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We introduce and explore near-complete external difference families, a partitioning of the nonidentity elements of a group so that each nonidentity element is expressible as a difference of elements from distinct subsets a fixed number of times. We show that the existence of such an object implies the existence of a near-resolvable design. We provide examples and general constructions of these objects, some of which lead to new parameter families of near-resolvable designs on a non-prime-power number of points. Our constructions employ cyclotomy, partial difference sets, and Galois rings. Postprint

This paper concerns the problem of the existence of Hadamard difference sets in nonabelian groups of order 400. By introducing a new construction method, we construct new difference sets in 20 groups. We survey non-existence results, verifying non-existence in 45 groups.

Cyclic codes are a subclass of linear codes and are widely used in many applications as they have efficient encoding and decoding algorithms. In this paper, we investigate two classes of cyclic codes over źźq$\mathbb {F}_{q}$ with prime length using cyclotomy of order 4. Both the dimensions and the generator polynomials are explicitly determined. The method serves as a connection between cyclic codes and sequences over źźq$\mathbb {F}_{q}$ whose supports are the unions of certain cyclotomic classes of order 4. The main results thus partially answer two open problems in Ding (2015).

In cognitive radio networks, the process for any two unlicensed (secondary) users (SU) to establish links through a common channel is called a rendezvous. It is hard to employ a common control channel to solve the rendezvous problem, because of the bottleneck of the single control channel. Asynchronous channels hopping (ACH) systems have been proposed and investigated to guarantee rendezvous without requirement of global synchronization and common control channels. An ACH system with N channels can be mathematically interpreted as a set of sequences of the same period T on the alphabet $$\{0,1,\ldots , N-1\}$${0,1,ź,N-1} satisfying certain rotation closure properties. For each $$l\in \{0,1,\ldots , T-1 \}$$lź{0,1,ź,T-1}, each $$j\in \{0,1,\ldots , N-1 \}$$jź{0,1,ź,N-1} and any two distinct sequences $$\mathbf u$$u, $$\mathbf v$$v in an ACH system H, if there always exists i such that the i-th entries of $$\mathbf u$$u and $$L^{l}(\mathbf v)$$Ll(v) are both identical to j where $$L^{l}(\mathbf v)$$Ll(v) denotes the cyclic shift of $$\mathbf v$$v by l, then H is called a complete ACH system, which guarantees rendezvous between any two SUs who share at least one common channel. In this paper, we prove some properties of such systems. Moreover, we investigate the complete ACH systems, in each of which all SUs repeat the same (global) sequence associated with the system. One challenging research problem is to construct such ACH systems of period $$T=\beta N^2 + o(N^2)$$T=βN2+o(N2) such that $$\beta $$β is as small as possible. By applying mathematical tools such as group rings and (relative, relaxed) difference sets from combinatorial design theory, we obtain two constructions of them in which $$\beta =1,2$$β=1,2 respectively. Finally, we also present a simple and powerful approach to combining known ACH systems to produce new ones.

The area within a closed convex plane curve C may be estimated by enlarging C by a factor R , translating, counting the set J of integer points inside, and scaling back to the original size. This estimate is accurate when C is three times continuously differentiable in a certain sense. The set J is very sensitive to translations of the curve. We show that as R tends to infinity, the domains in which each set J occurs tend to uniform distribution modulo the integer lattice; this was only known for the special case of the circle.

In the early 1980′s Bloom and Hsu extended the notation of graceful labelings to directed graphs, and gave a relationship between graceful digraphs and a variety of algebraic structures. In this paper using a cyclic (v, k, λ) difference set with λ copies of elements of Zv\ {0}, we construct graceful digraphs of k vertices and v – 1 arcs. It is known that if gracefully labelled graph has e edges then its symmetric digraph is graceful with the same vertex labels. Although, the cycle Cm is not graceful for m≡1, 2 (mod 4) we show that the symmetric digraph based on cycle Cm i.e the double cycle, DCm which is constructed from a m-cycle by replacing each edge by a pair of arcs, edge xy gives rise to arcs (x, y) and (y, x), is graceful for any m vertices specifically for m≡1, 2 (mod 4).

