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In the previous work, we introduce a notion of pre-difference sets in a finite group G defined by weaker conditions than the difference sets. In this paper we gave a construction of a pre-difference set in $$G=NA$$ with A an abelian subgroup and N a subgroup satisfying $$N\cap A=\{e\}$$ , from a difference set in $$N\times A$$ . This gives a (16, 6, 2) pre-difference set in $$D_{16}$$ and a (27, 13, 6) pre-difference set in UT(3, 3), where no non-trivial difference sets exist. We also give a product construction of pre-difference sets similar to Kesava Menon construction, which provides infinite series of pre-difference sets that are not difference sets. We show some necessary conditions for the existence of a pre-difference set in a group with index 2 subgroup. For the proofs, we use a rather simple framework “relation partitions,” which is obtained by dropping an axiom from association schemes. Most results are proved in that frame work.

A $$(v,k,\lambda )$$ symmetric design is said to have the symmetric difference property (SDP) if the symmetric difference of any three blocks is either a block or the complement of a block. The designs associated to the symplectic difference sets introduced by Kantor (J Algebra 33:43–58, 1975) have the SDP. Parker (J Comb Theory Ser A 67:23–43, 1994) claimed that the symplectic design on 64 points is the only SDP design on 64 points admitting an abelian regular automorphism group (an abelian difference set). We show in this paper that there is an SDP design on 64 points that is not isomorphic to the symplectic design and yet admits the group $$C_8 \times C_4 \times C_2$$ as a regular automorphism group. This abelian difference set is the first in an infinite family of abelian difference sets whose designs have the SDP and yet are not isomorphic to the symplectic designs of the same order. We define a new method for establishing the non-isomorphism of the two families.

In a 1989 paper \cite{arasu2}, Arasu used an observation about multipliers to show that no $(352,27,2)$ difference set exists in any abelian group. The proof is quite short and required no computer assistance. We show that it may be applied to a wide range of parameters $(v,k,\lambda)$, particularly for small values of $\lambda$. With it a computer search was able to show that the Prime Power Conjecture is true up to order $2 \cdot 10^{10}$, extend Hughes and Dickey's computations for $\lambda=2$ and $k \leq 5000$ up to $10^{10}$, and show nonexistence for many other parameters., Comment: 8 pages

In this note we prove the non-existence of two types of partial difference sets in Abelian groups of order 216 . This completes the classification of parameters for which a partial difference set of size at most 100 exists in an Abelian group.

This paper addresses the question of whether triple arrays can be constructed from Youden squares developed from difference sets. We prove that if the difference set is abelian, then having −1 as multiplier is both a necessary and sufficient condition for the construction to work. Using this, we are able to give a new infinite family of triple arrays. We use an alternative version of the construction to analyze the case of non-abelian difference sets, for which we prove a sufficient condition for giving triple arrays. We do a computer search for such non-abelian difference sets, but have not found any examples satisfying the given condition.

Generalizing a result of Ko and Ray-Chaudhuri (Discrete Math. 39 (1982), 37–58), we show the following: Assume the existence of an affine difference set in G relative to N of even order n≠2. If G is of the form G = N ⊕ H, where N is abelian, then n is actually a multiple of 4, say n = 4k, and there exists a (4k − 1, 2k − 1, k − 1)-Hadamard difference set in N. More detailed considerations lead to variations of this result (under appropriate assumptions) which yield even stronger non-existence theorems. In particular, we show the non-existence of abelian affine difference sets of order n ≡ 4 mod 8 (with the exception n = 4) and of nilpotent affine difference sets of order n ≡ 2 mod 4 (n ≠ 2). The latter result is the first general non-existence theorem in the non-abelian case.

A finite group G is called a Schur group if every Schur ring over G is the transitivity module of a point stabilizer in a subgroup of Sym(G) that contains all permutations induced by the right multiplications in G. It is proved that the group $$ {\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_{2^n} $$ is Schur, which completes the classification of Abelian Schur 2-groups. It is also proved that any non-Abelian Schur 2-group of order larger than 32 is dihedral (the Schur 2-groups of smaller orders are known). Finally, the Schur rings over a dihedral 2-group G are studied. It turns out that among such rings of rank at most 5, the only obstacle for G to be a Schur group is a hypothetical ring of rank 5 associated with a divisible difference set.

