# On the Existence of(v,k,λ) Difference Sets with k<1250 and k−λ Is a Square

Authors: Adegoke Solomon Osifodunrin
Journal: ISRN Algebra
 (2012)

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## Abstract

We combine Turyn's self-conjugacy result, variance technique, Dillon dihedral trick, and Sylow theorem to investigate the existence of (v,k,λ) difference sets in which k−λ is a square and k<1250.


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## Body

## 1. Introduction

LetG be a multiplicative group of order v and let D be a subset of G consisting of k elements, where 1<k<v-1. D is a nontrivial (v,k,λ) difference set if every nonidentity element can be reproduced λ times by the multiset {d1d2-1:d1,d2∈D,d1≠d2}. The natural number n=k-λ>1 is known as the order of the difference set. The group structure determines the nature of the difference set. For instance, if the underlying group G is abelian (resp., nonabelian or cyclic), then D is abelian (resp., nonabelian or cyclic) difference set. The study of difference sets integrates various techniques ranging from algebraic number theory to geometry, algebra, and combinatorics [1]. There are many classical results on constructions and nonexistence of difference sets in the literature [2–14]. These results are mainly based on Hall’s multiplier concept [15] or Turyn’s self-conjugacy method [14]. Recently, Schmidt [13] developed a new method for studying combinatorial structures using group ring equations without any restrictive assumptions. Arasu [2], Arasu and Sehgal [3, 16], Baumert [4], Hughes [17], Iiams [7], Kibler [18], Kopilovich [10], Lander [11], and López and Sánchez [19] among other authors studied the existence of abelian (v,k,λ) difference sets with k≤150. They were able to either indicate the existence or otherwise of difference sets. Some of these authors also listed parameter sets that were open, whose existence or otherwise has been concluded by other authors. This paper mainly uses Turyn’s self-conjugacy approach to study a class of (v,k,λ) difference sets in which n=m2, where m is a positive integer. We illustrate with examples where 2≤m≤25 and the ideal generated by prime divisors of m factors trivially in the respective cyclotomic rings. This assumption along with Dillon dihedral trick and Sylow theorems provide sufficient information required to decide the nonesxistence of the difference sets in some or all groups of order v.We assume thatG is a finite group of order v. Section 2 gives a brief description of some basic results which include materials from group theory, representation, and algebraic number theories. Section 3 lists difference sets parameters that do not exist and examples of partial results of nonexistence of difference sets in groups of order v.


## 2. Preliminaries

### 2.1. Difference Sets

Letℤ be the ring of integers and ℂ be the field of complex numbers. Suppose that G is a group of order v and D is a (v,k,λ) difference set in G. We sometimes view the elements of D as members of the group ring ℤ[G], which is a subring of the group algebra ℂ[G]. Thus, D represents both subset of G and element ∑g∈Dg of ℤ[G]. The sum of inverses of elements of D is D(-1)=∑g∈Dg-1. Consequently, D is a difference set if and only if
(2.1)DD(-1)=n+λG,DG=kG.
Suppose that D is a difference set in a group G of order v and N is a normal subgroup of G. Suppose that ψ:G→G/N is a homomorphism. We can extend ψ by linearity, to the corresponding group rings. Thus, the difference set image in G/N (also known as the contraction of D with respect to the kernel N) is the multiset D/N=ψ(D)={dN:d∈D}. Let T*={1,t1,…,th} be a left transversal of N in G. We can write ψ(D)=∑tj∈T*djtjN, where the integer dj=|D∩tjN| is known as the intersection number of D with respect to N. In this work, we will always use the notation D̂ for ψ(D) and mi≥0 denotes the number of times di equals i.


### 2.2. Representation and Algebraic Number Theories

Aℂ-representation of G is a homomorphism, χ:G→GL(d,ℂ), where GL(d,ℂ) is the group of invertible d×d matrices over ℂ. The positive integer d is the degree of χ. A linear representation (character) is a representation of degree one. The set of all linear representations of G is denoted by G*.  G* is an abelian group under multiplication and if G′ is the derived group of G, then G* is isomorphic to G/G′. Define ζm′:=e(2π/m′)i to be a primitive m′th root of unity and Km′:=ℚ(ζm′) to be the cyclotomic extension of the field of rational numbers, ℚ, where m′ is the exponent of G. Without loss of generality, we may replace ℂ by the field Km′. Thus, the central primitive idempotents in ℂ[G] is
(2.2)eχi=χi(1)|G|∑g∈Gχi(g)g-1=1|G|∑g∈Gχi(g)¯g,
where χi is an irreducible character of G.Aliases are members of group ring which enable us to transfer information fromℂ[G] to group algebra ℚ[G] and then to ℤ[G]. Let G be an abelian group and let Ω={χ1,χ2,…,χh}, be the set of characters of G. The element β∈ℤ[G] is known as Ω-alias if for A∈ℤ[G] and all χi∈Ω, χi(A)=χi(β). Since A=∑χ∈G*χ(A)eχ, we can replace the occurrence of χ(A), which is a complex number by Ω-alias, β, an element of ℤ[G]. Furthermore, two characters of G are algebraic conjugate if and only if they have the same kernel and we denote the set of equivalence classes of G* by G*/~. The central rational idempotents in ℚ[G] are obtained by summing over the equivalence classes Xi={eχi∣χi~χj}∈G*/~ on the eχ's under the action of the Galois group of Km′ over ℚ. That is, [eχi]=∑eχj∈Xieχj, i=1,…,s.For instance, supposeG=Cpm′=〈x:xpm′=1〉 (p is prime) is a cyclic group whose characters are of the form χi(x)=ζpm′i, i=0,…,pm′-1. Then the rational idempotents are
(2.3)[eχ0]=1pm′〈x〉,[eχpj]=1pj+1(p〈xpm′-j〉-〈xpm′-j-1〉),0≤j≤m′-1.
The following is the general formula employed in the search of difference set [22].Theorem 2.1.
LetG be an abelian group and G*/~ be the set of equivalence classes of characters. Suppose that {χ0,χ1,…,χs} is a system of distinct representatives for the equivalence classes of G*/~. Then for A∈ℤ[G], one has
(2.4)A=∑i=0sαi[eχi],
where αi is any χi-alias for A.Equation (2.4) is known as the rational idempotent decomposition of A.Suppose thatχ is any nontrivial representation of degree d and χ(D̂)∈ℤ[ζ], where ζ is the primitive root of unity. Suppose that x∈G is a nonidentity element. Then, χ(xG)=χ(x)χ(G)=χ(G). This shows that (χ(x)-1)χ(G)=0. Since x is not an identity element, (χ(x)-1)≠0 and χ(G)=0 (ℤ[ζ] is an integral domain). Consequently, χ(D)χ(D)¯=n·Id+λχ(G)=n·Id, where Id is the d×d identity matrix. Furthermore, if χ is a nontrivial representation of G/N of degree d then D̂D̂(-1)=n·1G/N+|N|λ(G/N) and χ(D̂)χ(D̂)¯=n·Id.Recall that the ring of integers of the cyclotomic fieldℚ[ζm′] is ℤ[ζm′]. This ring is also an integral domain. Let p, a, b∈ℤ[ζm′]. The number p is irreducible if p=ab implies one of a or b is a unit. The element p is prime if p∣ab implies p∣a or p∣b [23]. A domain is a unique factorization domain (UFD) if factorization into irreducibles is possible and unique. In UFD, the irreducibles are also primes. In order to successfully obtain the difference set images, we need the aliases. Suppose that G/N is an abelian factor group of exponent m′ and D̂ is a difference set image in G/N. If χ is not a principal character of G/N, then χ(D̂)χ(D̂)¯=n is an algebraic equation in ℤ[ζm′]. The determination of the alias requires the knowledge of how the ideal generated by χ(D̂) factors in cyclotomic ring ℤ[ζm′], where ζm′ is the m′th root of unity. If δ:=χ(D̂), then by (2.4), we seek α∈ℤ[G/N] such that χ(α)=δ. The task of solving the algebraic equation δδ¯=n is sometimes made easier if we consider the factorization of principal ideals 〈δ〉〈δ¯〉=〈n〉. Suppose we are able to find δ=∑i=0ϕ(m′)-1diζm′i∈ℤ[ζm′] such that δδ¯=n, where ϕ is the Euler ϕ-function. A theorem due to Kronecker [12, 13] states that any algebraic integer whose all conjugates have absolute value 1 must be a root of unity. We use this theorem to characterize the solutions. If there is any other solution to the algebraic equation, then it must be of the form δ′=δu, where u=±ζm′j is a unit.The following result is used to determine the number of factors of an ideal in a ring: supposep is any prime and m′ is an integer such that gcd(p,m′)=1. Suppose that d is the order of p in the multiplicative group ℤm′* of the modular number ring ℤm′. Then the number of prime ideal factors of the principal ideal 〈p〉 in the cyclotomic integer ring ℤ[ζm′] is ϕ(m′)/d, where ϕ is the Euler ϕ-function, that is, ϕ(m′)=|ℤm′*| [25]. For instance, the ideal generated by 2 has two factors in ℤ[ζ7], the ideal generated by 7 has two factors in ℤ[ζ20], while the ideal generated by 3 has four factors in ℤ[ζ40]. On the other hand, since 2s is a power of 2, the ideal generated by 2 is said to completely ramifies as power of 〈1-ζ2s〉=〈1-ζ2s〉¯ in ℤ[ζ2s].According to Turyn [14], an integer n is said to be semi-primitive modulo m′ if for every prime factor p of n, there is an integer i such that pi≡-1modm′. In this case, −1 belongs to the multiplicative group generated by p. Furthermore, n is self-cosnjugate modulo m′ if every prime divisor of n is semi primitive modulo mp′, where mp′ is the largest divisor of m′ relatively prime to p. This means that all prime ideals over n in ℤ[ζm′] are fixed by complex conjugation. For instance, 72≡-1(modm′), where m′=2, 5, 10 and 7≡-1(modm′), m′=2, 4, 8. Thus, 〈7〉 is fixed by conjugation in ℤ[ζm′], m′=2, 4, 5, 8, 10, 50.Remark 2.2.
If〈n〉=Πi=1sθi in cyclotomic ring ℤ[ζm′], where θi is an ideal and s is an odd integer, then there is no solution to δδ¯=n. To see this, assume that a δ exist such that 〈δ〉=Πi=1kαi. Then 〈n〉=〈δ〉〈δ¯〉 has 2k factors but 〈n〉 has odd factors.Remark 2.3.
Let us consider the ideal generated by 2 which has two factors in the cyclotomic ringℤ[ζ23]. We claim that the algebraic number 2 is prime in this ring. Since (23, 11, 5) difference sets exist, and there exists θ such that θθ¯=6 and θ+θ¯=-1. This implies that θ2+θ+6=0 and θ=(-1±-23)/2. Consequently, θ∈ℤ[-23]. Suppose that the algebraic number 2 is not prime in ℤ[-23]. As -23≡1(mod4)([23], chapter 3), we seek a, b∈ℤ such that δ=(a+b-23)/2 and δδ¯=(a2+23b2)/4=2. The equation a2+23b2=8 has no integer solution. Thus, there is no algebraic number such that δδ¯=2. In fact, δδ¯=p has no solution, where p=2, 3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22. However, the equation a2+23b2=4m2 has trivial solutions (a,b)=(-2m,0) and (2m,0), where m=2, 3, 4, 5, 7, 9, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 25. We noticed that since the class number of the cyclotomic ring ℤ[ζ23] is 3, the equation a2+23b2=4·23 has nontrivial solutions (a,b)=(-3,-1), (-3,1), (3,-1) and (3,1). Also, a2+23b2=4m has nontrivial solutions for m=6, 8, 12, 18, 62, 82, 122, 162, 182 and 242.In this paper, we will use the phasemfactors triviallyin ℤ[ζm′] if the ideal generated by m is prime or ramifies in ℤ[ζm′]; m is self-conjugate modulo m′; the ideal generated by m has odd factors or the algebraic equation δδ¯=m2 has no solution or has trivial solutions. In summary, suppose that D̂ is the difference set image of order n=m2 in the cyclic factor group G/N, where G/N is a group with exponent m′. Suppose that m factors trivially in ℤ[ζm′] and χ is a nontrivial representation of G/N. Then χ(D̂)=±mζm′i, ζm′ is the m′th root of unity [13].