In this article I survey some recent results on planar functions, almost planar functions and modified planar functions from the perspective of difference sets.

In this paper, a new algorithm to approximate real particles using multiple overlapping spheres as numerical models for the discrete element method is introduced. First, we convert the issue of approximating particles with a cluster of multiple overlapping spheres to a set-covering problem. Then, we use an algorithm to solve the set-covering problem in detail. This manuscript presents three different solution schemes based on a modified greedy heuristic algorithm, namely, a body-covering scheme, a surface-covering scheme and a triangular surface-covering scheme. To evaluate the algorithm, we calculated the amount of multiple overlapping spheres, the intersection error of volume or area, the difference set error of volume or area between multiple overlapping spheres and the real particles, and the error of the moment of inertia of the multiple overlapping spheres and the real particles. The parameters used to evaluate the precision of the three different schemes indicated that all three schemes are excellent. It is understood that different schemes offer different levels of precision for different particles with different numbers of multiple spheres. Therefore, it is important to choose the scheme best suited to represent a particular objective. Besides, the computational time is considered as the efficiency of the algorithm. In general, the body-covering scheme approximates complicated particles with the fewer spheres and better accuracy, while the surface-covering scheme realizes the representation with less time for a few particles. However, if a particle is generated by fewer than several thousand triangles, the triangular surface-covering scheme may finish the approximation in shorter time and with fewer multiple spheres.

Let [Formula: see text] be a simple undirected graph. [Formula: see text] is a circulant graph defined on [Formula: see text] with difference set [Formula: see text] provided two vertices [Formula: see text] and [Formula: see text] in [Formula: see text] are adjacent if and only if [Formula: see text]. For convenience, we use [Formula: see text] to denote such a circulant graph. A function [Formula: see text] is an integer [Formula: see text]-domination function if for each [Formula: see text], [Formula: see text] By considering all [Formula: see text]-domination functions [Formula: see text], the minimum value of [Formula: see text] is the [Formula: see text]-domination number of [Formula: see text], denoted by [Formula: see text]. In this paper, we prove that if [Formula: see text], [Formula: see text], then the integer [Formula: see text]-domination number of [Formula: see text] is [Formula: see text].

In next-generation wireless networks, high data rates, improvement of Spectral Efficiency (SE), meeting the Quality of Service (QoS) requirements, increasing Energy Efficiency (EE), reducing cost, and lower complexity of cellular networks systems have become essential design metrics. Use of resource allocation and interference mitigation techniques achieve these targets, especially for users located in the cell edge region. FFR schemes with relays are used to minimize the impact of ICI in Orthogonal Frequency Division Multiple Access (OFDMA) cellular networks and to enhance the QoS. The frequency patterns (sets) in the FFR scheme are designed such that the interference from adjacent cells is minimized. Instead of using the cell system with Frequency Reuse Factor (FRF) of 1 $\mathbf{TRF} \pmb{=1}$ ), $\mathbf{FRF} \pmb{=3}$ , and $\mathbf{FRF} \pmb{=(1,3)}$ , this paper proposes new frequency patterns for $\mathbf{FRF} \pmb{=(1,7/3)}$ using (7,3,1) difference set and $\mathbf{FRF} \pmb{=(1,7/4)}$ using (7,4,2) difference set to provide an enhancement of the system performance in terms of SE, EE, Signal-to-Interference plus Noise Ratio (SINR), Cumulative Distribution Function (CDF) of SINR of the received signal, and Outage Probability (OP). Simulation results show that the proposed schemes have more significant ICI reduction for cell edge users compared to those of previous work.

In this paper, we investigate the indistinguishability of generalized Bell states (GBSs) by one-way local operations and classical communication (LOCC). We first introduce two sets, i.e., the difference set and the testing set, and show that the GBSs cannot always be distinguished by one-way LOCC if the testing set is a subset of the difference set of GBSs. Then we give a unified upper bound about indistinguishable GBSs by one-way LOCC. The newly derived bound is smaller than those presented in Zhang et al. [Phys. Rev. A 91, 012329 (2015)] and Wang et al. [Quantum Inf. Process. 15, 1661 (2016).]. More precisely, the minimum number of one-way locally indistinguishable GBSs is not more than $2\ensuremath{\lceil}\sqrt{d}\ensuremath{\rceil}+\ensuremath{\lceil}\frac{\sqrt{d}}{4}\ensuremath{\rceil}+1$.