Let $(G, +)$ be an abelian group. In 2004, Eliahou and Kervaire found an explicit formula for the smallest possible cardinality of the sumset $A+A$, where $A \subseteq G$ has fixed cardinality $r$. We consider instead the smallest possible cardinality of the difference set $A-A$, which is always greater than or equal to the smallest possible cardinality of $A+A$ and can be strictly greater. We conjecture a formula for this quantity and prove the conjecture in the case that $G$ is a cyclic group or a vector space over a finite field. This resolves a conjecture of Bajnok and Matzke on signed sumsets., 19 pages

In this paper, we present some construction methods for punctured binary array/sequence pairs (PBAPs/PBSPs) with ideal/optimal correlation constant using their algebraic counterparts punctured difference set pairs in Abelian groups. In addition, we provide new construction techniques of PBAPs/PBSPs via geometry and also using the embeddable sequence pairs of smaller lengths to obtain larger ones. PBAPs/PBSPs find a plethora of applications in radar systems, cryptography, frame synchronization, mismatched filtering, and various other engineering fields.

The combinatorial theory of difference sets has prior applications in the field of mathematical music theory. The theory of almost difference sets, however, has not received similar attention from music scholars. Nevertheless, these types of structures also have significant musical applications. For instance, the well known all-interval tetrachords of pitch-class set theory are almost difference sets. To that end, we investigate the various categories of almost difference sets (cyclic, abelian, and non-abelian) in terms of their representations in Lewinian music-transformational groups.

On Isoperimetric Stability, Discrete Analysis 2018:14, 11 pp. Let $A$ be a subset of the Hamming cube $\{0,1\}^n$. If we regard the cube as a graph in the usual way, by joining two points $x$ and $y$ if they differ in exactly one coordinate, then the _edge boundary_ of $A$ is defined to be the set of all edges that join a point in $A$ to a point in its complement $\{0,1\}^n\setminus A$. The edge-isoperimetric inequality in the cube tells us that if $A$ has cardinality $2^d$, then the size of the edge boundary of $A$ is minimized when $A$ is a subcube of dimension $d$, in which case the number of edges between $A$ and its complement is $2^d(n-d)$. We can regard the graph as a Cayley graph, where the group is $\mathbb F_2^n$ and the generating set $S$ consists of the $n$ standard basis vectors. Then we find that the size of the edge boundary is always at least $|A|(|S|-\log_2|A|)$, an inequality that continues to hold when the size of $A$ is not a power of 2. As a consequence, if the size of the edge boundary is at most $(1-\gamma)|A||S|$, then $A$ must have size at least $2^{\gamma|S|}$. The purpose of this paper is to prove a similar statement for Abelian groups in general. Given an Abelian group $G$, a subset $A\subset G$, and a generating set $S$ of $G$, define the edge boundary to be the set of pairs $(a,s)\in A\times S$ such that $a+s\notin A$. The precise statement proved is that if the size of the edge boundary is at most $(1-\gamma)|A||S|$, then $A$ must have size at least $4^{(1-1/d)\gamma|S|}$, where $d$ is the exponent of $G$ -- that is, the smallest order of any element of $G$. Note that when $G=\mathbb F_2^n$ , then $d=2$, and we recover the statement above. Note also that in this case we can rewrite the lower bound for the size of $A$ as $|G|^\gamma$, a bound that is shown to be valid for groups of exponent 3 or 4 as well. However, for larger exponents the lower bound of $|G|^\gamma$ does not hold, and examples show that we have to settle for the weaker bound $4^{(1-1/d)\gamma|S|}$ proved in this paper. The reason that groups of low exponent are a special case is connected with the notion of a _dissociated set_, which is a subset $X$ of an Abelian group $G$ with the property that if $Y$ and $Z$ are finite subsets of $X$ with $\sum_{x\in Y}x=\sum_{x\in Z}x$, then $Y=Z$. Equivalently, it is a subset $X$ with a kind of linear independence property: if $x_1,\dots,x_n$ are distinct elements of $X$ and $\epsilon_1,\dots,\epsilon_n$ are coefficients taken from the set $\{-1,0,1\}$ such that $\sum_i\epsilon_ix_i=0$, then $\epsilon_1=\dots=\epsilon_n=0$. For groups of exponent 2 or 3, all possible integer coefficients can be replaced by ones that belong to the set $\{-1,0,1\}$, so this is just normal independence over the integers (and therefore, when these groups are considered as vector spaces, linear independence), but for groups of higher exponent that ceases to be the case. For example, in the group $\mathbb F_5^n$, the set $\{e_1,\dots,e_n,2e_1,\dots,2e_n\}$ is dissociated but not independent. These notions lead naturally to notions of dimension for subsets of Abelian groups: given a finite subset $E$, we define $\mathrm{dim}_D(E)$ to be the size of the largest dissociated subset of $E$ and $\mathrm{dim}_I(E)$ to be the size of the largest independent subset. A corollary of the main results of this paper is a sharp bound for the dimension of the _popular difference set_ of a set $A$, which is defined as follows. Given a parameter $\gamma>0$, we set $P_\gamma(A)$ to be the set of all group elements $g$ such that $|A\cap(A-g)|\geq\gamma|A|$. Popular difference sets play an important role in additive combinatorics. A result of Shkredov and Yekhanin states that $\mathrm{dim}_D(P_\gamma(A))$ is at most $C\gamma^{-1}\log|A|$, for some absolute constant $C$. (To see why the factor $\gamma^{-1}$ is necessary, consider an arbitrary union of $\gamma^{-1}$ subspaces of $\mathbb F_2^n$ of the same dimension.) In this paper it is shown that if $p$ is the smallest order of a non-zero element of $G$, then $\mathrm{dim}_I(P_\gamma(A)) \le (2(1-1/p))^{-1}\gamma^{-1}\log_2|A|$, which is refined to $\mathrm{dim}_I(P_\gamma(A)) \le \gamma^{-1}\log_3|A|$ in the special case $p=3$. These estimates are sharp when $G$ is homocyclic of exponent $p=2$ or $3$, when the two notions of dimension coincide. One of the proof ingredients in the paper is a result that stands on its own. A _down-set_ in $\mathbb Z_{\geq 0}^n$ is a subset $A$ with the property that if $x\in A$ and $y_i\leq x_i$ for every $i$, then $y\in A$. The paper gives an upper bound of $\frac 12\log_2|A|$ for the average number of non-zero coordinates of a vector in a down-set. It is a simple exercise to prove that this bound is sharp.