### 2.3. Characteristics of Difference Set Images in Subgroup of a Group

In this subsection, we use the attributes of subgroups of a group to obtain information about the difference set image in the subgroups. Dillon [5] proved the following results which will be used to obtain difference set images in dihedral group of a certain order if the difference images in the cyclic group of same order are known.Theorem 2.4 (dillon dihedral trick).
LetH be an abelian group and let G be the generalized dihedral extension of H. That is, G=〈q,H:q2=1,qhq=h-1,∀h∈H〉. If G contains a difference set, then so does every abelian group which contains H as a subgroup of index 2.Corollary 2.5.
If the cyclic groupZ2m does not contain a (nontrivial) difference set, then neither does the dihedral group of order 2m.Remark 2.6.
We look at subgroup properties of a group that can aid the construction of difference set image. For the convenience of the reader, we reproduce the idea of Gjoneski et al. [26]. Suppose that H is a group of order 2h with a central involution z. We take T={ti:i=1,…,h} to be the transversal of 〈z〉 in H so that every element in H is viewed as tizj, 0≤i≤h, j=0,1. Denote the set of all integral combinations, ∑i=1haiti of elements of T, ai∈ℤ by ℤ[T]. Using the two representations of subgroup 〈z〉 and Frobenius reciprocity theorem [27], we may write any element X of the group ring ℤ[H] in the form
(2.5)X=X(1+z2)+X(1-z2).
Furthermore, let A be the group ring element created by replacing every occurrence of z in X by 1. Also, let B be the group ring element created by replacing every occurrence of z in H by −1. Then
(2.6)X=A(〈z〉2)+B(2-〈z〉2),
where A=∑i=1haiti and B=∑j=1hbjtj, ai, bj∈ℤ. As X∈ℤ[H], A and B are both in ℤ[T], and A≡Bmod2. We may equate A with the homomorphic image of X in G/〈z〉. Consequently, if X is a difference set, then the coefficients of ti in the expression for A will be intersection number of X in the coset 〈z〉[26]. In particular, it can be shown that if K is a subgroup of a group H such that
(2.7)H≅K×〈z〉,
then the difference set image in H is
(2.8)D̂=A(〈z〉2)+gB(2-〈z〉2),
where g∈H, A is a difference set in K, α=(k+m)/|K| or α=(k-m)/|K|, B=A-αK and k is the size of the difference set. Equation (2.8) is true as long as |K|∣(k+m) or |K|∣(k-m).


