Strongly regular graphs constructed from p-ary bent functions

Transcription

1 J Algebr Comb 011) 3:51 66 DOI /s Strongly regular grahs constructed from -ary bent functions Yeow Meng Chee Yin Tan Xian De Zhang Received: 8 January 010 / Acceted: 19 November 010 / Published online: December 010 Sringer Science+Business Media, LLC 010 Abstract In this aer, we generalize the construction of strongly regular grahs in Tan et al. J. Comb. Theory, Ser. A 117:668 68, 010) from ternary bent functions to -ary bent functions, where is an odd rime. We obtain strongly regular grahs with three tyes of arameters. Using certain non-quadratic -ary bent functions, our constructions can give rise to new strongly regular grahs for small arameters. Keywords Strongly regular grahs Partial difference sets -ary bent functions Weakly) regular bent functions 1 Introduction Boolean bent functions were first introduced by Rothaus in 1976 [15]. They have been extensively studied for their imortant alications in crytograhy. Such functions have the maximum Hamming distance to the set of all affine functions. In [10], the authors generalized the notion of a bent function to be defined over a finite field of arbitrary characteristic. Precisely, let f be a function from F n to F.TheWalsh Y.M. Chee Y. Tan ) X.D. Zhang Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 1 Nanyang Link, Singaore , Singaore itanyinmath@gmail.com Y.M. Chee ymchee@ntu.edu.sg X.D. Zhang xiandezhang@ntu.edu.sg Present address: Y. Tan Temasek Laboratories, National University of Singaore, 5A Engineering Drive 1, 09-0, Singaore 11711, Singaore

2 5 J Algebr Comb 011) 3:51 66 transform of f is the comlex valued function W f : F n C defined by W f b) := ζ fx)+trbx), b F n, x F n where ζ is a rimitive th root of unity and Trx) is the absolute trace function, i.e., Trx) := n 1 i=0 xi. The function f is called -ary bent if every Walsh coefficient W f b) has magnitude n/, i.e., W f b) = n/ for all b F n. Moreover, f is called regular if there exists some function f : F n F such that W f b) = n/ ζ f b), and f is called weakly regular if W f b) = μ n/ ζ f b) for some constant μ C with μ =1. Obviously, regularity imlies weak regularity. Itisshownin[7, 9] that quadratic bent functions and most monomial -ary bent functions are weakly regular, excet for one soradic non-weakly regular examle. Recently, a new family of non-quadratic weakly regular -ary bent functions has been constructed in [8] and a new soradic non-weakly regular bent function was given. However, there are still few non-quadratic bent functions known over the field F n when 5. There are many motivations to find more weakly regular bent functions, esecially non-quadratic ones. Recently, it is shown in [1, 16] that, under some conditions, weakly regular bent functions can be used to construct certain combinatorial objects, such as strongly regular grahs and association schemes. Precisely, let f : F 3 k F 3 be a weakly regular bent function. Define D i := { x : x F 3 k fx)= i }, 0 i. It is shown in [16] that D 0,D 1,D are all regular artial difference sets. The Cayley grahs generated by D 0,D 1,D in the additive grou of F 3 k are strongly regular grahs. Some non-quadratic bent functions seem to give rise to new families of strongly regular grahs u to isomorhism see [16, Tables, 3]). In this aer, we generalize the work in [16]byusing-ary bent functions to construct strongly regular grahs. We show that if f : F k F satisfies Condition A defined in Sect. 3), then the subsets D = { x F fx)= 0 }, k D S = { x F fx)is a non-zero square }, k D S = { x F } fx)is a square, and k D N = { x F } fx)is a non-square k 1) are regular artial difference sets. Using this construction, it seems that the -ary bent functions in [8] may give rise to new negative Latin square tye strongly regular grahs. For small arameters, we have verified that the grahs are new. The aer is organized as follows. In Sect., we give necessary definitions and results. The constructions of strongly regular grahs will be given in Sect. 3. In Sect., we discuss the newness of the grahs obtained.