A linking system of difference sets is a collection of mutually related group difference sets, whose advantageous properties have been used to extend classical constructions of systems of linked symmetric designs. The central problems are to determine which groups contain a linking system of difference sets, and how large such a system can be. All previous constructive results for linking systems of difference sets are restricted to 2-groups. We use an elementary projection argument to show that neither the McFarland/Dillon nor the Spence construction of difference sets can give rise to a linking system of difference sets in non-2-groups. We make a connection to Kerdock and bent sets, which provides large linking systems of difference sets in elementary abelian 2-groups. We give a new construction for linking systems of difference sets in 2-groups, taking advantage of a previously unrecognized connection with group difference matrices. This construction simplifies and extends prior results, producing larger linking systems than before in certain 2-groups, new linking systems in other 2-groups for which no system was previously known, and the first known examples in nonabelian groups., Comment: 25 pages. Better description of relationship to previous work, especially in Section 3. Shorter proof of Proposition 1.7. Correction of minor errors

In this paper, we present constructions of abelian Paley type sets by using multiplicative characters of finite fields and Arasu---Dillon---Player difference sets. The constructions produce many new Paley type sets and their configurations that were previous unknown in our classification of Paley type sets in finite fields of small orders.

Equiangular tight frames (ETFs) and biangular tight frames (BTFs) - sets of unit vectors with basis-like properties whose pairwise absolute inner products admit exactly one or two values, respectively - are useful for many applications. A well-understood class of ETFs are those which manifest as harmonic frames - vector sets defined in terms of the characters of finite abelian groups - because they are characterized by combinatorial objects called difference sets. This work is dedicated to the study of the underlying combinatorial structures of harmonic BTFs. We show that if a harmonic frame is generated by a divisible difference set, a partial difference set or by a special structure with certain Gauss summing properties - all three of which are generalizations of difference sets that fall under the umbrella term "bidifference set" - then it is either a BTF or an ETF. However, we also show that the relationship between harmonic BTFs and bidifference sets is not as straightforward as the correspondence between harmonic ETFs and difference sets, as there are examples of bidifference sets that do not generate harmonic BTFs. In addition, we study another class of combinatorial structures, the nested divisible difference sets, which yields an example of a harmonic BTF that is not generated by a bidifference set., 29 pages; 3 tables of BTF families

Relatively few constructions are known of negative Latin square type Partial Difference Sets (PDSs), and most of the known constructions are in elementary abelian groups. We present a product construction that produces negative Latin square type PDSs, and we apply this product construction to generate examples in p-groups of exponent bigger than p.