### 2.4. Amalgamation of Results

In this paper, we study(v,k,λ) difference sets in which n=k-λ=m2 and the ideal generated by m factors trivially in the cyclotomic ring ℤ[ζm′]. That is, if n=m2, then (n)=(m)(m) up to units in ℤ[ζm′]. This method is very useful in the investigation of difference sets in solvable groups. A group G is solvable if the sequence G⊇G′⊇G′′⋯⊇⋯⊇G(i)⋯ terminates in the identity, G(e)=1, in a finite number of steps, each G(i) is the derived group of the preceding one [28]. Consequently, each i, the factor group G(i)/G(i+1) is Abelian. We now state the extended Sylow theorem in solvable groups ([28], page 141).Theorem 2.7.
LetG be a solvable group of order mn, in which gcd(m,n)=1. Then(1)
G possesses at least one subgroup of order m;(2)
any two subgroups of orderm are conjugates;(3)
any subgroup whose orderm′ divides m is contained in a subgroup of order m;(4)
the numbernm of subgroups of order m may be expressed as a product of factors, each of which (a) is congruent to 1 modulo some prime factor of m, and (b) is a power of a prime and divides one of the chief factors of G.The next three criteria enable us to rule out the existence of difference sets.Criterion 1.
Suppose thatG is a group of order v=p′s, where p′ is prime, s and t are integers. Then G does not admit (v,k,λ) if there exists a normal subgroup N of G such that(1)
k-λ=m2, m is a natural number,(2)
|G/N|=p′,(3)
m factors trivially in the cyclotomic ring ℤ[ζp′], where ζp′ is the p′th root of unity,(4)
the difference set solution inG/N is one of the forms α(G/N)+m, α+m>|N| or α(G/N)-m, α<m.Proof.
The nonexistence of viable difference set image inG/N implies that G does not admit (v,k,λ) difference set.In this criterion, we may replace|G/N|=p′ with |G/N|=p′q if q>2 is prime power, q∣s, gcd(p′,q)=1, and the ideal generated by p factors trivially in ℤ[ζp′q], where p is a prime divisor of m (see Remark 2.3).Criterion 2.
Suppose thatG is a group of even order v and H is a factor group of G with |H|=2q, where q is prime. Let g be an element of order 2 in H. Then G does not admit (v,k,λ) if(1)
k-λ=m2, m is a natural number,(2)
m factors trivially in the cyclotomic rings ℤ[ζq], where ζq is qth root of unity,(3)
the difference set solution inH/〈g〉 is one of the forms α(H/〈g〉)+m,  α+m>|G/(H/〈g〉)| or α(H/〈g〉)-m,  α<m; alternatively, the difference set image in H is one of the forms α(H)+m,  α+m>|G/H| or α(H)-m,  α<m.Proof.
The proof follows from Criterion1 and the fact that if |H|≡2(mod4), the cyclotomic rings ℤ[ζ2q] and ℤ[ζq] are the same.Criterion 3.
Suppose thatG is a group of order v=22×q×s, where q≥3 is prime and s is an integer. Suppose that H is a factor group of G of order 2q. The group G does not admit (v,k,λ) difference set if there exists a normal subgroup N such that G/N≅H×C2 and(1)
k-λ=m2,  m is a natural number,(2)
every prime divisorm′ of m factors trivially in the cyclotomic rings ℤ[ζq], where ζq is qth root of unity, gcd(m′,q)=1(3)
the difference set solution inH is of the form αH+m, and α is an odd integer or(4)
the difference set solution inH is of the form αH-m, α is an even integer, m is an odd integer, and m>α/2.Proof.
There are two groups of order2q, cyclic and dihedral groups. Since every prime divisor m′ of m factors trivially in the cyclotomic rings ℤ[ζq], gcd(m′,q)=1, it follows that m factors trivially in ℤ[ζq]. Consequently, the (v,k,λ) difference set image in H is of the form αH+m or αH-m. Suppose that the (v,k,λ) difference set image in H is of the form αH+m and α is an odd integer. Using (2.8), the difference set image in H×C2 is
(2.9)D̂=A(〈z〉2)+gB(2-〈z〉2),
where g∈H×C2, A=αH+m is the difference set image in H, B=A-αH with α=(k+m)/2q or (k-m)/2q and z is the generator of C2. Since α is odd, A(〈z〉/2) consists of at least 2q-2 odd entries while B((2-〈z〉)/2) consists of at least 2q-2 even entries. Thus, (2.8) has no integer solutions. On the other hand, suppose that the (v,k,λ) difference set image in H is of the form αH-m, α is an even integer, m is odd an odd integer, and m>α/2. The difference set image is of the form (2.9) with A=αH-m. Since α is even and m is odd, A(〈z〉/2) and B((2-〈z〉)/2) have two entries that are fractions. In particular, we can translate if necessary, to ensure that the coefficients of the identity in both components are (α-m)/2 and -m/2, respectively. The sum and difference of these two entries are, respectively, (α-2m)/2 and α/2. But m>α/2 and 2m>α. Hence, (α-2m)/2 is a negative integer. Thus, there is no difference set image in H×C2 and the criterion follows.Notice that there are five factor groups of order22×q if q≡1(mod4) and four factor groups if q≡3(mod4). Criterion 3 rules out the existence of difference set images in Cq×C2×C2 and D2q≅Dq×C2. In addition to conditions of Criterion 3, if m factors trivially also in ℤ[ζ22×q], then three of the four or five factor groups (C2q, Cq×C2×C2 and D2q) of order 22×q do not admit difference sets.


## 2.1. Difference Sets

Letℤ be the ring of integers and ℂ be the field of complex numbers. Suppose that G is a group of order v and D is a (v,k,λ) difference set in G. We sometimes view the elements of D as members of the group ring ℤ[G], which is a subring of the group algebra ℂ[G]. Thus, D represents both subset of G and element ∑g∈Dg of ℤ[G]. The sum of inverses of elements of D is D(-1)=∑g∈Dg-1. Consequently, D is a difference set if and only if
(2.1)DD(-1)=n+λG,DG=kG.
Suppose that D is a difference set in a group G of order v and N is a normal subgroup of G. Suppose that ψ:G→G/N is a homomorphism. We can extend ψ by linearity, to the corresponding group rings. Thus, the difference set image in G/N (also known as the contraction of D with respect to the kernel N) is the multiset D/N=ψ(D)={dN:d∈D}. Let T*={1,t1,…,th} be a left transversal of N in G. We can write ψ(D)=∑tj∈T*djtjN, where the integer dj=|D∩tjN| is known as the intersection number of D with respect to N. In this work, we will always use the notation D̂ for ψ(D) and mi≥0 denotes the number of times di equals i.


