s + 1. Then, by Theorem 8, for any t such that gcd (t , 3) = 1 and m t we have
G2 f
t = G f χm 3t G f χm −t . χ3m
(3.1)
χm3 (γ ) = χm 3 (ω), by Theorem 6 we have
Furthermore, since
G2 f
2
3t = −G f χm 3t . χ3m
(3.2)
Moreover, we will use the following:
G f
χm t =
χm t (x) −
Tr
(x)=0 2 f /2
Tr
χm t (x) = 2
(x)=1 2 f /2
χm t (x).
Tr
(x)=0 2 f /2
3.1. Lemmas In this subsection, we give a few lemmas as preparation for proving our main theorem. Let χ1 and χ2 be multiplicative characters of Fq . We define the sum
J (χ1 , χ2 ) =
χ1 (x)χ2 (1 − x),
x∈Fq , x =0,1
the so-called Jacobi sum of Fq . We will use the following propositions.
(3.3)
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Proposition 9. (See Theorem 5.21 [13].) Let χ1 and χ2 be nontrivial multiplicative characters of F p f . Then, if χ1 χ2 is nontrivial, it holds that
J (χ1 , χ2 ) =
G f (χ1 )G f (χ2 ) G f (χ1 χ2 )
.
Proposition 10. (See Theorem 2.51 [2].) For i , j = 0, 1, . . . , k − 1, it holds that k −1 k −1 (k, p f ) 1 u (k, p f ) C ∩ + 1 C χ (−1)ζk−iu− jv J χ u , χ v , = i j 2
k
(k, p f )
where C i = ωi ωk with F p f such that χ (ω) = ζk .
u =0 v =0
ω a primitive element of F p f and χ is a multiplicative character of order k of
Throughout the rest of this paper, we use the notation of Theorem 3. Lemma 11. Let f a (x) := x3 + x2 + ωa ∈ Fq [x]. Then, f a (x) = 0 has exactly one solution in Fq if and only if a ∈ − S. Proof. Since x3 + x2 + ωa = 0 does not have any solution with multiplicity three in Fq , it is enough to consider the equations of the form (x + c )(x2 + dx + e ) = 0, where c = d + 1 and cd + e = 0, i.e., (x + d + 1)(x2 + dx + d(d + 1)) = 0, and x2 + dx + d(d + 1) is irreducible in Fq [x]. It is well known that the polynomial x2 + ux + v ∈ Fq [x] with u = 0 and q an even prime power is irreducible in Fq [x] if and only if Trq/2 (u −2 v ) = 1 (cf. [13, Corollary 5.35]). Hence, x2 + dx + d(d + 1) is irreducible in Fq [x] if and only if Trq/2 (d−1 ) = 0 by noting that Trq/2 (1) = 1. Furthermore, we have
Trq/2 d−1 (d + 1)−2 = Trq/2
1 d
+
1 d+1
+
1 d+1
2
= Trq/2 d−1 .
Thus, x2 + dx + d(d + 1) = 0 is irreducible in Fq [x] if and only if Trq/2 (d−1 (d + 1)−2 ) = 0, i.e., f a (x) = 0 has exactly one solution in Fq if and only if ωa = d(d + 1)2 and it satisfies Trq/2 (d−1 (d + 1)−2 ) = 0. For d, d ∈ Fq∗ with d = d such that Trq/2 (d−1 ) = Trq/2 (d −1 ) = 0, if ωa = d(d + 1)2 = d (d + 1)2 , then we have (x + d + 1)(x2 + dx + d(d + 1)) = (x + d + 1)(x2 + d x + d (d + 1)) = x3 + x2 + ωa . This means that f a (x) has two different factorizations in Fq [x], which is impossible. Hence, we have {d(d + 1)2 ∈ Fq∗ | Trq/2 (d−1 ) = 0} = {d ∈ Fq∗ | Trq/2 (d−1 ) = 0}. This shows the assertion of the lemma. 2 Lemma 12. There are subsets S 1 , S 2 ⊆ Zm such that S 1 ∪ S 2 = Zm \ (− S ), S 1 ∩ S 2 = ∅, and m −1
G f
−t 2
χm
G f
3t
t
χm χm ω
a
t =1
⎧ f ⎪ if a (mod m) ∈ − S ; ⎨22 f = −2 f + 2 2 +1 if a (mod m) ∈ S 1 ; ⎪ ⎩ f +1 f 2 −22 if a (mod m) ∈ S 2 .