3 J Algebr Comb 011) 3: Preliminaries Grou rings and character theory are useful tools to study difference sets. We refer to [13] for the basic facts of grou rings and [11] for the character theory on finite fields. Let G be a multilicative grou of order v. Ak-subset D of G is a v,k,λ,μ) artial difference set PDS) if each non-identity element in D can be reresented as gh 1 g, h D,g h) in exactly λ ways, and each non-identity element in G\D can be reresented as gh 1 g, h D,g h) in exactly μ ways. We shall always assume that the identity element 1 G of G is not contained in D. Using the grou ring language, a k-subset D of G with 1 G D is a v,k,λ,μ)-pds if and only if the following equation holds: DD 1) = k μ)1 G + λ μ)d + μg. ) Combinatorial objects associated with artial difference sets are strongly regular grahs. A grah Γ with v vertices is called a v,k,λ,μ) strongly regular grah SRG) if each vertex is adjacent to exactly k other vertices, any two adjacent vertices have exactly λ common neighbors, and any two non-adjacent vertices have exactly μ common neighbors. GivenagrouG of order v and a k-subset D of G with 1 G D and D 1) = D, the grah Γ = V, E) defined as follows is called the Cayley grah generated by D in G: 1. The vertex set V is G.. Two vertices g,h are joined by an edge if and only if gh 1 D. The following result oints out the relationshi between SRGs and PDSs. Result 1 [1] Let Γ be the Cayley grah generated by a k-subset D of a multilicative grou G with order v. Then Γ is a v,k,λ,μ)strongly regular grah if and only if D is a v,k,λ,μ)-pds with 1 G D and D 1) = D. Strongly regular grahs or artial difference sets) with arameters n,rn+ ε), εn + r + 3εr,r + εr) are called of Latin Square tye if ε = 1, and of negative Latin Square tye if ε = 1. There are many constructions of SRGs of Latin square tye any collection of r 1 mutually orthogonal Latin squares gives rise to such a grah, see [1], for instance), but only a few constructions of negative Latin square tye are known. We will show that certain weakly regular -ary bent functions can be used to construct SRGs of Latin square and of negative Latin square tye. Next we introduce the concet of association schemes. Let V be a finite set of vertices, and let {R 0,R 1,...,R d } be binary relations on V with R 0 := {x, x) : x V }. The configuration V ; R 0,R 1,...,R d ) is called an association scheme of class d on V if the following holds: 1. V V = R 0 R 1 R d and R i R j = for i j.. t R i = R i for some i {0, 1,...,d}, where t R i := {x, y) y, x) R i }.Ifi = i, we call R i is symmetric.

4 5 J Algebr Comb 011) 3: For i, j, k {0, 1,...,d} and for any air x, y) R k, the number {z V x, z) R i,z,y) R j } is a constant, which is denoted by k ij. An association scheme is said to be symmetric if every R i is symmetric. Given an association scheme V ;{R l } 0 l d ), we can take the union of classes to form grahs with larger sets this is called fusion), but it is not necessarily guaranteed that the fused collection of grahs will form an association scheme on V. If an association scheme has the roerty that any of its fusions is also an association scheme, then we call the association scheme amorhic. VanDam[17] roved the following result. Result Let V be a set of size v, and let {G 1,G,...,G d } be an edge-decomosition of the comlete grah on V, where each G i is a strongly regular grah on V. If G i, 1 i d, are all of Latin square tye or all of negative Latin square tye, then the decomosition is a d-class amorhic association scheme on V. We conclude this section by recording the bent function in [8] as below. Result 3 Let n = k. Then the -ary function fx)maing F n to F given by fx)= Tr n x + x 3k + k k +1 ) is a weakly regular bent function. Moreover, for b F n coefficient of fx)is equal to the corresonding Walsh W f b) = k ζ Tr kx 0 )/, where x 0 is a unique solution in F k of the equation b k +1 + b + x ) k +1)/ + b k k +1) + b + x ) k k +1)/ = 0. 3 The construction In this section, we construct SRGs using -ary bent functions. First, we introduce some notations used throughout this section. Let f : F n F be a weakly regular -ary bent function satisfying f x) = fx). Without loss of generality, we may assume f0) = 0. If not, we can relace fx)with fx) f0). For each b F n, assume that W f b) = μ ) n ζ f b), where μ =±1 and = 1) 1. Suose that there exists an integer l with l 1, 1) = 1 such that for each α F and x F n, fαx)= α l fx)holds. Let D i := { x F n fx)= i }, 0 i 1. Denote by G and H the additive grous of F n and F, resectively. Clearly, we have 1 i=0 D i = G. Moreover, D 1) i = D i for each 0 i 1 since f x) = fx).