## 2.2. Representation and Algebraic Number Theories

Aℂ-representation of G is a homomorphism, χ:G→GL(d,ℂ), where GL(d,ℂ) is the group of invertible d×d matrices over ℂ. The positive integer d is the degree of χ. A linear representation (character) is a representation of degree one. The set of all linear representations of G is denoted by G*.  G* is an abelian group under multiplication and if G′ is the derived group of G, then G* is isomorphic to G/G′. Define ζm′:=e(2π/m′)i to be a primitive m′th root of unity and Km′:=ℚ(ζm′) to be the cyclotomic extension of the field of rational numbers, ℚ, where m′ is the exponent of G. Without loss of generality, we may replace ℂ by the field Km′. Thus, the central primitive idempotents in ℂ[G] is
(2.2)eχi=χi(1)|G|∑g∈Gχi(g)g-1=1|G|∑g∈Gχi(g)¯g,
where χi is an irreducible character of G.Aliases are members of group ring which enable us to transfer information fromℂ[G] to group algebra ℚ[G] and then to ℤ[G]. Let G be an abelian group and let Ω={χ1,χ2,…,χh}, be the set of characters of G. The element β∈ℤ[G] is known as Ω-alias if for A∈ℤ[G] and all χi∈Ω, χi(A)=χi(β). Since A=∑χ∈G*χ(A)eχ, we can replace the occurrence of χ(A), which is a complex number by Ω-alias, β, an element of ℤ[G]. Furthermore, two characters of G are algebraic conjugate if and only if they have the same kernel and we denote the set of equivalence classes of G* by G*/~. The central rational idempotents in ℚ[G] are obtained by summing over the equivalence classes Xi={eχi∣χi~χj}∈G*/~ on the eχ's under the action of the Galois group of Km′ over ℚ. That is, [eχi]=∑eχj∈Xieχj, i=1,…,s.For instance, supposeG=Cpm′=〈x:xpm′=1〉 (p is prime) is a cyclic group whose characters are of the form χi(x)=ζpm′i, i=0,…,pm′-1. Then the rational idempotents are
(2.3)[eχ0]=1pm′〈x〉,[eχpj]=1pj+1(p〈xpm′-j〉-〈xpm′-j-1〉),0≤j≤m′-1.
The following is the general formula employed in the search of difference set [22].Theorem 2.1.
LetG be an abelian group and G*/~ be the set of equivalence classes of characters. Suppose that {χ0,χ1,…,χs} is a system of distinct representatives for the equivalence classes of G*/~. Then for A∈ℤ[G], one has
(2.4)A=∑i=0sαi[eχi],
where αi is any χi-alias for A.Equation (2.4) is known as the rational idempotent decomposition of A.Suppose thatχ is any nontrivial representation of degree d and χ(D̂)∈ℤ[ζ], where ζ is the primitive root of unity. Suppose that x∈G is a nonidentity element. Then, χ(xG)=χ(x)χ(G)=χ(G). This shows that (χ(x)-1)χ(G)=0. Since x is not an identity element, (χ(x)-1)≠0 and χ(G)=0 (ℤ[ζ] is an integral domain). Consequently, χ(D)χ(D)¯=n·Id+λχ(G)=n·Id, where Id is the d×d identity matrix. Furthermore, if χ is a nontrivial representation of G/N of degree d then D̂D̂(-1)=n·1G/N+|N|λ(G/N) and χ(D̂)χ(D̂)¯=n·Id.Recall that the ring of integers of the cyclotomic fieldℚ[ζm′] is ℤ[ζm′]. This ring is also an integral domain. Let p, a, b∈ℤ[ζm′]. The number p is irreducible if p=ab implies one of a or b is a unit. The element p is prime if p∣ab implies p∣a or p∣b [23]. A domain is a unique factorization domain (UFD) if factorization into irreducibles is possible and unique. In UFD, the irreducibles are also primes. In order to successfully obtain the difference set images, we need the aliases. Suppose that G/N is an abelian factor group of exponent m′ and D̂ is a difference set image in G/N. If χ is not a principal character of G/N, then χ(D̂)χ(D̂)¯=n is an algebraic equation in ℤ[ζm′]. The determination of the alias requires the knowledge of how the ideal generated by χ(D̂) factors in cyclotomic ring ℤ[ζm′], where ζm′ is the m′th root of unity. If δ:=χ(D̂), then by (2.4), we seek α∈ℤ[G/N] such that χ(α)=δ. The task of solving the algebraic equation δδ¯=n is sometimes made easier if we consider the factorization of principal ideals 〈δ〉〈δ¯〉=〈n〉. Suppose we are able to find δ=∑i=0ϕ(m′)-1diζm′i∈ℤ[ζm′] such that δδ¯=n, where ϕ is the Euler ϕ-function. A theorem due to Kronecker [12, 13] states that any algebraic integer whose all conjugates have absolute value 1 must be a root of unity. We use this theorem to characterize the solutions. If there is any other solution to the algebraic equation, then it must be of the form δ′=δu, where u=±ζm′j is a unit.The following result is used to determine the number of factors of an ideal in a ring: supposep is any prime and m′ is an integer such that gcd(p,m′)=1. Suppose that d is the order of p in the multiplicative group ℤm′* of the modular number ring ℤm′. Then the number of prime ideal factors of the principal ideal 〈p〉 in the cyclotomic integer ring ℤ[ζm′] is ϕ(m′)/d, where ϕ is the Euler ϕ-function, that is, ϕ(m′)=|ℤm′*| [25]. For instance, the ideal generated by 2 has two factors in ℤ[ζ7], the ideal generated by 7 has two factors in ℤ[ζ20], while the ideal generated by 3 has four factors in ℤ[ζ40]. On the other hand, since 2s is a power of 2, the ideal generated by 2 is said to completely ramifies as power of 〈1-ζ2s〉=〈1-ζ2s〉¯ in ℤ[ζ2s].According to Turyn [14], an integer n is said to be semi-primitive modulo m′ if for every prime factor p of n, there is an integer i such that pi≡-1modm′. In this case, −1 belongs to the multiplicative group generated by p. Furthermore, n is self-cosnjugate modulo m′ if every prime divisor of n is semi primitive modulo mp′, where mp′ is the largest divisor of m′ relatively prime to p. This means that all prime ideals over n in ℤ[ζm′] are fixed by complex conjugation. For instance, 72≡-1(modm′), where m′=2, 5, 10 and 7≡-1(modm′), m′=2, 4, 8. Thus, 〈7〉 is fixed by conjugation in ℤ[ζm′], m′=2, 4, 5, 8, 10, 50.Remark 2.2.
If〈n〉=Πi=1sθi in cyclotomic ring ℤ[ζm′], where θi is an ideal and s is an odd integer, then there is no solution to δδ¯=n. To see this, assume that a δ exist such that 〈δ〉=Πi=1kαi. Then 〈n〉=〈δ〉〈δ¯〉 has 2k factors but 〈n〉 has odd factors.Remark 2.3.
Let us consider the ideal generated by 2 which has two factors in the cyclotomic ringℤ[ζ23]. We claim that the algebraic number 2 is prime in this ring. Since (23, 11, 5) difference sets exist, and there exists θ such that θθ¯=6 and θ+θ¯=-1. This implies that θ2+θ+6=0 and θ=(-1±-23)/2. Consequently, θ∈ℤ[-23]. Suppose that the algebraic number 2 is not prime in ℤ[-23]. As -23≡1(mod4)([23], chapter 3), we seek a, b∈ℤ such that δ=(a+b-23)/2 and δδ¯=(a2+23b2)/4=2. The equation a2+23b2=8 has no integer solution. Thus, there is no algebraic number such that δδ¯=2. In fact, δδ¯=p has no solution, where p=2, 3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22. However, the equation a2+23b2=4m2 has trivial solutions (a,b)=(-2m,0) and (2m,0), where m=2, 3, 4, 5, 7, 9, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 25. We noticed that since the class number of the cyclotomic ring ℤ[ζ23] is 3, the equation a2+23b2=4·23 has nontrivial solutions (a,b)=(-3,-1), (-3,1), (3,-1) and (3,1). Also, a2+23b2=4m has nontrivial solutions for m=6, 8, 12, 18, 62, 82, 122, 162, 182 and 242.In this paper, we will use the phasemfactors triviallyin ℤ[ζm′] if the ideal generated by m is prime or ramifies in ℤ[ζm′]; m is self-conjugate modulo m′; the ideal generated by m has odd factors or the algebraic equation δδ¯=m2 has no solution or has trivial solutions. In summary, suppose that D̂ is the difference set image of order n=m2 in the cyclic factor group G/N, where G/N is a group with exponent m′. Suppose that m factors trivially in ℤ[ζm′] and χ is a nontrivial representation of G/N. Then χ(D̂)=±mζm′i, ζm′ is the m′th root of unity [13].


## 2.3. Characteristics of Difference Set Images in Subgroup of a Group

In this subsection, we use the attributes of subgroups of a group to obtain information about the difference set image in the subgroups. Dillon [5] proved the following results which will be used to obtain difference set images in dihedral group of a certain order if the difference images in the cyclic group of same order are known.Theorem 2.4 (dillon dihedral trick).
LetH be an abelian group and let G be the generalized dihedral extension of H. That is, G=〈q,H:q2=1,qhq=h-1,∀h∈H〉. If G contains a difference set, then so does every abelian group which contains H as a subgroup of index 2.Corollary 2.5.
If the cyclic groupZ2m does not contain a (nontrivial) difference set, then neither does the dihedral group of order 2m.Remark 2.6.
We look at subgroup properties of a group that can aid the construction of difference set image. For the convenience of the reader, we reproduce the idea of Gjoneski et al. [26]. Suppose that H is a group of order 2h with a central involution z. We take T={ti:i=1,…,h} to be the transversal of 〈z〉 in H so that every element in H is viewed as tizj, 0≤i≤h, j=0,1. Denote the set of all integral combinations, ∑i=1haiti of elements of T, ai∈ℤ by ℤ[T]. Using the two representations of subgroup 〈z〉 and Frobenius reciprocity theorem [27], we may write any element X of the group ring ℤ[H] in the form
(2.5)X=X(1+z2)+X(1-z2).
Furthermore, let A be the group ring element created by replacing every occurrence of z in X by 1. Also, let B be the group ring element created by replacing every occurrence of z in H by −1. Then
(2.6)X=A(〈z〉2)+B(2-〈z〉2),
where A=∑i=1haiti and B=∑j=1hbjtj, ai, bj∈ℤ. As X∈ℤ[H], A and B are both in ℤ[T], and A≡Bmod2. We may equate A with the homomorphic image of X in G/〈z〉. Consequently, if X is a difference set, then the coefficients of ti in the expression for A will be intersection number of X in the coset 〈z〉[26]. In particular, it can be shown that if K is a subgroup of a group H such that
(2.7)H≅K×〈z〉,
then the difference set image in H is
(2.8)D̂=A(〈z〉2)+gB(2-〈z〉2),
where g∈H, A is a difference set in K, α=(k+m)/|K| or α=(k-m)/|K|, B=A-αK and k is the size of the difference set. Equation (2.8) is true as long as |K|∣(k+m) or |K|∣(k-m).