2 ) = G (χ ). Then, by Proposition 9, we have Proof. First note that G f (χm f m m −1 t =1
G f
2
χm −t G f χm 3t χm t ωa = 2 f
m −1 t =1
J
χm −t , χm −2t χm t ωa .
(3.4)
K. Momihara / Finite Fields and Their Applications 25 (2014) 280–292
287
On the other hand, by Proposition 10, we have m −1 m −1 a (m,q) (m,q) 1 j (2u − v ) −u a u v ω C ∩ C +1 = ζm χ ω J χ ,χ ,
−2 j
j
where we can assume that
m
m2
m
m
u =0 v =0
χm (ω) = ζm . Hence, we further have
m −1
m −1 m −1 m −1 a (m,q) (m,q) 1 j (2u − v ) −u a u v ω C ∩ C +1 = ζm χ ω J χ ,χ
j =0
−2 j
j
m2
m
m
m
j =0 u =0 v =0
m −1 1 −u a u 2u χm ω J χm , χm m
=
u =0
1
=
m
m −1
χm −u ωa J χm u , χm 2u +
m−1
u =1
m
.
Then, (3.4) is reformulated as follows:
(3.4) = 2
f
−1 a (m,q) (m,q) m f 2 −1 ω C −2 j ∩ C j +1 − 2 −2 . f
j =0
Now, we must show that m −1
a (m,q) (m,q) ω C ∩ C +1 = −2 j
j =0
m−1
j
1 if a(mod m) ∈ − S ; 0 if a (mod m) ∈ S 1 ; 3 if a (mod m) ∈ S 2 .
+ 1)| is the number of solutions of ωa x−2 = x + 1, i.e., x3 + x2 + ω = 0 in Fq . Furthermore, it is impossible that x3 + x2 + ωa = 0 has exactly two distinct solutions. As shown in Lemma 11, x3 + x2 + ωa = 0 has exactly one solution in Fq if and only if a ∈ − S. This completes the proof. 2 It is clear that a
j =0
(m,q)
(m,q)
|ωa C −2 j ∩ (C j
3.2. Proof of Theorem 3 In this subsection, we give a proof of Theorem 3. Now we must compute the sum ψ(γ a D ) for 0 a q2 − 2, where ψ is the canonical additive character of F2 f . By the orthogonality of characters, we have
3m · ψ
γaD =
3m −1 j =0
G2 f
−j χ3m
j χ3m γ a +i .
i∈I
We compute this sum by partitioning the range of the outer sum in the right-hand side of the above equation into four parts: (i) j = 0; (ii) j = ms for s = 1, 2; (iii) j = 3t + ms for t = 1, . . . , m − 1 and s = 1, 2; and (iv) j = 3t for t = 1, . . . , m − 1.
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0 (i) By G 2 f (χ3m ) = −1, we have
G2 f
0 χ3m
0 χ3m γ a+i = −| I | = −2 f + 2.
i∈I
(ii) We denote
m η = χ3m , which is a multiplicative character of order 3 of Fq2 . Then, we have
G2 f
−ms χ3m
s=1,2
ms χ3m γ a +i =
G2 f
s=1,2
i∈I
η s γ a η s γ 3x+my
x∈ S y =1,2
= −| S |
η −s
G2 f
η −s η s γ a .
(3.5)
s=1,2
Then, by Theorem 4, we obtain
(3.5) = −(−1)
f −1
f
2 |S|
s
η γ
a
s=1,2
f +1 f −1 − 1) if 3 | a; = −f2 f −(21 2 (2 − 1) if 3 a.
(iii) By (3.1), we have −1 m
G2 f
−3t −ms χ3m
s=1,2 t =1
=
3t +ms χ3m γ a +i
i∈I
−1 m
G f
χm 3(−3t −ms) G f χm 3t +ms
s=1,2 t =1
=
m −1
G f
m −1
χm −9t G f χm 3t
m −1
3t +ms χ3m γ a+3x+my
s=1,2 x∈ S y =1,2
G f
χm −9t G f χm 3t χm 3t ωa
t =1
=−
i∈I
t =1
=−
3t +ms χ3m γ a +i
ηs γ a
χm −3t G f χm t χm t ωa
t =1
x∈ S
s=1,2
G f
χm 3t ω3x
ηs γ a
χm t ω3x
.