Let $G$ be a finite group. The set $D\subseteq G$with $|D|=k$ is called a $(n,k,\lambda,\mu)$-partial difference set(PDS) in $G$ if the differences $d_1d_2 ^{-1}, d_2,d_2\in D, d_1\neq d_2$, represent each non-identity element in $D$ exactly $\lambda$ times and each non-identity element in $G-\{D\}$ exactly $\mu$ times.In the present paper, we determine for which group $G\in \{D_{2n},T_{4n},U_{6n},V_{8n}\}$ the derangement set is a PDS. We also prove that the derangement set of a Frobenius group is a PDS.

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 work we present a new construction of quantum error correcting codes. It is based on the stabilizer formalism but does not use the CSS construction. It takes advantage of the properties of the difference set to construct the code. The technique may also be used to generate the quantum LDPC codes as well as the cyclic codes.

In this paper, we deal with the construction of disjoint difference set as combinatorial optimisation problem, which allows us to construct Quasi-cyclic majority logic decoding (QC MLD) codes. We will propose an algorithm based on genetic algorithm to construct many of these designs. Our algorithm was able to construct many new Quasi-cyclic MLD codes based on the constructed Disjoint Difference Set.

When constructing practical zero-knowledge proofs based on the hardness of the Ring-LWE or the Ring-SIS problems over polynomial rings \(\mathbb {Z}_p[X]/(X^n+1)\), it is often necessary that the challenges come from a set \(\mathcal {C}\) that satisfies three properties: the set should be large (around \(2^{256}\)), the elements in it should have small norms, and all the non-zero elements in the difference set \(\mathcal {C}-\mathcal {C}\) should be invertible. The first two properties are straightforward to satisfy, while the third one requires us to make efficiency compromises. We can either work over rings where the polynomial \(X^n+1\) only splits into two irreducible factors modulo p, which makes the speed of the multiplication operation in the ring sub-optimal; or we can limit our challenge set to polynomials of smaller degree, which requires them to have (much) larger norms.

According to the newly proposed nested MIMO (Multiple-Input Multiple-Input Multiple Output Multiple Array) array design method, we propose to replace the traditional nested array into an optimizing nested array, ie, to optimizing nested MIMO array design. It not only retains the original advantage of nested MIMO array design closed expression with array element position and degree of freedom(DOF), but also greatly improves the array aperture and DOF. Optimizing nested MIMO array firstly uses the optimizing nested array as the transmitting and receiving arrays, and then make the difference set processing for the coarray of MIMO array (coarray, CA). By properly designing the array spacing of the transmitting and receiving arrays, we can obtain a non-porous difference array. When the total number of array elements is given, by analyzing the characteristics of the array structure, the best array element number of the transmitting and receiving arrays can be obtained. Simulation experiments show that compared with the nested MIMO array design, the proposed method can effectively expand the array aperture, increase the DOF, and increase the DOA estimation accuracy of the MIMO radar without increasing the number of actual array elements.

A difference set tiling in a group is a collection of its Q2 ( , , ) difference sets that partition ⧵ {; ; 1}; ; . It can exist in an abelian as well as in a nonabelian group. A tiling is normalized if a product of elements in each difference set equals 1. All known cases in abelian groups are normalized. Ćustić, Krčadinac, and Zhou made a conjecture that this is necessary. We will call it a normalized tiling conjecture (NTC). Using character theory, we prove that NTC is true for ( , , 1) where is odd. Also, if ( , , ) difference set has a multiplier, we prove that NTC is also true.

There has been recent interest on designing double error correction (DEC) codes for 32-bit data words that support fast decoding as they can be useful to protect memories. To that end, solutions based on orthogonal Latin square codes have been recently presented that achieve fast decoding but require a large number of parity check bits. In this letter, a DEC code derived from difference set codes is presented. The proposed code is able to reduce the number of parity check bits needed at the cost of a slightly more complex decoding. Therefore, it provides memory designers with an additional option that can be useful when making trade-offs between memory size and speed.