## 2.4. Amalgamation of Results

In this paper, we study(v,k,λ) difference sets in which n=k-λ=m2 and the ideal generated by m factors trivially in the cyclotomic ring ℤ[ζm′]. That is, if n=m2, then (n)=(m)(m) up to units in ℤ[ζm′]. This method is very useful in the investigation of difference sets in solvable groups. A group G is solvable if the sequence G⊇G′⊇G′′⋯⊇⋯⊇G(i)⋯ terminates in the identity, G(e)=1, in a finite number of steps, each G(i) is the derived group of the preceding one [28]. Consequently, each i, the factor group G(i)/G(i+1) is Abelian. We now state the extended Sylow theorem in solvable groups ([28], page 141).Theorem 2.7.
LetG be a solvable group of order mn, in which gcd(m,n)=1. Then(1)
G possesses at least one subgroup of order m;(2)
any two subgroups of orderm are conjugates;(3)
any subgroup whose orderm′ divides m is contained in a subgroup of order m;(4)
the numbernm of subgroups of order m may be expressed as a product of factors, each of which (a) is congruent to 1 modulo some prime factor of m, and (b) is a power of a prime and divides one of the chief factors of G.The next three criteria enable us to rule out the existence of difference sets.Criterion 1.
Suppose thatG is a group of order v=p′s, where p′ is prime, s and t are integers. Then G does not admit (v,k,λ) if there exists a normal subgroup N of G such that(1)
k-λ=m2, m is a natural number,(2)
|G/N|=p′,(3)
m factors trivially in the cyclotomic ring ℤ[ζp′], where ζp′ is the p′th root of unity,(4)
the difference set solution inG/N is one of the forms α(G/N)+m, α+m>|N| or α(G/N)-m, α<m.Proof.
The nonexistence of viable difference set image inG/N implies that G does not admit (v,k,λ) difference set.In this criterion, we may replace|G/N|=p′ with |G/N|=p′q if q>2 is prime power, q∣s, gcd(p′,q)=1, and the ideal generated by p factors trivially in ℤ[ζp′q], where p is a prime divisor of m (see Remark 2.3).Criterion 2.
Suppose thatG is a group of even order v and H is a factor group of G with |H|=2q, where q is prime. Let g be an element of order 2 in H. Then G does not admit (v,k,λ) if(1)
k-λ=m2, m is a natural number,(2)
m factors trivially in the cyclotomic rings ℤ[ζq], where ζq is qth root of unity,(3)
the difference set solution inH/〈g〉 is one of the forms α(H/〈g〉)+m,  α+m>|G/(H/〈g〉)| or α(H/〈g〉)-m,  α<m; alternatively, the difference set image in H is one of the forms α(H)+m,  α+m>|G/H| or α(H)-m,  α<m.Proof.
The proof follows from Criterion1 and the fact that if |H|≡2(mod4), the cyclotomic rings ℤ[ζ2q] and ℤ[ζq] are the same.Criterion 3.
Suppose thatG is a group of order v=22×q×s, where q≥3 is prime and s is an integer. Suppose that H is a factor group of G of order 2q. The group G does not admit (v,k,λ) difference set if there exists a normal subgroup N such that G/N≅H×C2 and(1)
k-λ=m2,  m is a natural number,(2)
every prime divisorm′ of m factors trivially in the cyclotomic rings ℤ[ζq], where ζq is qth root of unity, gcd(m′,q)=1(3)
the difference set solution inH is of the form αH+m, and α is an odd integer or(4)
the difference set solution inH is of the form αH-m, α is an even integer, m is an odd integer, and m>α/2.Proof.
There are two groups of order2q, cyclic and dihedral groups. Since every prime divisor m′ of m factors trivially in the cyclotomic rings ℤ[ζq], gcd(m′,q)=1, it follows that m factors trivially in ℤ[ζq]. Consequently, the (v,k,λ) difference set image in H is of the form αH+m or αH-m. Suppose that the (v,k,λ) difference set image in H is of the form αH+m and α is an odd integer. Using (2.8), the difference set image in H×C2 is
(2.9)D̂=A(〈z〉2)+gB(2-〈z〉2),
where g∈H×C2, A=αH+m is the difference set image in H, B=A-αH with α=(k+m)/2q or (k-m)/2q and z is the generator of C2. Since α is odd, A(〈z〉/2) consists of at least 2q-2 odd entries while B((2-〈z〉)/2) consists of at least 2q-2 even entries. Thus, (2.8) has no integer solutions. On the other hand, suppose that the (v,k,λ) difference set image in H is of the form αH-m, α is an even integer, m is odd an odd integer, and m>α/2. The difference set image is of the form (2.9) with A=αH-m. Since α is even and m is odd, A(〈z〉/2) and B((2-〈z〉)/2) have two entries that are fractions. In particular, we can translate if necessary, to ensure that the coefficients of the identity in both components are (α-m)/2 and -m/2, respectively. The sum and difference of these two entries are, respectively, (α-2m)/2 and α/2. But m>α/2 and 2m>α. Hence, (α-2m)/2 is a negative integer. Thus, there is no difference set image in H×C2 and the criterion follows.Notice that there are five factor groups of order22×q if q≡1(mod4) and four factor groups if q≡3(mod4). Criterion 3 rules out the existence of difference set images in Cq×C2×C2 and D2q≅Dq×C2. In addition to conditions of Criterion 3, if m factors trivially also in ℤ[ζ22×q], then three of the four or five factor groups (C2q, Cq×C2×C2 and D2q) of order 22×q do not admit difference sets.