5 J Algebr Comb 011) 3: Define the grou ring elements in Z[ζ ]G): 1 L t := D i ζ it, 0 t 1, i=0 in articular L 0 = F n. The following result gives some roerties of the L t s see [1]). Result i) If s,t,s + t 0, then L t L s = μ tsv )n ) n L v with s 1 l + t 1 l = v 1 l. ii) L t L t = n for t {1,..., 1}. iii) 1 t=1 L tl 0 ζ at = D a n )F n. It is clear that we may comute D i s from L t s, namely D i = 1 Result,wehave D a D b = 1 s,t=0 = L s=1 L t L s ζ at bs s,t 0 s+t 0 = n F n + L s L 0 ζ bs 1 + L t L s ζ at bs 1 + t=1 s,t,s+t 0 s 1 l +t 1 l =v 1 l s=1 L t L 0 ζ at L s L s ζ sa b) + D b n) F n + D a n) F n 1 = s=1 + 1 ) tsv n μ ) 1 n ζ at bs L v + n ζ sa b) + D a + D b ) n) F n s,t,s+t 0 s 1 l +t 1 l =v 1 l tsv μ t=0 L tζ it.by s=1 n ζ sa b) ) n ) n ζ at bs L v. 3) In the following, we only work on the field F n with n = k. Note that in this case the Walsh coefficient of f can be written as the form W f b) = 1) 1)k μ k ζ f b). For each b F k,letχ b be the additive character of G defined by χ b x) = ζ Trbx),

6 56 J Algebr Comb 011) 3:51 66 and η be the additive character of H defined by ηx) = ζ x. Now for each b F n, W f b) = x F k 1 ζ fx)+trbx) = i=0 ζ Trbx) x D i 1 )ζ i = χ b D i )ζ i = χ bηr), where R ={x, f x)) : x F k}. Since f is a bent function, we know that χ b ηr) = W f b) = k. First, we determine the cardinalities of D i s. Lemma 1 Let f : F k F be the bent function as above. Then i) D 1 = D = = D 1. ii) D 0 = k 1 + ɛ k k 1 ) and D i = k 1 ɛ k 1 for each 1 i 1, where ɛ = 1) 1)k μ. Proof i) For any 1 a,b 1, by Result iii) we have Da k) 1 F k = t=1 L t L 0 ζ at 1 = t=1 i=0 L t L 0 ζ ab 1 ) bt = Db k) F k. The last equality holds since ζ ab 1 is also a rimitive th root of unity. Thus D a = D b. ii) Let χ 0 be the rincial character of F k,wehave 1 χ 0 ηr) = D 0 + D i ζ i = D 0 + D 1 ζ + ζ i=1 1) + +ζ = D0 D 1. Since χ b ηr) = W f b) = 1) 1)k μ k ζ f b) = ɛ k ζ f b) rational integer, we have and D 0 D 1 is a D 0 D 1 =ɛ k. ) On the other hand, D 0 + 1) D 1 = k. 5) By solving ) and 5), the result follows. Next, we define Condition A for a function f as follows. Condition A Let f : F k F be a weakly regular bent function with f0) = 0 and f x) = fx), where is an odd rime. There exists an integer l with l 1, 1) = 1 such that fαx)= α l fx) for any α F and x F k. For each b F k, W f b) = ɛ k ζ f b), where ɛ = 1) 1)k μ with μ =±1.