A celebrated theorem of Kleitman in extremal combinatorics states that a collection of binary vectors in { 0 , 1 } n with diameter d has cardinality at most that of a Hamming ball of radius d / 2 . In this paper, we give an algebraic proof of Kleitman's Theorem, by carefully choosing a pseudo-adjacency matrix for certain Hamming graphs, and applying the Cvetkovic bound on independence numbers. This method also allows us to prove several extensions and generalizations of Kleitman's Theorem to other allowed distance sets, in particular blocks of consecutive integers of width much smaller than n. We also improve on a theorem of Alon about subsets of F p n whose difference set does not intersect { 0 , 1 } n nontrivially.

The problems studied in this paper resemble in spirit partially the following two theorems due to Stanton and Sprott (Can J Math 10:73---77, 1958) and Whiteman (Ill J Math 6:107---121, 1962) and are based on ideas given in Storer (Cyclotomy and difference sets. Lectures in advanced mathematics, 1967). Theorem 1 (Stanton, Sprott) Let g be a primitive root of both$$p$$pand$$p -2$$p-2, where$$p$$pand$$p-2$$p-2are a pair of twin primes. Then the numbers$$\begin{aligned} 1, g, g^2, \ldots g^{(p^2 -3)/2}\, together\, with \,0, p+2, , 2(p +2), \ldots , (p- 1)(p +2) \end{aligned}$$1,g,g2,źg(p2-3)/2togetherwith0,p+2,,2(p+2),ź,(p-1)(p+2)form a difference set modulo$$p(p+2)$$p(p+2)with parameters$$ v=p(p + 2), k= (v - 1)/2, \lambda =(v- 3)/4$$v=p(p+2),k=(v-1)/2,ź=(v-3)/4 Theorem 2 (Whiteman) Let$$p$$pand$$q$$qbe two primes such hat$$(p- 1, q- 1)= 4$$(p-1,q-1)=4, and let$$ d = (p- 1)(q -1)/4$$d=(p-1)(q-1)/4. Then there are exactly two subgroups$$U_1,U_2$$U1,U2generated by a common primitive root of$$p$$pand$$q$$q, and one (but not both) of he sets$$\begin{aligned}&U_1\, together\, with \,0, q, 2q,\ldots , (p - 1)q\\&U_2\, together\, with \,0, q, 2q,\ldots , (p- 1)q \end{aligned}$$U1togetherwith0,q,2q,ź,(p-1)qU2togetherwith0,q,2q,ź,(p-1)qis a difference set modulo$$p q $$pqwith parameters$$v = pq, k= (v- 1)/4, \lambda = (v -5)/16 $$v=pq,k=(v-1)/4,ź=(v-5)/16if and only if$$ q = 3p +2$$q=3p+2and$$(v- 1)/4$$(v-1)/4is an odd square. However, we do not follow the classical method of cyclotomic numbers to generalize these results but rather present in a modern setting the connection between difference sets or partial difference sets and Jacobi sums. With the help of Jacobi sums we are able to rephrase the existence of difference sets, partial difference sets and so called $$G$$G-sets in Galois domains into a fundamental equation on Jacobi sums presented in Theorem 14. As an application we determine all cyclotomic difference and partial difference sets of small order or which are semiprimitive modulo its order by restricting the theorem to a product of two finite fields.

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.

Let $$({\mathbb {P}},{\mathbb {B}})$$(P,B) be an $$(m,u,k,\lambda )$$(m,u,k,?)-divisible design and let $$G$$G be a subgroup of $$\mathrm{Aut}({\mathbb {P}},{\mathbb {B}})$$Aut(P,B). We say $$G$$G is an SCT$$(m,u,k,\lambda )$$(m,u,k,?) automorphism group of $$({\mathbb {P}},{\mathbb {B}})$$(P,B) if $$G$$G is semiregular on $${\mathbb {P}}\cup {\mathbb {B}}$$P?B and regular on the set of point classes of $$({\mathbb {P}},{\mathbb {B}})$$(P,B). In this paper we show that each $$\hbox {SCT}(m,u,k,\lambda )$$SCT(m,u,k,?) automorphism group corresponds to a certain special kind of matrix, which we call an $$\hbox {SCT}(m,u,k,\lambda )$$SCT(m,u,k,?) matrix over $$G$$G. Using such matrices we construct $$(m,u,k,\lambda )$$(m,u,k,?)-divisible designs. We also consider the close connection between SCT automorphism groups and relative difference sets. As an application we generalize the notion of planar functions and show that an arbitrary $$p$$p-group may be the forbidden subgroup of a semiregular relative difference set.