## 3. Some Difference Sets Parameters (Tables1–5)

Table 1
Parameter sets that do not exist by Criterion1. C|G/N|=〈x〉 and parameters with asterisk indicate new results.
(v, k, λ)mp|G/N|Factoring ofp in ℤ[ζ|G/N|]No. of groups of ordervSolutions inG/N1(115, 19, 3)4223Remark2.31-4+〈x〉2(1333, 37, 1)62, 343a≡-1(mod43)a=27,3211-6+〈x〉3(221, 45, 9)62, 317a≡-1(mod17)a=24,381-6+3〈x〉4(145, 64, 28)62, 329a14≡-1(mod29),
a=2,316+2〈x〉5(1463, 86, 5)931939≡-1(mod19)1-9+5〈x〉6(583, 97, 16)9353326≡-1(mod53)1-9+2〈x〉7(345, 129, 48)9323Remark2.31-9+6〈x〉8(3503, 103, 3)102, 5113a≡-1(mod113)a=214,5561-10+〈x〉9(2185, 105, 5)102, 523Remark2.31-10+5〈x〉10(1309, 109, 9)102, 517a≡-1(mod17)a=24,581-10+7〈x〉11(1037, 112, 12)102, 561a≡-1(mod61)a=230,5151-10+2〈x〉12*(621, 125, 25)102, 569511≡-1(modb)b=23,69 2 factors trivially  in ℤ[ζa], a=23,69510+5〈x〉 in C23; None in C6913(469, 144, 44)102, 567a≡-1(mod67)a=233,511110+2〈x〉14(407, 175, 75)102, 537a18≡-1(mod37)a=2,51-10+5〈x〉15(3151, 126, 5)11111371134≡-1(mod137)1-11+〈x〉16*(483, 241, 120)1111231111≡-1(modb)b=23,69211+10〈x〉 in C23; None in C6917(561, 176, 55)1111187118≡-1(mod17)11 factors trivially in ℤ[ζa], a=17,1871-11+11〈x〉 in C17; None in C18718(20881, 145, 1)122, 3157a≡-1(mod157)a=226,3391-12+〈x〉19(1591, 160, 16)122, 343a≡-1(mod43)a=27,3211-12+4〈x〉20(9805, 172, 3)1313371318≡-1(mod37)1-13+5〈x〉21*(3895, 177, 8)131319139≡-1(mod19)2-13+10〈x〉22*(1711, 190, 21)131329137≡-1(mod29)2-13+7〈x〉23(2323, 216, 20)1414231411≡-1(mod23)1-14+10〈x〉24*(9951, 200, 4)142, 7107a53≡-1(mod107)a=2,72-14+2〈x〉25(8041, 201, 5)142, 743a≡-1(mod43)a=27,731-14+5〈x〉26(793, 352, 156)142, 761a30≡-1(mod61)a=2,71-14+6〈x〉27(50851, 226, 1)153, 5241a≡-1(mod241)a=360,5201-15+〈x〉28*(2871, 246, 21)153, 529a≡-1(mod29)a=314,572-15+9〈x〉29(2491, 250, 25)153, 553a26≡-1(mod53)a=3,51-15+5〈x〉30*(13573, 261, 5)162277246≡-1(mod277)2-16+〈x〉31(4879, 271, 15)16241210≡-1(mod41)1-16+7〈x〉32*(26815, 328, 4)182, 3173a86≡-1(mod173)a=2,32-18+2〈x〉33*(4551, 351, 27)182, 341a≡-1(mod41)a=210,342-18+9〈x〉34*(16975, 369, 8)1919971916≡-1(mod97)2-19+4〈x〉35*(15171, 370, 9)19193891997≡-1(mod389)2-19+〈x〉36(2599, 433, 72)19191131956≡-1(mod113)1-19+4〈x〉37(11455, 415, 15)202, 529a≡-1(mod29)a=214,571-20+15〈x〉38(3657, 457, 57)202, 553a26≡-1(mod53)a=2,51-20+9〈x〉39(194923, 442, 1)213, 7463a≡-1(mod463)a=3231,7771-21+〈x〉40(28609, 448, 7)213, 767a≡-1(mod67)a=311,7331-21+7〈x〉41*(18533, 452, 11)213, 743a≡-1(mod43)a=321,732-21+11〈x〉42*(13833, 456, 15)213, 753a≡-1(mod53)a=326,7132-21+9〈x〉43(4891, 490, 49)213, 773a≡-1(mod73)a=36,7121-21+7〈x〉44(3649, 513, 72)213, 789a44≡-1(mod89)a=3,71-21+6〈x〉45(2941, 540, 99)213, 7173a86≡-1(mod173)a=3,7121+3〈x〉46(1919, 686, 245)213, 7101a50≡-1(mod101)a=3,71-21+7〈x〉47(1769, 833, 392)213, 761a≡-1(mod61)a=35,7301-21+14〈x〉48*(78895, 487, 3)222, 11509a≡-1(mod509)a=2254,111272-22+〈x〉49(20461, 496, 12)222, 1137a≡-1(mod37)a=218,1131-22+14〈x〉50(4081, 561, 77)222, 1153a≡-1(mod53)a=226,11131-22+11〈x〉51(3835, 568, 84)222, 1159a29≡-1(mod59)a=2,111-22+10〈x〉52(3601, 576, 92)222, 11277a≡-1(mod277)a=246,11138122+2〈x〉53(94165, 532, 3)232337236≡-1(mod37)1-23+15〈x〉54*(18531, 545, 16)232371237≡-1(mod71)2-23+8〈x〉55(8557, 621, 45)242, 343a≡-1(mod43)a=27,3211-24+15〈x〉56(2959, 783, 207)242, 3269a134≡-1(mod269)a=2,31-24+3〈x〉57*(131253, 628, 3)2556535326≡-1(mod653)2-25+〈x〉58(11289, 664, 39)25553526≡-1(mod53)1-25+13〈x〉59(6205, 705, 80)25573536≡-1(mod73)1-25+10〈x〉60(3115, 865, 240)25589522≡-1(mod89)1-25+10〈x〉Table 2
Partial results in groups of orderv by Criterion 1. C|G/N|=〈x〉 and parameters with asterisk indicate new results. ? means the number of groups of order v is unknown.
(v, k, λ)mp|G/N|Factoring ofp in ℤ[ζ|G/N|]No. of groups of ordervNo. of groups ruled outSolutions inG/N1(171, 51, 15)62,319a9≡-1(mod19)a=2,352-6+3〈x〉2(155, 56, 20)62,331315≡-1(mod31) 2 factors trivially [20]21-6+2〈x〉3(231, 70, 21)777775≡-1(mod11) 7 factors trivially in ℤ[ζb], b=11,7721-7+7〈x〉 in C7; None in C774*(2325, 84, 3)9331315≡-1(mod31)103-9+3〈x〉,5*(10101, 101, 1)102,537a18≡-1(mod37)a=2,3145-10+3〈x〉6(715, 154, 33)1111143116≡-1(mod13) 11 factors trivially in ℤ[ζb], b=13,1432111+11〈x〉 in C13; None in C1437*(7155, 147, 3)122,353a26≡-1(mod53)a=2,3??-12+3〈x〉8*(38613, 197, 1)142,7211a105≡-1(mod211)a=2,752-14+〈x〉9*(5859, 203, 7)142,731715≡-1(mod31)2 factors trivially [20]??-14+7〈x〉10*(903, 287, 91)142,743a≡-1(mod43)a=27,7372-14+7〈x〉11*(2255, 392, 68)182,341a≡-1(mod41)a=210,3472-18+10〈x〉12*(160401, 401, 1)202,5421a≡-1(mod421)a=2210,510552-20+〈x〉13*(23607, 407, 7)202,561a≡-1(mod61)a=230,515115-20+7〈x〉14*(22451, 450, 9)213,7157a≡-1(mod157)a=339,72621-21+3〈x〉15*(2619, 561, 120)213,797a≡-1(mod97)a=324,748135-21+6〈x〉16(2211, 715, 231)222,1167a33≡-1(mod67)a=2,1141-22+11〈x〉17(7450, 573, 44)23231492374≡-1(mod149)105-23+4〈x〉18(111555, 579, 3)242,367a≡-1(mod67)a=233,311??-24+9〈x〉19(37961, 585, 9)242,329a14≡-1(mod29)a=2,321-24+21〈x〉20(23247, 591, 15)242,341a≡-1(mod41)a=210,34??-24+15〈x〉21(25641, 641, 16)25537518≡-1(mod37)144-25+18〈x〉Table 3
Parameter sets that do not exist by Criterion2. C|H|=〈x,y:xq=y2=[x,y]〉 and parameters with asterisk indicate new results. No diff. set image in H implies no difference set image in Dihedral group of same order.