7 J Algebr Comb 011) 3: Two functions f,g : F n F are called affine equivalent if there exist an affine ermutation A 1 of F and an affine ermutation A of F n such that g = A 1 f A. Furthermore, they are called extended affine equivalent EA-equivalent) if g = A 1 f A + A, where A : F n F is an affine function. A olynomial L of the form Lx) = n i=0 a i x i F [x] is called a linearized olynomial. Note that the affine ermutations of F are cx + d, where c F and d F. We have the following result. Proosition 1 Let L 1,L F [x] be linearized olynomials, where L 1 is a ermutation of F and L is a ermutation of F k. If L 1 1) 0 and f satisfies Condition A, then the function g = L 1 f L satisfies Condition A. Proof That g0) = 0,g x) = gx) and gαx) = α l gx) for α F are easy to be verified. We only need to rove that for each b F k, W g b) = ɛ k ζ g b), where ɛ = 1) 1)k μ with μ =±1. First assume that L 1 x) = cx. Note that L 1 1) 0 imlies that c 0 and ζ c is also a rimitive th root of unity. Now W g b) = x F k = y F k = W f L 1 ζ L 1f L x))+trbx)) = x F k ζ c ) fy)+trc 1 bl 1 y)) = y F k ) c 1 b )) = ɛ k ζ f L 1 ζ c ) fl x)+trc 1 bx)) ) c 1 b)), ζ c ) fy)+trl 1 ) c 1 b)y) where L 1 ) is the adjoint oerator of L 1. We finish the roof. Remark 1 Clearly, the function g in the above roosition is affine equivalent to f. However, for a function g which is EA-equivalent but not affine equivalent to f,it may be seen that g does not satisfy Condition A. Indeed, assume that g = A 1 f A + A, then gαx) α l 1 gx) for α F when l>. Now assume that a function f : F k F satisfies Condition A, then clearly the functions L 1 f L all satisfy Condition A, where L 1,L F [x] are linearized olynomials, and L 1 is a ermutation of F, L is a ermutation of F k.wesee that the functions of the form L 1 f L are affine equivalent to f. However, for a function g which is EA-equivalent but not affine equivalent to f, it may be seen that g does not satisfy Condition A. Now we will rove the first result in this section. Theorem 1 Let f be a function satisfying Condition A. Let D := { x : x F k fx)= 0 }.

8 58 J Algebr Comb 011) 3:51 66 Then D is a v,d,λ 1,λ )-PDS, where v = k, d = k ɛ ) k 1 + ɛ ), λ 1 = k 1 + ɛ ) 3ɛ k 1 + ɛ ) + ɛ k, λ = k 1 + ɛ ) k 1. 6) Proof Recall that L t = 1 i=0 D iζ it.by3), we have: D 0 D 0 = 1) k + D 0 k) F k + ɛ Now we comute the last term of 7). s,t,s+t 0 s 1 l +t 1 l =v 1 l Substituting 8)into7), we have 1 k L v = k )L i i=1 i=1 j=1 s,t,s+t 0 s 1 l +t 1 l =v 1 l ) 1 1 = ) k D 0 + D j ζ ji 1 = ) k 1)D 0 + j=1 1 D j ) 1 = ) k 1)D 0 D j j=1 i=1 = ) k 1)D 0 F k D 0 ) ) ζ ji k L v. 7) )) = ) k D 0 F k). 8) D 0 D 0 = 1) k + D 0 k) F k + ɛ ) k D 0 F k) = 1) k + ɛ k+1 )D 0 + D 0 k ɛ ) k) F k. 9) Note that D + 0 = D 0 and D 0 = k 1 + ɛ k k 1 ) Lemma 1), by 9)wehave D + 0) = 1) k + ɛ k+1 )D + 0) + D 0 k ɛ ) k) F k.

9 J Algebr Comb 011) 3: After simlifying, we get the equation D = k 1 k + ɛ k ɛ k 1 1 ) + ɛ k ɛ k 1 ) D + ɛ k 1 + k ) F k. By ), the roof is done. Remark When ɛ = 1 in Theorem 1, we get negative Latin square tye SRGs. When = 3, some new SRGs arise using non-quadratic ternary bent functions; see [16, Tables, 3]. Unfortunately, when 5, for the known bent functions, we don t get new grahs. To give another construction of SRGs using -ary bent functions, we show two lemmas first. Lemma Let S and T be the sets of non-zero squares and non-squares in F, resectively, where is an odd rime. Then we have i) When 1 mod ), S = 1 T = ST = TS= 1 S + T). ii) When 3 mod ), S = 3 T = + 1 ST = TS= + 1 S T, S + 3 T, S + 1 T, S + 5 T, + 3 F. Proof i) Note that 1 is a square when 1 mod ) and S 1) = S,T 1) = T in this case. By [1, Theorem ], S is a, 1, 5, 1 ) almost difference set in F, which imlies that SS 1) = S = S + 1 T. It is easy to see that T = F 0 S, and hence TT 1) = T = F 0 S).Now the results can be followed by direct comutation.