Complex periodical sequences with lower autocorrelation values are used in CDMA communication systems and cryptography. In this paper we present new nonexistence results on perfect p-ary sequences and almost p-ary sequences and related difference sets by using some knowledge on cyclotomic fields and their subfields.

In constructing the cognitive radio networks, a fundamental process is to establish a communication link on a same channel for every two users, which is referred to as rendezvous. Most studies focus on minimizing the time to rendezvous after they start the process synchronously or asynchronously. However, to the best of our knowledge, no work has considered the possibility that the users may achieve different utilities when they rendezvous on different channels for communication. The utility originates from the quality of the specific channel that two users rendezvous on, and it is influenced by the channel's bandwidth, transmission rate, stability, etc. In this paper, we formally formulate the problem of maximizing rendezvous utilities and propose a novel method by extending the channels to promote the users to achieve higher rendezvous utilities. We propose channel extension algorithms for both symmetric and asymmetric rendezvous scenarios, where the users may have the same or different sets of rendezvous channels respectively. These algorithms are built on the construction of Disjoint Relaxed Difference Set (DRDS) in [5], and we show the efficiency of achieving rendezvous on the channel with high utility theoretically. Moreover, we conduct thorough simulations to evaluate our algorithms and the results also corroborate our theoretical analyses.

An asymmetric multiple access channel (AMAC) is a multiple-access channel where a portion of the users can observe the messages of other users. We first propose three 2-user uniquely decodable (UD) codes for two-user AMAC, which are shown to achieve higher sum-rate than the previous 2-user codes. Then, we consider multiuser error correcting codes for T-user noisy AMAC. A theorem shows that given a T a -user δ a -decodable k-ary code A and a T d -subset δa-decodable difference set D a priori, a larger error-correcting T-user code C is obtained by Hadamard matrices. For practical construction, we give 2-user difference sets based on the 2-user UD codes, and obtain multiuser correcting codes for multiuser AMAC. The proposed correcting codes have increasing sum-rate and error-correcting capability with an increasing code length.

In this paper, we give new constructions of disjoint difference families from Galois rings. The constructions are based on choosing cosets of the unit group of a subring in the Galois ring $\mathrm{GR}(p^2,p^{2s})$. Two infinite families of disjoint difference families are obtained from the Galois rings $\mathrm{GR}(p^{2},p^{4n})$ and $\mathrm{GR}(2^2,2^{2s})$.

Co-prime arrays have gained in popularity as an efficient way to estimate second order statistics at the Nyquist rate from sub-Nyquist samples without any sparsity constraint. We derive an expression for the degrees of freedom and the number of consecutive values in the difference set for the prototype co-prime array. This work shows that, under the wide sense stationarity (WSS) condition, larger consecutive difference values can be achieved by using the union of all the difference sets. We provide a closed-form expression in order to determine the number of sample pairs that are available for estimating the statistics for each value of the difference set, also known as the weight function. The estimation accuracy and latency depends on the number of sample pairs used for estimating the second order statistic. We also obtain the closed-form expression for the bias of the correlogram spectral estimate. Simulation results show that the co-prime based periodogram and biased correlogram estimate are equivalent, and the reconstruction using our proposed formulation provides lower latency.

Co-prime arrays and samplers are popular sub-Nyquist schemes for estimating second order statistics at the Nyquist rate. This paper focuses on the perturbations in the array locations or sampling times, and analyzes its effect on the difference set. Based on this analysis we propose a method to estimate the autocorrelation which makes best use of the sampled data in order to improve the estimation accuracy of the autocorrelation and hence the spectral estimate. Our analysis indicates that such an advantage is limited only to samplers, and does not carry over to the antenna arrays. In addition, we obtain expressions for the computational complexity of the autocorrelation estimation and provide an upper bound on the number of multiplications and additions required for its hardware implementation.