Ding and Feng (IEEE Trans Inform Theory 52(9):4229---4235, 2006, IEEE Trans Inform Theory 53(11):4245---4250, 2007) constructed series of (N, K) codebooks which meet or nearly meet the Welch bound $${\sqrt{\frac{N-K}{(N-1)K}}}$$ by using difference set (DS) or almost difference set (ADS) in certain finite abelian group respectively. In this paper, we generalize the cyclotomic constructions considered in (IEEE Trans Inform Theory 52(9):4229---4235, 2006, IEEE Trans Inform Theory 53(11):4245---4250, 2007) and (IEEE Trans Inform Theory 52(5), 2052---2061, 2006) to present more series of codebooks which nearly meet the Welch bound under looser conditions than ones required by DS and ADS.

Davis, Dillon, and Jedwab all showed the existence of difference sets in groups $${C_{2^{r+2}}\times C_{2^{r}}}$$ . Turyn's bound had previously shown that abelian 2-groups with higher exponents could not admit difference sets. We give a new construction technique that utilizes character values, rational idempotents, and tiling structures to produce Hadamard difference sets in the group $${C_{2^{r+2}}\times C_{2^{r}}}$$ to replicate the result.

A condition is introduced on the abelian difference set [math] of an abelian projective plane of odd order so that the oval [math] is the set of absolute points of a polarity, with the consequence that any such abelian projective plane is Desarguesian.

In an abelian group G, a more sums than differences (MSTD) set is a subset A of G such that |A+A|>|A-A|. We provide asymptotics for the number of MSTD sets in finite abelian groups, extending previous results of Nathanson. The proof contains an application of a recently resolved conjecture of Alon and Kahn on the number of independent sets in a regular graph., 17 pages

Let p be a prime and let b be a positive integer. If a (v, k, ?, n) difference set D of order n = p b exists in an abelian group with cyclic Sylow p-subgroup S, then $${p\in\{2,3\}}$$ and |S| = p. Furthermore, either p = 2 and v ? ? ? 2 (mod 4) or the parameters of D belong to one of four families explicitly determined in our main theorem.

This article introduces a new approach to studying difference sets via their additive properties. We introduce the concept of special subsets, which are interesting combinatorial objects in their own right, but also provide a mechanism for measuring additive regularity. Skew Hadamard difference sets are given special attention, and the structure of their special subsets leads to several results on multipliers, including a categorisation of the full multiplier group of an abelian skew Hadamard difference set. We also count the number of ways to write elements as a product of any number of elements of a skew Hadamard difference set.

Let D be an affine difference set of order n in an abelian group G relative to a subgroup N. We denote by @p(s) the set of primes dividing an integer s(>0) and set H^*[email protected]?{@w}, where H=G/N and @[email protected]?"@s"@?"[email protected] In this article, using D we define a map g from H to N satisfying for @t,@[email protected]?H^*,g(@t)=g(@r) iff {@t,@t^-^1}={@r,@r^-^1} and show that ord"o"("@s")(m)/ord"o"("g"("@s")")(m)@?{1,2} for any @[email protected]?H^* and any integer m>0 with @p(m)@[email protected](n). This result is a generalization of J.C. Galati's theorem on even order n [J.C. Galati, A group extensions approach to affine relative difference sets of even order, Discrete Mathematics 306 (2006) 42-51] and gives a new proof of a result of Arasu-Pott on the order of a multiplier modulo exp(H) ([K.T. Arasu, A. Pott, On quasi-regular collineation groups of projective planes, Designs Codes and Cryptography 1 (1991) 83-92] Section 5).

Let D be a (v, k, λ)-difference set in an abelian group G, and (v, 31) = 1. If n = 5p r with p a prime not dividing v and r a positive integer, then p is a multiplier of D. In the case 31|v, we get restrictions on the parameters of such difference sets D for which p may not be a multiplier.

Let D be an affine difference set of order n in an abelian group G relative to a subgroup N. Set $${\tilde{H}}$$ = H \ {1, ?}, where H = G/N and $${\omega=\prod_{\sigma\in H}\sigma}$$ . Using D we define a two-to-one map g from $${\tilde{H}}$$ to N. The map g satisfies g(? m ) = g(?) m and g(?) = g(? ?1) for any multiplier m of D and any element ? ? $${\tilde{H}}$$ . As applications, we present some results which give a restriction on the possible order n and the group theoretic structure of G/N.