10 60 J Algebr Comb 011) 3:51 66 ii) When 3 mod ), 1 is a non-square, and S 1) = T,T 1) = S. Itis well known that the subset S is a, 1, 3 ) difference set in F. Hence SS 1) = ST = F. The comutations are similar to those in the roof of i). Lemma 3 Let ζ be a rimitive th root of unity, S and T be the sets of non-zero squares and non-squares of F, resectively. Define m = i S ζ i, then i) When 1 mod ), m1 + m) = 1. ii) When 3 mod ), m1 + m) = +1. Proof We only rove the case 1 mod ), the roof of ii) is similar. Note that 1 + m) = i T ζ i, hence m1 + m) = i S,j T ζ i+j). By Lemma i), we know that i S,j T ζ i+j) = 1 i F ζ i = 1. Now we rove the following result. Theorem Let f be a function satisfying Condition A. Let D S := { x : x F k fx)is a non-zero square }, then D S is a v,d,λ 1,λ )-PDS, where v = k, d = 1 k k 1) k ɛ ), λ 1 = 1 k k 1) 3ɛ k k 1) + k ɛ, λ = 1 k k 1) 1 k k 1) ) ɛ. 10) Proof We only rove the case 1 mod ), the roof of the case 3 mod ) is similar. Let S and T be the sets of non-zero squares and non-squares of F, resectively. Now by 3), we have DS = D a D b = a,b S a,b S 1 s=1 + ɛ k k ζ sa b) + D 1 k) F k s,t,s+t 0 s 1 l +t 1 l =v 1 l ζ at bs L v ). 11)

11 J Algebr Comb 011) 3: Now a,b S 1 s=1 k ζ sa b) D1 k) 1) F k = a,b S ) = k 1) + 1), D 1 k 1) ) F k. 1) To comute the third term in 11), for s,t F, we define δs,t) as follows m if s,t both are squares; δs,t) = 1 + m) if s,t both are non-squares; m1 + m) otherwise, where m = i S ζ i. For convenience, denote the set {s, t) : s,t F s 0,t 0,s+ t 0} by Ω. Fors,t,s + t 0, define the function σs,t)= v, where s 1 l + t 1 l = v 1 l. Since l 1, 1) = 1, σ is well defined. Now we comute the third term of 11): a,b S s,t) Ω = = s,t) Ω ζ at bs L σs,t) a S s,t) Ω S S) ζ t s,t) Ω\S S) T T)) ) ) a ) ζ s b )L σs,t) = b S m L σs,t) + s,t) Ω T T) s,t) Ω 1 + m) L σs,t) δs,t)l σs,t) m1 + m)l σs,t). 13) By Lemma i), we know that the multiset { σs,t) : s, t) Ω S S) } = 5 S + 1 T, then s,t) Ω S S) m L σs,t) = m 5 v S L v + 1 L v ). Using similar argument, we can comute the last two terms of 13). Then we have ζ at bs L σs,t) a,b S s,t) Ω = m 5 v S m) 1 L v + 1 v S ) L v v T L v + 5 v T ) L v v T

12 6 J Algebr Comb 011) 3:51 66 m1 + m) 1 ) L v v F = A L v + B L v, 1) v S v T where A = 5 m + 1 m) 1 m1 + m). Now we see that a,b S s,t) Ω Denote a i = A v S ζ vi 1 + m) 1 ζ at bs L σs,t) = A L v + B L v v S v T ) = A v S i=0 1 = A v S i=0 + B v T ζ vi a 0 = 1 A + B) = 1 Similarly, we get the following: m1 + m) and B = 1 m D i ζ vi ζ vi + B 1 ) D i ζ vi v T i=0 + B ) ζ vi D i. 15) v T for 0 i 1. Clearly, m m + 3 ) 1) 3) = 1)m1 + m) + 1) 1) 3) = + = 1. a i = { 1)/ i {0} T, + 1)/ i S. Now ζ at bs a,b S s,t) Ω L σs,t) = 1 F k D S ) D S. 16)