(v, k, λ)mp|H|Factoring ofpin ℤ[ζ|H/〈g〉|]No. of groups of ordervSolutions inH1(56, 11, 2)331433≡-1(mod7)13-3+2〈x〉 in H/〈g〉2(154, 18, 2)422225≡-1(mod11)4-4+2〈x〉 in H/〈g〉3(66, 26, 10)422225≡-1(mod11)24+〈x〉〈y〉; 〈x〉〈y〉+2(1+x+y-xy)4(112, 37, 2)551453≡-1(mod7)43-5+3〈x〉〈y〉5*(690, 53, 4)771072≡-1(mod5)8-7+6〈x〉〈y〉6(496, 55, 6)77627 factors trivially see [21]42-7+2〈x〉 in H/〈g〉7*(306, 61, 12)773478≡-1(mod17)10-7+4〈x〉 in H/〈g〉8*(2146, 66, 2)8274218≡-1(mod37)4-8+2〈x〉 in H/〈g〉9*(806, 70, 6)822626≡-1(mod13)4-8+6〈x〉 in H/〈g〉10*(430, 78, 14)828627≡-1(mod43)4-8+2〈x〉 in H/〈g〉11*(370, 82, 18)8274218≡-1(mod37)48+〈x〉〈y〉; 〈x〉〈y〉+4(1+x+y-xy)12*(266, 106, 42)823829≡-1(mod19)4-8+6〈x〉 in H/〈g〉13*(3404, 83, 2)9346Remark2.311-9+4〈x〉 in H/〈g〉14*(714, 93, 12)933438≡-1(mod17)12-9+6〈x〉 in H/〈g〉15*(2668, 127, 6)111146Remark2.311-11+6〈x〉 in H/〈g〉16*(1704, 131, 10)11111421135≡-1(mod71)39-11+2〈x〉 in H/〈g〉17*(1450, 162, 18)122,358a14≡-1(mod29)a=2,310-12+6〈x〉 in H/〈g〉18*(760, 253, 84)131338139≡-1(mod19)39-13+7〈x〉〈y〉19*(13054, 229, 4)153,5122a5≡-1(mod61)a=3,54-15+4〈x〉 in H/〈g〉20*(4064, 239, 14)153, 5254a≡-1(mod127)a=363,521195-15+2〈x〉 in H/〈g〉21*(3268, 243, 18)153,586a21≡-1(mod43)a=3,59-15+6〈x〉 in H/〈g〉22*(2278, 253, 28)153,5134a11≡-1(mod67)a=3,54-15+4〈x〉 in H/〈g〉23*(1886, 261, 36)153,546Remark2.34-15+12〈x〉 in H/〈g〉24*(1406, 281, 56)153,574a≡-1(mod37)a=39,5184-15+8〈x〉 in H/〈g〉25*(1054, 325, 100)153,534a8≡-1(mod17)a=3,54-15+10〈x〉〈y〉26*(918, 393, 168)153,534a8≡-1(mod17)a=3,530-15+12〈x〉〈y〉27*(902, 425, 200)153,582a≡-1(mod41)a=34,510415+5〈x〉〈y〉28*(33154, 258, 2)162274234≡-1(mod137)10-16+2〈x〉 in H/〈g〉29*(11398, 262, 6)162278269≡-1(mod139)4-16+2〈x〉 in H/〈g〉30*(2466, 290, 34)162274234≡-1(mod137)1016+〈x〉〈y〉; 〈x〉〈y〉+8(1+x+y-xy)31*(1660, 316, 60)162166241≡-1(mod83)11-16+4〈x〉 in H/〈g〉32*(1066, 426, 170)16282210≡-1(mod41)416+5〈x〉〈y〉; 5〈x〉〈y〉+8(1+x+y-xy)33*(7526, 301, 12)17171061713≡-1(mod53)4-17+6〈x〉 in H/〈g〉34*(5796, 305, 16)1717461711≡-1(mod23)111-17+14〈x〉 in H/〈g〉35*(20758, 408, 8)202,5214a53≡-1(mod107)a=2,54-20+4〈x〉 in H/〈g〉36*(7474, 424, 24)202,574a18≡-1(mod37)a=2,54-20+12〈x〉 in H/〈g〉37*(5038, 438, 38)202,5458a≡-1(mod229)a=238,5574-20+2〈x〉 in H/〈g〉38*(2014, 550, 150)202,5106a26≡-1(mod53)a=2,5420+5〈x〉〈y〉; 5〈x〉〈y〉+10(1+x+y-xy)39*(1918, 568, 168)202,5274a≡-1(mod137)a=234,568420+4〈x〉 in H/〈g〉40*(24346, 541, 12)2323942323≡-1(mod47)8-23+12〈x〉 in H/〈g〉41*(34282, 586, 10)242, 3122a≡-1(mod61)a=230,354-24+10〈x〉 in H/〈g〉42*(20770, 645, 20)255134511≡-1(mod67)12-25+10〈x〉 in H/〈g〉Table 4
Partial results in groups of orderv by Criterion 2. C|H|=〈x,y:xq=y2=[x,y]〉 and parameters with asterisk indicate new results. No difference set image in H implies no diff. set image in Dihedral group of same order. ? means the number of groups of order v is unknown.
(v, k, λ)mp|H|Factoring ofpin ℤ[ζ|H/〈g〉|]No. of groups of ordervNo. of groups ruled outSolutions inH1(78, 22, 6)422626≡-1(mod13)64-4+2〈x〉 in H/〈g〉2(204, 29, 4)553458≡-1(mod17)1210-5+2〈x〉 in H/〈g〉3*(1140, 68, 4)823829≡-1(mod19)4129-8+4〈x〉 in H/〈g〉4*(396, 80, 16)826625≡-1(modb)b=11,333015None inH/〈g〉5*(300, 92, 48)8250210≡-1(mod25)499-8+4〈x〉 in H/〈g〉6*(980, 89, 8)931433≡-1(mod7)3431-9+7〈x〉〈y〉7*(456, 105, 24)933839≡-1(mod19)5439-9+6〈x〉 in H/〈g〉8*(1856, 106, 6)102, 558a≡-1(mod29)a=214,5716301387-10+4〈x〉 in H/〈g〉9*(7504, 123, 2)11111341133≡-1(mod67)??-11+2〈x〉 in H/〈g〉10*(3876, 125, 4)111134118≡-1(mod17)4034-11+8〈x〉 in H/〈g〉11*(2870, 152, 8)122, 382a≡-1(mod41)a=210,34128-12+4〈x〉 in H/〈g〉12*(666, 210, 66)122, 374a≡-1(mod37)a=218,391810-12+6〈x〉 in H/〈g〉13*(610, 232, 88)122, 3122a≡-1(mod61)a=230,3564-12+4〈x〉 in H/〈g〉14*(14536, 171, 2)131346Remark2.3??-13+8〈x〉 in H/〈g〉15*(7440, 173, 4)1313621315≡-1(mod31)??-13+6〈x〉 in H/〈g〉16*(5076, 175, 6)1313941323≡-1(mod47)??-13+4〈x〉 in H/〈g〉17*(2716, 181, 12)13131941348≡-1(mod97)119-13+2〈x〉 in H/〈g〉18*(19504, 198, 2)142, 7106a≡-1(mod53)a=226,713??-14+4〈x〉 in H/〈g〉19*(1696, 226, 30)142, 7212226≡-1(mod53)713≡-1(modb)b=53,106,21223519414+〈x〉〈y〉20*(8856, 231, 6)153, 582a≡-1(mod41)a=34,520??-15+6〈x〉 in H/〈g〉21*(1508, 275, 50)153, 558a≡-1(mod29)a=314,571511-15+10〈x〉 in H/〈g〉22*(976, 351, 126)153, 5122a≡-1(mod61)a=35,5155114-15+6〈x〉 in H/〈g〉23*(3256, 280, 24)16274218≡-1(mod37)??-16+8〈x〉 in H/〈g〉24*(1036, 460, 204)16274218≡-1(mod37)11916+6〈x〉〈y〉; 6〈x〉〈y〉+8(1+x+y-xy)25*(21390, 293, 4)1717621715≡-1(mod31)3616-17+10〈x〉 in H/〈g〉26*(52976, 326, 2)182, 386321≡-1(mod43) See Remark 2.2 for 〈2〉??-18+8〈x〉 in H/〈g〉27*(18096, 330, 6)182, 358a14≡-1(mod29)a=2,3??-18+12〈x〉 in H/〈g〉28*(1776, 426, 102)182, 374a≡-1(mod37)a=218,39260170-18+12〈x〉 in H/〈g〉29*(65704, 363, 2)19193821995≡-1(mod191)??-19+2〈x〉 in H/〈g〉30*(22388, 367, 6)19193861996≡-1(mod193)1511-19+2〈x〉 in H/〈g〉31*(13728, 371, 10)191926196≡-1(mod13)??-19+15〈x〉〈y〉32*(7960, 379, 18)1919398199≡-1(mod199)??-19+2〈x〉 in H/〈g〉33*(5084, 391, 30)1919821920≡-1(mod41)119-19+10〈x〉 in H/〈g〉34*(2256, 451, 90)1919941923≡-1(mod47)??-19+10〈x〉 in H/〈g〉35*(40704, 404, 4)202, 5106a26≡-1(mod53)a=2,5??-20+8〈x〉 in H/〈g〉36*(16770, 410, 10)202, 586a≡-1(mod43)a=27,5214824-20+10〈x〉 in H/〈g〉37*(1830, 590, 190)202, 5122a≡-1(mod61)a=230,515188-20+10〈x〉inH/〈g〉38*(49369, 445, 4)213, 7466a≡-1(mod233)a=3116,7581511-21+2〈x〉 in H/〈g〉39*(17064, 453, 12)213, 7158a39≡-1(mod79)a=3,7??-21+6〈x〉 in H/〈g〉40*(14756, 455, 14)213, 734a8≡-1(mod17)a=3,72723-21+14〈x〉〈y〉41*(10604, 461, 20)213, 7482a≡-1(mod241)a=360,7120119-21+2〈x〉 in H/〈g〉42*(7380, 471, 30)213, 782a≡-1(mod41)a=34,72014989-21+12〈x〉 in H/〈g〉43*(5336, 485, 44)213, 746Remark2.3??-21+11〈x〉〈y〉44*(3128, 531, 90)213, 746Remark2.3??-21+12〈x〉〈y〉45*(2408, 581, 140)213, 786a≡-1(mod43)a=321,73??-21+14〈x〉 in H/〈g〉46*(70890, 533, 4)23232782323≡-1(mod139)2416-23+4〈x〉 in H/〈g〉47*(47616, 535, 8)232362235≡-1(mod31)??-23+18〈x〉 in H/〈g〉48*(13776, 551, 22)232382235≡-1(mod41)??-23+7〈x〉〈y〉49*(166754, 578, 2)242, 386a≡-1(mod43)a=27,321128-24+14〈x〉 in H/〈g〉50*(56358, 582, 6)242, 3202a50≡-1(mod101)a=2,3??-24+6〈x〉 in H/〈g〉51*(10388, 612, 36)242, 3106a26≡-1(mod53)a=2,33428-24+12〈x〉 in H/〈g〉52*(4844, 668, 92)242, 3346a86≡-1(mod173)a=2,3119-24+4〈x〉 in H/〈g〉53*(3690, 714, 138)242, 382a≡-1(mod41)a=210,343020-24+18〈x〉 in H/〈g〉54*(3172, 756, 180)242, 3122a≡-1(mod61)a=230,35151124+6〈x〉〈y〉,