Dillon and Dobbertin proved that if L := GF(2 m ), gcd(k, m) = 1, d := 4 k ? 2 k + 1 and Δ k (x) := (x + 1) d + x d + 1, then B k := L\Δ k (L) is a difference set in the cyclic multiplicative group L × of L. Used in the proof were the auxiliary functions $$c_k^{\gamma}(x) := b_k(\gamma x^{2^k+1})$$ , where ? is in L × and b k is the characteristic function of B k on L. When m is odd $$c_k^{\gamma}$$ is itself the characteristic function of a cyclic difference set which is equivalent to B k . In this paper we point out that when m is even and ? is not a cube in L then $$c_k^{\gamma}$$ is the characteristic function of a difference set in the elementary abelian additive group of L; i.e. $$c_k^{\gamma}$$ is a bent function.

We show that difference sets satisfying the condition n|? have a product property which can be exploited to construct more difference sets. Many of the newly discovered difference sets arise in precisely this way.

A subset S ={ s1 ,...,s k} of an Abelian group G is called an St -set of size k if all sums of t different elements in S are distinct. Let s(G) denote the cardinality of the largest S2-set in G .L etv(k) denote the order of the smallest Abelian group for which s(G) k. In this article, bounds for s(G) are developed and v(k) is determined for k 15 by computing s(G) for Abelian groups of order up to 183 using exhaustive backtrack search with isomorph rejection. © 2006 Elsevier Inc. All rights reserved.

We study tilings of groups with mutually disjoint difference sets. Some necessary existence conditions are proved and shown not to be sufficient. In the case of tilings with two difference sets we show the equivalence to skew Hadamard difference sets, and prove that they must be normalized if the group is abelian. Furthermore, we present some constructions of tilings based on cyclotomy and investigate tilings consisting of Singer difference sets.

We theoretically establish the existence status of some previously open abelian difference sets. More precisely, we show the nonexistence of all the following difference sets: (841,120,17) in Z"2"9xZ"2"9, (364,121,40) in Z"2xZ"2xZ"7xZ"1"3, (837,133,21) in Z"3xZ"3xZ"3xZ"3"1,Z"3xZ"9xZ"3"1 and Z"8"3"7.

In this paper, we continue our investigation of relative difference sets fixed by inversion. We exclusively focus our attention on abelian groups. New necessary conditions are obtained and a new family of such relative difference sets with forbidden subgroup Z/4Z is constructed. The methods we use are character theory of abelian groups and Galois rings over Z/4Z.

A subset S={s"1,...,s"k} of an abelian group G is called an S"t-set of size k if all sums of t different elements in S are distinct. A function with applications in coding theory, v"@c(k) denotes the order of the smallest cyclic group in which an S"2-set of size k exists. A lower bound for v"@c(k) is given in this study, and exact values of v"@c(k) are obtained for k=

We present three constructions of partial difference sets (PDS) using different types of finite local rings. The first construction uses homomorphic images of finite Frobenius local rings and generalizes a previous result by the author. The second construction uses finite Frobenius local rings. The third construction uses finite (noncommutative) chain rings and generalizes a recent construction of partial difference sets by Leung and Ma. The three constructions provide many new PDS in nonelementary abelian groups.

Kantor [5] proved an exponential lower bound on the number of pairwise inequivalent difference sets in the elementary abelian group of order 22s+2. Dillon [3] generalized a technique of McFarland [6] to provide a framework for determining the number of inequivalent difference sets in 2-groups with a large elementary abelian direct factor. In this paper, we consider the opposite end of the spectrum, the rank 2 group ℤ, and compute an exponential lower bound on the number of pairwise inequivalent difference sets in this group. In the process, we demonstrate that Dillon difference sets in groups ℤ can be constructed via the recursive construction from [2] and we show that there are exponentially many pairwise inequivalent difference sets that are inequivalent to any Dillon difference set. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 249–259, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/jcd.10046

Let G be an abelian group. A collection of (G,k,λ) disjoint difference families, { F 0 , F 1 ,…, F s−1 } , is a complete set of disjoint difference families if ⋃ 0⩽i⩽s−1 ⋃ B∈ F i B form a partition of G−{0}. In this paper, several construction methods are provided for complete sets of disjoint difference families. Applications to one-factorizations of complete graphs and to cyclically resolvable cyclic Steiner triple systems are also described.