13 J Algebr Comb 011) 3: By 11), 1) and 16), we know that D S D 1) S = D S = k 1) + 1) + ɛ k 1 = C 1 + C D S + C 3 F k, 1) + D 1 k 1) ) F k F k D S ) ) D S where C 1 = 1 k 1) + 1), C = k+1 ɛ and C 3 = 1 k 1) 1 k+1 1)ɛ. The roof follows by ). Remark 3 i) Using similar roof, we may also rove that the set D N := { x : x F k fx)is a non-square } is a PDS with the same arameters as in Theorem. ii) In [16], it is shown that for weakly regular ternary bent function f : F 3 k F 3, the set D i := {x F 3 k fx)= i} is a artial difference set for each 0 i. We may see that Theorem is the generalization of the result in [16]. With a small modification, we may get the following result. Theorem 3 Let f be a function satisfying Condition A. Let then D S is a v,d,λ 1,λ )-PDS, where v = k, D S := { x : x F k fx)is a square }, d = 1 k + k 1 + ɛ ) k ɛ ), λ 1 = 1 k + k 1 + ɛ ) 3ɛ k + k 1 + ɛ ) + k ɛ, λ = 1 k + k 1) k + k 1 + ɛ ). 17) Proof Clearly, D S = D + D S, where D ={x : x F fx) = 0}. Then k D S D S ) 1) = D S + D)D S + D). The result follows from Theorems 1, and similar grou ring comutations as those in Theorem. Remark By using MAGMA, we know that Theorems 1,, 3 are not true for nonweakly regular bent function fx)= Trξ 7 x 98 ) over F 3 6, where ξ is a rimitive element of F 3 6. This imlies that the weakly regular condition is necessary.

14 6 J Algebr Comb 011) 3:51 66 We conclude this section by a result on association schemes. Combining Result and Theorems 1,, 3, we have the following result. Theorem Let f be a function satisfying Condition A. Define the following sets: D = { x : x F k fx)= 0 }, D S = { x : x F k fx)is a non-zero square }, D N = { x : x F k fx)is a non-square }. Then {F k;{0},d,d S,D N } is an amorhic association scheme of class 3. Newness In this section, we discuss the known constructions of the SRGs with arameters 10) and 17). Since there are many constructions of the Latin square tye SRGs, we only discuss the newness of the negative Latin square tye SRGs with the above arameters, namely, ɛ = 1 in arameters 10) and 17). One may check the known constructions of the SRGs with the arameters 10), 17) via the online database []. It is well known that rojective two-weight codes can be used to construct SRGs [3], one can check the known constructions of twoweight codes via the online database []. Next we give the following constructions of SRGs with arameters 10) and 17). Result 5 [3] Let k = m and let Q be a non-degenerate quadratic form on F q with q odd. Let and Then M = { v F k q \{0} Tr Qv) ) is a non-zero square }, M = { v F k q \{0} Tr Qv) ) is a square }. i) The Cayley grah generated by M in F k q, +) is a k, 1 k k 1 ) k ɛ), 1 k k 1 ) 3ɛ k k 1 ) + k ɛ, 1 k k 1 ) 1 k k 1 ) ɛ)) strongly regular grah, where ε =±1 and deends on Q. ii) The Cayley grah generated by M in F k q, +) is a k, 1 k + k 1 +ɛ) k ɛ), 1 k + k 1 +ɛ) 3ɛ k + k 1 +ɛ)+ k ɛ, 1 k + k 1 ) k + k 1 + ɛ)) strongly regular grah, where ε =±1 and deends on Q. The SRGs constructed by Result 5i) are called FE1, and the SRGs from Result 5ii) are called RT in [3]. They both are also called affine olar grahs in [6,. 85]. To the best of our knowledge, affine olar grahs are the only known infinitive constructions of the SRGs with arameters 10) and 17) when 5.