6〈x〉〈y〉+12(1+x+y-xy)55*(2332, 1036, 460)242, 3106a26≡-1(mod53)a=2,3119-24+20〈x〉 in H/〈g〉56*(196252, 627, 2)255326527≡-1(mod163)??-25+4〈x〉 in H/〈g〉57*(66256, 631, 6)25582510≡-1(mod41)??-25+16〈x〉 in H/〈g〉58*(50008, 633, 8)25594523≡-1(mod47)??-25+14〈x〉 in H/〈g〉59*(17524, 649, 24)255674556≡-1(mod337)1511-25+2〈x〉〈y〉60*(14280, 655, 30)2553458≡-1(mod17)??-25+20〈x〉〈y〉61*(11040, 665, 40)255125≡-1(modb)b=2,3??-25+115〈x〉 in H/〈g〉62*(11040, 665, 40)25546Remark2.3??-25+15〈x〉〈y〉63*(5104, 729, 104)2555857≡-1(mod29)??-25+13〈x〉〈y〉64*(2812, 937, 312)25574518≡-1(mod37)119-25+13〈x〉〈y〉Table 5
Partial results in groups of orderv by Criterion 3. Parameters with asterisk indicate new results.? means the number of groups of order v is unknown.
(v, k, λ)mpFactoring ofpin ℤ[ζq]No. of groups of ordervNo. of groups ruled outSolutions inH1(40, 13, 4)3332≡-1(modb)b=5,1014103+H, |H|=102*(400, 57, 8)7772≡-1(modb)b=5,102211667+5H,
|H|=103*(280, 63, 14)7772≡-1(modb)b=5,104030-7+7H,
|H|=104*(220, 73, 24)7772≡-1(modb)b=5,10157-7+8H,
|H|=105*(820, 91, 10)9332≡-1(modb)b=5,10207-9+10H,
|H|=106*(540, 99, 18)9332≡-1(modb)b=5,10119569+9H,
|H|=107*(3876, 125, 4)111111≡-1(modb)b=3,6,12403211+19H,
|H|=68*(1464, 133, 12)1111112≡-1(mod61)613011+H,
|H|=1229*(988, 141, 30)1111116≡-1(modb)b=13,2611511+5H,
|H|=2610*(756, 151, 30)111111≡-1(modb)b=3,6,1218996-11+27H,
|H|=611*(2380, 183, 14)1313132≡-1(mod17)351513+5H,
|H|=3412*(1056, 211, 42)1313135≡-1(mod11)102899513+9H,
|H|=2213*(1456, 195, 26)131313≡-1(mod7)17917113+13H,
|H|=1414*(1380, 197, 28)1313132≡-1(modb)b=5,102915-13+21H,
|H|=1015*(1548, 273, 48)153, 5a21≡-1(mod43)a=3,546615+3H,
|H|=8616*(1160, 305, 80)153, 5a≡-1(mod29)a=314,57493315+5H,
|H|=5817*(1012, 337, 112)153, 5511≡-1(mod23)Remark 2.313515+7H,
|H|=4618*(1300, 433, 144)1717172≡-1(modb)b=5,105016-17+45H,
|H|=1019*(5220, 307, 18)1717172≡-1(modb)b=5,101135017+29H,
|H|=1020*(5220, 307, 18)1717172≡-1(mod29)1135017+5H,
|H|=5821*(5220, 307, 18)171717≡-1(modb)b=3,611350-17+27H,
|H|=1222*(33216, 365, 4)19191986≡-1(mod173)??19+H,
|H|=34623*(11564, 373, 12)1919193≡-1(mod7)2816-19+28H,
|H|=1424*(7240, 381, 20)1919192≡-1(mod181)??19+H, |H|=36225*(4368, 397, 36)1919193≡-1(mod7)??19+27H, |H|=1426*(4180, 399, 38)1919195≡-1(mod11)3615-19+19H, |H|=2227*(2508, 437, 76)1919195≡-1(mod11)342019+19H, |H|=2228*(1976, 475, 114)1919196≡-1(mod13)3930-19+19H,
|H|=2629*(1624, 541, 180)19191914≡-1(mod29)563019+9H, |H|=5830*(1520, 589, 228)1919193≡-1(modb)b=5,1017814719+57H,
|H|=1031*(97904, 443, 2)213, 7a105≡-1(mod211)a=3,7??21+H,
|H|=42232*(11680, 459, 18)213, 7a≡-1(mod73)a=36,712??21+3H,
|H|=14633*(9724, 463, 22)213, 7a8≡-1(mod17)a=3,7351521+13H,
|H|=3434*(7840, 469, 28)213, 7a2≡-1(mod5)a=3,7??-21+49H,
|H|=1035*(7380, 471, 30)213, 7a2≡-1(mod5)a=3,71496621+45H,
|H|=1036*(3128, 531, 90)213, 7a8≡-1(mod17)a=3,7??21+15H,
|H|=3437*(2756, 551, 110)213, 7a≡-1(mod53)a=326,71320521+5H,
|H|=10638*(2296, 595, 154)213, 7a≡-1(mod41)a=34,720??21+7H,
|H|=8239*(1904, 693, 252)213, 7a8≡-1(mod17)a=3,7186147-21+21H, |H|=3440*(1836, 735, 294)213, 7a≡-1(mod17)a=32,721175621+21H,
|H|=3441*(1820, 749, 308)213, 7a≡-1(mod5)a=32,723515-21+77H,
|H|=1042*(1800, 771, 330)213, 7a≡-1(mod5)a=32,7274941221+75H,
|H|=1043*(47616, 535, 6)232323≡-1(mod3)??-23+93H,
|H|=644*(35980, 537, 8)23232332≡-1(mod257)351523+H,
|H|=51445*(12720, 553, 24)2323232≡-1(mod5)??23+53H, |H|=1046*(12720, 553, 24)2323232≡-1(mod53)??23+5H,
|H|=10647*(7176, 575, 46)2323233≡-1(mod13)??-23+23H,
|H|=2648*(4320, 617, 88)232323≡-1(mod3)??23+99H,
|H|=649*(3220, 667, 138)2323232≡-1(mod5)2715-23+69H,
|H|=1050*(2760, 713, 184)232323≡-1(mod3)??23+115H,
|H|=651*(2760, 713, 184)2323232≡-1(mod5)??23+69H,
|H|=1052*(2380, 793, 264)2323238≡-1(mod17)3515-23+24H, |H|=3453*(2380, 793, 264)2323232≡-1(mod5)351523+77H,
|H|=1054*(196252, 627, 2)25553≡-1(mod7)??25+43H,
|H|=1455*(196252, 627, 2)255521≡-1(mod43)??25+7H,
|H|=8656*(40260, 635, 10)255515≡-1(mod61)3706625+5H,
|H|=12257*(16276, 651, 26)25554≡-1(mod313)20525+H,
|H|=62658*(14280, 655, 30)25553≡-1(mod7)??25+45H,
|H|=1459*(11040, 665, 40)2555≡-1(mod3)??-25+115H,
|H|=660*(9100, 675, 50)25552≡-1(mod13)1185025+25H,
|H|=2661*(6328, 703, 78)255556≡-1(mod113)??25+3H,
|H|=22662*(4620, 745, 120)25553≡-1(mod7)14066-25+55H,
|H|=1463*(4380, 755, 130)255536≡-1(mod73)531525+5H,
|H|=14664*(3400, 825, 200)25558≡-1(mod17)??-25+25H,
|H|=3465*(3060, 875, 250)25558≡-1(mod17)1135025+25H,
|H|=3466*(2640, 1015, 390)2555≡-1(mod3)??25+165H,
|H|=667*(2520, 1145, 520)2555≡-1(mod3)??-25+195H,
|H|=6We list some parameter sets (both known and new) that do not exist. In each of these cases,G is a group of order v and φ:G→H is a group homomorphism. Suppose that D is a k-subset of G and n=k-λ=m2 such that m factors trivially in the cyclotomic ring ℤ[ζ|H|]. We use Criteria 1, 2 and 3 to rule out the existence of (v,k,λ) difference set. Examples of such parameters are listed in Tables 1 and 3. We also listed partial results in Tables 2, 4, and 5.



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