A new nonexistence theorem is proved for abelian difference sets. The proof technique generalizes to yield similar theorems for partial and divisible difference sets. As applications, we establish the nonexistence of two previously open difference sets: (243, 121, 60) difference sets in Z 9 × Z 27 and (351, 126, 45) difference sets in Z 351 , thereby filling two missing entries in a recent table of Vera Lopez and Sanchez (1997) both with answer ‘no’.

An n-subset D of a group G of order n^2-1 is called an affine difference set of G relative to a normal subgroup N of G of order n-1 if the list of differences d_1d_2^{-1} ( d_1, d_2 \in D, d_1 \neq d_2) contains each element of G-N exactly once and no element of N. It is a well-known conjecture that if D is an affine difference set in an abelian group G, then for every prime p, the Sylow p-subgroup of G is cyclic. In Arasu and Pott [1], it was shown that the above conjecture is true when p=2. In this paper we give some conditions under which the Sylow p-subgroup of G is cyclic.

Nontrivial difference sets in 2-groups are part of the family of Hadamard difference sets. An abelian group of order 2^{2d+2} has a difference set if and only if the exponent of the group is less than or equal to 2^{d+2}. We provide an exponent bound for a more general type of 2-group which has a Hadamard difference set. A recent construction due to Davis and Iiams shows that we can attain this bound in at least half of the cases.

This paper contains a discussion of cocyclic Hadamard matrices, their associated relative difference sets, and regular group actions. Nearly all central extensions of the elementary abelian 2-groups by Z2 are shown to act regularly on the associated group divisible design of the Sylvester Hadamard matrices. Cocyclic orthogonal designs are then introduced, and the construction and classification questions for cocyclic M-concordant systems of orthogonal designs are addressed. (M-concordance generalises the concepts of amicability and anti-amicability.) We give an algebraic procedure for constructing and classifying these designs when each indeterminate is constrained to appear just once in each row and column of the orthogonal designs. This paper also gives a general (but apparently not comprehensive) method for constructing systems with no zero entries. In particular, we obtain a cocyclic pair of amicable OD(16; 8, 8). Using this pair of designs, we prove there is a cocyclic Hadamard matrix of order 2ts for any odd integer s>1 and any t⩾⌊8log2s⌋. A consequence of our argument is the theorem, “Let S be any group of odd order s containing k prime order subgroups Si of S such that (1) for i=0, …, k−1, the sets Ui=Si+1Si+2…Sk are subgroups of S, (2) for i=0, …, k−1, Si∩Ui=〈1〉, and (3) U0=S. Let T be any group of order 2t+1 containing a central involution z such that (1) T/〈z〉 is elementary abelian, and (2) the largest elementary abelian direct factor of T has rank between 4log2s and t−2−4log2s. Then there is a normal relative difference set of size 2ts with forbidden subgroup 〈z〉 in the group T×S.” The condition on S holds for any group of odd square-free order or any direct product of odd order elementary abelian groups.

In this paper we study a certain type of so-called tight intercalation pairs of subsets of a finite (abelian) group. We show that in two important special cases the problem of finding such tight intercalation pairs can be reduced to the problem of finding a difference set, or of finding a so-called partial difference set in the given group.

We recursively construct a new family of ( 2^{6d+4}, 8, 2^{6d+4}, 2^{6d+1}) semi-regular relative difference sets in abelian groups G relative to an elementary abelian subgroup U. The initial case d=0 of the recursion comprises examples of ( 16,8,16,2) relative difference sets for four distinct pairs ( G,U).

A ( v, k, \lambda)-difference set D in a group G can be used to create a symmetric 2-( v, k, \lambda) design, \cal D, from which arises a code C, generated by vectors corresponding to the characteristic function of blocks of \cal D. This paper examines properties of the code C, and of a subcode, C^o=JC, where J is the radical of the group algebra of G over {\Bbb Z}_2. When G is a 2-group, it is shown that C^o is equivalent to the first-order Reed-Muller code, {\cal R}(1, 2s+2), precisely when the 2-divisor of C^o is maximal. In addition, ifD is a non-trivial difference set in an elementary abelian 2-group, and if D is generated by a quadratic bent function, then C^o is equal to a power of the radical. Finally, an example is given of a difference set whose characteristic function is not quadratic, although the 2-divisor of C^o is maximal.