15 J Algebr Comb 011) 3: Table 1 SRGs with arameters 10) v,k,λ,μ) Tye Rank of M AutG) Note 65, 60, 105, 110) n.l affine olar 65, 60, 105, 110) n.l new 01, 1050, 55, 6) n.l affine olar 01, 1050, 55, 6) n.l new Table SRGs with arameters 17) v,k,λ,μ) Tye Rank of M AutG) Note 65, 36, 13, 10) n.l affine olar 65, 36, 13, 10) n.l new 01, 1350, 761, 756) n.l affine olar 01, 1350, 761, 756) n.l new For small arameters, now we discuss the known constructions of the SRGs with arameters 10) and 17). When = 3, by Theorem and the bent function in Result 3, in the field F 3 8, we get an SRG with arameter 6561, 1, 79, 756). Comuted by MAGMA, the order of its automorhism grou and the 3-rank of the adjacent matrix of the SRG are 3 8, 566). By comaring to [16, Table ], we know that this grah is new. When = 5, in the field F 5, the known constructions of the SRGs with arameters 65, 60, 105, 110), 65, 36, 13, 10) are affine olar grahs, or from Theorems, 3, or from the rojective two weight codes in Chen [5]. It is verified that Chen s SRGs are isomorhic to affine olar grahs and the SRGs from the bent functioninresult3 are new. When = 7, in the field F 7, the known constructions of the SRGs with arameters 01, 1050, 55, 6), 01, 1350, 761, 756) are the same as the case = 5. Using MAGMA, it is verified that Chen s SRGs [5] are also isomorhic to affine olar grahs and the SRGs from the bent function in Result 3 are new. In the following two tables, we give some comutational results of the SRGs from different constructions. In the first column, we list the arameters of the SRGs. The grou AutG) is the full automorhism grou of the SRG G, and M denotes an adjacency matrix of G, to be considered in F. The abbreviation n.l. means the SRG is of negative Latin square tye. The symbol means the SRG is constructed by the bent function in Result 3. Remark 5 From Tables 1 and, we conjecture that the bent functions in Result 3 can give a family of SRGs of negative Latin square tye which are not isomorhic to the affine olar grahs. It is difficult to rove it in general cases. We may see that the automorhism grous of the Cayley grahs generated by the PDSs in 1) have the subgrou of order k coming from translations τ c : x x + c for any c F k and the subgrou of order k coming from the Galois automorhism

16 66 J Algebr Comb 011) 3:51 66 of F k. This means that k k divides the order of the automorhism grous of the Cayley grahs of 1). For the bent functions given by Result 3, we conjecture that AutGX)) is not divisible by k+1, where X = D S or D S. This is a ossible method to rove that they are new. Acknowledgements Research suorted by the National Research Foundation of Singaore under Research Grant NRF-CRP and by the Nanyang Technological University under Research Grant M The authors would like to thank Professor Alexander Pott for helful discussions. They are also indebted to one of the anonymous referees to oint out the result in Proosition 1 and the two subgrous of the automorhism grous of the SRGs in this aer. References 1. Arasu, K., Ding, C., Helleseth, T., Kumer, P., Martinsen, H.: Almost difference sets and their sequences with otimal autocorrelation. IEEE Trans. Inf. Theory 7, ). Brouwer, A.: Web database of strongly regular grahs. htt:// html online) 3. Calderbank, R., Kantor, W.: The geometry of two-weight codes. Bull. Lond. Math. Soc. 18), ). Chen, E.: Web database of two-weight codes. htt://moodle.tec.hkr.se/~chen/research/-weight-codes/ search.h online) 5. Chen, E.: Construction of two-weight codes. Internal Reorts 008) 6. Colbourn, C., Dinitz, J.: Handbook of Combinatorial Designs, Discrete Mathematics and its Alications, nd edn. Chaman & Hall/CRC, Boca Raton 007) 7. Helleseth, T., Hollmann, H., Kholosha, A., Wang, Z., Xiang, Q.: Proofs of two conjectures on ternary weakly regular bent functions. IEEE Trans. Inf. Theory 555), ) 8. Helleseth, T., Kholosha, A.: New binomial bent functions over the finite fields of odd characteristic. IEEE Trans. Inf. Theory 569), ) 9. Helleseth, T., Kholosha, A.: Monomial and quadratic bent functions over the finite fields of odd characteristic. IEEE Trans. Inf. Theory 55), ) 10. Kumar, P., Scholtz, R., Welch, L.: Generalized bent functions and their roerties. J. Comb. Theory, Ser. A 01), ) 11. Lidl, R., Niederreiter, H.: Finite Fields, nd edn. Encycloedia of Mathematics and Its Alications, vol. 0. Cambridge University Press, Cambridge 1997) 1. Ma, S.: A survey of artial differential sets. Des. Codes Crytogr. 3), ) 13. Passman, D.: The Algebraic Structure of Grou Rings. Krieger, Melbourne 1985). Rerint of the 1977 original 1. Pott, A., Tan, Y., Feng, T., Ling, S.: Association schemes arising from bent functions. In: Preroceedings of the International Worksho on Coding and Crytograhy, Bergen, ) 15. Rothaus, O.: On bent functions. J. Comb. Theory, Ser. A 03), ) 16. Tan, Y., Pott, A., Feng, T.: Strongly regular grahs associated with ternary bent functions. J. Comb. Theory, Ser. A 1176), ) 17. van Dam, E.R.: Strongly regular decomositions of the comlete grah. J. Algebr. Comb. 17), )