(3.6)
x∈ S
s=1,2
) = G (χ 2 ), we have Furthermore, by (3.3) and G f (χm f m
(3.6) = −
m −1 1
2
= −2 f
G f
χm −3t G f χm t G f χm 3t χm t ωa
t =1 −1
ηs γ a
s=1,2
−1 m t =1
G f
χm t χm t ωa ·
2 if 3 | a; −1 if 3 a.
(3.7)
K. Momihara / Finite Fields and Their Applications 25 (2014) 280–292
289
t ), we have Let ψ denote the canonical additive character of Fq . By the definition of G f (χm m −1
G f
χm t χm t ωa =
t =1
m −1 m −1
ψ ω s χm t ω s χm t ωa
t =1 s =0
=
m −1
−1 −1 m m ψ ωs χm t ω s+a − ψ ωs
s =0
t =0
s =0
if a (mod m) ∈ − S ; = m · ψ ω−a + 1 = 2 f / −S. −2 + 2 if a (mod m) ∈ f
Therefore, we obtain
⎧ −2 f ⎪ ⎪ ⎨ f 2 − 2 f +1 (3.7) = f −1 2 ⎪ ⎪ ⎩ −2 f −1 + 2 f
if 3 | a and a (mod m) ∈ − S ; if 3 | a and a (mod m) ∈ / −S; if 3 a and a (mod m) ∈ − S ; if 3 a and a (mod m) ∈ / −S.
(iv) By (3.2) and (3.3), we have m −1
G2 f
−3t χ3m
t =1
3t χ3m γ a +i = −
m −1
G f
χm −3t
2
t =1
i∈I
=−
m −1
χm 3t ωa+i
i∈I
G f
2
χm −3t χm 3t ωa
m −1
G f
2
χm −3t χm 3t ωa
m −1
G f
χm 9t ωx
x∈ S
t =1
=−
χm 3t ω3x+my
x∈ S y =1,2
t =1
= −2
2
χm −t G f χm 3t χm t ωa .
(3.8)
t =1
Furthermore, by Lemma 12, there exist subsets S 1 , S 2 ⊆ Z2 f −1 such that S 1 ∪ S 2 = Z2 f −1 \ (− S ), S 1 ∩ S 2 = ∅, and
⎧ f ⎪ ⎨ −2 2 f (3.8) = 2 f − 2 2 +1 ⎪ ⎩ f −2 f +1 + 2 2
if a (mod m) ∈ − S ; if a (mod m) ∈ S 1 ; if a (mod m) ∈ S 2 .
Thus, we finally obtain
⎧ −2 f +1 + 2 f ⎪ ⎪ ⎪ f f +1 ⎪ 2 −2 ⎪ ⎪ ⎨ a −2 f +1 + 2 f f 3m · ψ γ D + 2 − 2 = ⎪ 2 f − 2 f +1 ⎪ ⎪ ⎪ ⎪ 2 f − 2 f +1 ⎪ ⎩ −2 f +1 + 2 f
if 3 | a and a (mod m) ∈ − S ; if 3 | a and a (mod m) ∈ S 1 ; if 3 | a and a (mod m) ∈ S 2 ; if 3 a and a (mod m) ∈ − S ; if 3 a and a (mod m) ∈ S 1 ; if 3 a and a (mod m) ∈ S 2 ,
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K. Momihara / Finite Fields and Their Applications 25 (2014) 280–292
which says that the sums ψ(γ a D ), a = 0, 1, . . . , 3m − 1, take exactly two values. This completes the proof of the theorem. Remark 13. In this remark, we discuss the “isomorphism” problem of strongly regular graphs obtained by Proposition 2 and Theorem 3. Consider the case where ( p , , s) = (2, 1, 2) in Proposition 2. For any partition { S 0 , S 1 , S 2 } of Z/9Z such that | S 0 | = | S 1 | = | S 2 | = 3, the graph Cay (F26 × F24 , D ) with
D =
2 i =0
(9,26 )
Ch
∪ {0}
h∈ S i
3,24 ) × C i(+ j
or
D =
2 i =0
(9,26 )
Ch
h∈ S i
4
(3,2 ) ∪ { 0} × C − , i+ j
is a strongly regular graph with parameters (1024, 330, 98, 110) since every Cay (F26 ,
0 j 2,
h∈ S j
(9,26 )
Ch
)
forms a Latin square type strongly regular graph (cf. [6, Theorem 2]). On the other hand, by Theo-
(93,210 )
rem 3, the graph Cay (F210 , D ) with D = x∈ S y =1,2 C 3x+31 y is a strongly regular graph with the same parameters, where S = {logω (x)(mod 31) | Tr210 /2 (x) = 0, x ∈ F∗210 }. Now, we are interested in the graph isomorphism of Cay (F210 , D ) and Cay (F26 × F24 , D ). (It seems that only one construction of strongly regular graphs with these parameters was known in [18] as listed in Brouwer’s table [4].) By using Magma [3], the author checked that Cay (F210 , D ) and Cay (F26 × F24 , D ) for any { S 0 , S 1 , S 2 } are nonisomorphic. The author believes that the strongly regular graphs obtained by Proposition 2 and Theorem 3 are nonisomorphic for every s 2. 