We introduce a uniform technique for constructing a family of symmetric designs with parameters (v(q^{m+1}-1)/(q-1),kq^m,\lambda q^m), where m is any positive integer, (v,k,\lambda ) are parameters of an abelian difference set, and q=k^2/(k-\lambda ) is a prime power. We utilize the Davis and Jedwab approach to constructing difference sets to show that our construction works whenever (v,k,\lambda ) are parameters of a McFarland difference set or its complement, a Spence difference set or its complement, a Davis–Jedwab difference set or its complement, or a Hadamard difference set of order 9\cdot 4^d, thus obtaining seven infinite families of symmetric designs.

It is shown that there does not exist a reversible abelian Hadamard difference set in Z 2 2 x Z 9 2 or Z 4 2 x Z 3 2 . This settles the last two open cases for the existence of reversible (4N 2 ,2N 2 -N,N 2 -N)-abelian Hadamard difference sets with N

Nontrivial difference sets in groups of order a power of 2 are part of the family of difference sets called Hadamard difference sets. In the abelian case, a group of order 2 2 t + 2 has a difference set if and only if the exponent of the group is less than or equal to 2 t + 2 . In a previous work (R. A. Liebler and K. W. Smith, in “Coding Theory, Design Theory, Group Theory: Proc. of the Marshall Hall Conf.,” Wiley, New York, 1992), the authors constructed a difference set in a nonabelian group of order 64 and exponent 32. This paper generalizes that result to show that there is a difference set in a nonabelian group of order 2 4 t + 2 with exponent 2 3 t + 2 . Thus a nonabelian 2-group G with a Hadamard difference set can have exponent | G | 3/4 asymptotically. Previously the highest known exponent of a nonabelian 2-group with a Hadamard difference set was | G | 1/2 asymptotically. We use representation theory to prove that the group has a difference set.

We obtain some results that are useful to the study of abelian difference sets and relative difference sets in cases where the self-conjugacy assumption does not hold. As applications we investigate McFarland difference sets, which have parameters of the form v=q^d+1\left( q^d+ q^d-1 +\cdots + q+2\right) ,k=q^d\left( q^d+q^d-1+\cdots +q+1\right) , \lambda = q^d \left( q^{d-1}+q^{d-2}+\cdots +q+1\right) , where q is a prime power andd a positive integer. Using our results, we characterize those abelian groups that admit a McFarland difference set of order k-\lambda =81. We show that the Sylow 3-subgroup of the underlying abelian group must be elementary abelian. Our results fill two missing entries in Kopilovich‘s table with answer ’’no‘‘.

The main result of this paper is the proof of aconnection between abelian groups and difference sets.From this fact we can show that any difference set canbe organized to a difference poset as a class of equivalence. We give an example of adifference set as a conditional probability space in thesense of Kolmogoroff.

In this paper, we will give a construction of a family of(4q^{2n+2}\frac{q^{2n+2}-1}{q^2-1} ,q^{2n+1}[\frac{2(q^{2n+2}-1)}{q+1}+1] , ( q^{2n+2}-q^{2n+1})\frac{q^{2n+1}+1}{q+1})-difference sets in the group K\times G, where q is any power of2 , K is any group with|K|=\frac{q^{2n+2}-1}{q^2-1} and G is an abelian2 -group of order 4q^{2n+2} which contains an elementary abelian subgroup of index 2.

Let D be a (v,k,λ) difference set over an abelian group G with even n = k - λ. Assume that t ∈ N satisfies the congruences t ≡ q^fi_i (mod exp(G)) for each prime divisor q_i of n/2 and some integer f_i. In [4] it was shown that t is a multiplier of D provided that n > λ, (n/2, λ) = 1 and (n/2, v) = 1. In this paper we show that the condition n > λ may be removed. As a corollary we obtain that in the case of n= 2p^a when p is a prime, p should be a multiplier of D. This answers an open question mentioned in [2].

We show that under the self-conjugacy condition a McFarland difference set with p=2 and f≥ in an abelian group G can only exist,if the exponent of the Sylow 2-subgroups does not exceed 4.The method also works for odd p(where the exponent bound is p and is necessary and sufficient),so that we obtain a unified proof of the exponent bounds for MacFarland difference sets.We also correct a mistake in the proof of an exponent bound for (320,88,24)-difference sets in a previous paper. Accepted version