If D is a $\phantom {\dot {i}\!}(4u^{2},2u^{2}-u,u^{2}-u)$ Hadamard difference set (HDS) in G, then $\phantom {\dot {i}\!}\{G,G\setminus D\}$ is clearly a $\phantom {\dot {i}\!}(4u^{2},[2u^{2}-u,2u^{2}+u],2u^{2})$ partitioned difference family (PDF). Any $\phantom {\dot {i}\!}(v,K,\lambda )$ -PDF will be said a Hadamard PDF if $\phantom {\dot {i}\!}v=2\lambda $ as the one above. We present a doubling construction which, starting from any Hadamard PDF, leads to an infinite class of PDFs. As a special consequence, we get a PDF in a group of order $\phantom {\dot {i}\!}4u^{2}(2n+1)$ and three block-sizes $\phantom {\dot {i}\!}4u^{2}-2u$ , $\phantom {\dot {i}\!}4u^{2}$ and $\phantom {\dot {i}\!}4u^{2}+2u$ , whenever we have a $\phantom {\dot {i}\!}(4u^{2},2u^{2}-u,u^{2}-u)$ -HDS and the maximal prime power divisors of $\phantom {\dot {i}\!}2n+1$ are all greater than $\phantom {\dot {i}\!}4u^{2}+2u$ .

A solution to the existence problem of $G$-designs with given subdesigns is known when ?$G$? is a triangle with ?$p = 0, 1$?, or ?$2$? disjoint pendent edges: for ?$p = 0$?, it is due to J. Doyen and R. M. Wilson Discrete Math. 5, 229-239 (1973), the first to pose such a problem for Steiner triple systems; for ?$p = 1$? and ?$p = 2$?, the corresponding designs are kite systems and bull designs, respectively. Here, a complete solution to the problem is given in the remaining case where ?$G$? is a 3-sun, i.e. a graph on six vertices consisting of a triangle with three pendent edges which form a 1-factor. Rešitev eksistenčnega problema ?$G$?-načrtov z danimi podnačrti je znana za primer, ko je ?$G$? trikotnik s ?$p = 0, 1$?, ali ?$2$? disjunktnimi nanj obešenimi povezavami: za ?$p = 0$? sta rešitev podala Doyen in Wilson, ki sta prva zastavila takšen problem za Steinerjeve sisteme trojic; za ?$p = 1$? so ustrezni načrti sistemi zmajev, za ?$p = 2$? pa sistemi bikov. Tukaj je dana popolna rešitev problema v preostalem primeru, ko je ?$G$? 3-sonce, tj. graf na šestih vozliščih, sestoječ iz trikotnika s tremi nanj obešenimi povezavami, ki tvorijo 1-faktor.

Let $F(x)=(f_1(x), \dots, f_m(x))$ be such that $1, f_1, \dots, f_m$ are linearly independent polynomials with real coefficients. Based on ideas of Bachoc, DeCorte, Oliveira and Vallentin in combination with estimating certain oscillatory integrals with polynomial phase we will show that the independence ratio of the Cayley graph of $\mathbb{R}^m$ with respect to the portion of the graph of $F$ defined by $a\leq \log |s| \leq T$ is at most $O(1/(T-a))$. We conclude that if $I \subseteq \mathbb{R}^m$ has positive upper density, then the difference set $I-I$ contains vectors of the form $F(s)$ for an unbounded set of values $s \in \mathbb{R}$. It follows that the Borel chromatic number of the Cayley graph of $\mathbb{R}^m$ with respect to the set $\{ \pm F(s): s \in \mathbb{R} \}$ is infinite. Analogous results are also proven when $\mathbb{R}$ is replaced by the field of $p$-adic numbers $\mathbb{Q}_p$. At the end, we will also the existence of real analytic functions $f_1, \dots, f_m$, for which the analogous statements no longer hold., Minor corrections and modifications

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.