4. Concluding remarks In this paper, we constructed negative Latin square type strongly regular Cayley graphs on F22(2s+1) with parameters (22(2s+1) , 2(22s − 1)(22s+1 + 1)/3, 4(22s − 1)2 /9 − 2, 2(22s+1 + 1)(22s − 1)/9) based on choosing suitable cyclotomic classes of order 3(22s+1 − 1) of F22(2s+1) . In this section, we point out an advantage in obtaining strongly regular graphs with these parameters from cyclotomy of F22(2s+1) . Let h = p 1 p 2 · · · p be a positive integer, where the p i ’s are distinct odd primes. Let p be a prime satisfying the following: for any divisor d = p i 1 · · · p im of h, if p is of index u modulo d, then so is p modulo d = p ix1 · · · p ixm for all xi 1, 1 i m. Let e denote the index of p modulo h. Let p 1 1
m
be a prime factor of k = p 11 p 22 · · · p and set k = kp 1 . Then, by the assumption, p is again of index e
e
e
e in both of (Z/kZ)∗ and (Z/k Z)∗ . Let q = p f and q = p f , where f = φ(k)/e and f = φ(k )/e. In [16], the following recursive construction of strongly regular graphs was given. Theorem 14. Let q = p f and q = p f , where f = φ(k)/e and f = φ(k )/e. Let
J = x x divides k and x is not divisible by p 11 ⊆ N. e
Let I ⊆ {0, 1, . . . , k − 1}. Assume that there is a subset J 1 ⊆ J satisfying the following conditions: (i)
ij i ∈ I ζk
= 0 for all j ∈ J 1 . φ(k )−φ(k)
(ii) G f (χ j ) = p 2e G f (χ j ) for all j ∈ J \ J 1 (=: J 2 ), where = 1 or −1 not depending on j. Here, χ is a multiplicative character of order k of Fq and χ is its restriction to Fq . e 1 +1
Furthermore, assume that (iii) if 2, it holds that G f (χ p 1 e k/ p 11 − 1. Let
v
) = p
φ(k )−φ(k) 2e
e1
G f (χ p 1 v ) for all 1 v
K. Momihara / Finite Fields and Their Applications 25 (2014) 280–292
D=
(k,q)
Ci
and
D =
i∈I
1 −1 p
i ∈ I j =0
C
291
(kp 1 ,q ) e . ip 1 + jk/ p 11
∗
∗
q } contains exactly two values, then so does {ψ ( D ) | ψ ∈ F If {ψ( D ) | ψ ∈ F q }. The following theorem was also given in [16]. Theorem 15. Assume that k is odd and gcd (k , p − 1) = 1. Let χ be a multiplicative character of order k of Fq and χ its restriction to Fq . Then, it holds that
G f
χ j = p
φ(k )−φ(k) 2e
Gf
χj
e
for any j such that p 11 j, where e 1 is the highest power of p 1 dividing k. We now consider the strongly regular graph Cay (F210 , D ) with the parameters (210 , 330, 98, 110) obtained from Theorem 3. Apply Theorem 14 to this strongly regular graph as (, p 1 , p 2 , k) = (2, 31, 3, 93) and J = {1, 3} and J 1 = ∅. Note that 2 is of order 10 × 31i −1 modulo 3 · 31i for every i 1. In order to apply Theorem 14, we need to check that (1) G 10·31 (χ3 ·312 ) = 2(10·31−10)/2 G 10 (χ3·31 ),
) = 2(10·31−10)/2 G (χ ), (2) G 10·31 (χ31 10 31 2
(3) G 10·31 (χ3 ) = 2(10·31−10)/2 G 10 (χ3 ),
where χn is a multiplicative character of order n of F210·31 and χm appeared in the right-hand side of each equation is the restriction of χn to F210 , which is of order m. By Theorem 15, it is clear that (1) and (2) hold. Furthermore, by Theorem 4 we have the equality of (3). Thus, the graph Cay (F210·31 , D ) with D =
i∈ S
30 j =0
h=1,2
(3·312 ,210·31 )
C 31(3i +31h)+3 j becomes a strongly regular graph. By applying Theorem 14
recursively, it follows that the graph Cay (F210·31r −1 , D ) with r −1
D =
31−1 i∈ S
j =0
h=1,2
r −1
(3·31r ,210·31
)
C 3( j +31r −1 i )+31r h
is a strongly regular graph for all r 1. Thus, we obtain an infinite series of strongly regular Cayley graphs on F210·31r −1 . This example points out an advantage in obtaining strongly regular graphs using cyclotomy of F22(2s+1) . Interesting problems which are worth studying in future work are (1) Determine whether the strongly regular graphs obtained from Proposition 2 and Theorem 3 are nonisomorphic for every s 2. (2) Generalize the construction of Theorem 3 to general p and for obtaining strongly regular graphs with the parameters of (1.1). Acknowledgments The author would like to thank the referees and the editor for their helpful and constructive comments which improved the readability of this paper. The author is also very grateful to Professor Qing Xiang, University of Delaware, for his helpful comments and improvements.
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K. Momihara / Finite Fields and Their Applications 25 (2014) 280–292
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
L.D. Baumert, W.H. Mills, R.L. Ward, Uniform cyclotomy, J. Number Theory 14 (1982) 67–82. B. Berndt, R. Evans, K.S. Williams, Gauss and Jacobi Sums, Wiley, 1997. W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user language, J. Symb. Comput. 24 (1997) 235–265. A.E. Brouwer, Parameters of strongly regular graphs, http://win.tue.nl/~aeb/graphs/srg/srgtab.html. A.E. Brouwer, W.H. Haemers, Spectra of Graphs, Springer, Universitext, 2012. A.E. Brouwer, R.M. Wilson, Q. Xiang, Cyclotomy and strongly regular graphs, J. Algebr. Comb. 10 (1999) 25–28. R. Calderbank, W.M. Kantor, The geometry of two-weight codes, Bull. Lond. Math. Soc. 18 (1986) 97–122. T. Feng, Q. Xiang, Strongly regular graphs from union of cyclotomic classes, J. Comb. Theory, Ser. B 102 (2012) 982–995. T. Feng, K. Momihara, Q. Xiang, Constructions of strongly regular Cayley graphs and skew Hadamard difference sets from cyclotomic classes, arXiv:1201.0701. G.A. Fernández-Alcober, R. Kwahira, L. Martínez, Cyclotomy over products of finite fields and combinatorial applications, Eur. J. Comb. 31 (2010) 1520–1538. G. Ge, Q. Xiang, T. Yuan, Construction of strongly regular Cayley graphs using index four Gauss sums, J. Algebr. Comb. 37 (2013) 313–329. C.L.M. de Lange, Some new cyclotomic strongly regular graphs, J. Algebr. Comb. 4 (1995) 329–330. R. Lidl, H. Niederreiter, Finite Fields, Cambridge Univ. Press, 1997. J.H. van Lint, A. Schrijver, Construction of strongly regular graphs, two-weight codes and partial geometries by finite fields, Combinatorica 1 (1981) 63–73. S.L. Ma, A survey of partial difference sets, Des. Codes Cryptogr. 4 (1994) 221–261. K. Momihara, Cyclotomic strongly regular graphs, skew Hadamard difference sets, and rationality of relative Gauss sums, Eur. J. Comb. 34 (2013) 706–723. K. Momihara, Q. Xiang, Lifting constructions of strongly regular Cayley graphs, arXiv:1212.5752. J.B. Polhill, Generalizations of partial difference sets from cyclotomy to nonelementary abelian p-groups, Electron. J. Comb. 15 (2008) R125. B. Schmidt, C. White, All two-weight irreducible cyclic codes?, Finite Fields Appl. 8 (2002) 1–17. J. Yang, L. Xia, Complete solving of explicit evaluation of Gauss sums in the index 2 case, Sci. China Ser. A 53 (2010) 2525–2542.