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A novel method is proposed to construct arbitrary length perfect integer sequences based on all-pass polynomial. It uses a fact that associative sequence of an all-pass polynomial is guaranteed to be perfect. In addition, geometric series method is our special case. Moreover, perfect Gaussian integer sequences can also be constructed. Furthermore, this approach can be applied to symmetric perfect sequences design. Finally, perfect sequence with low degree of freedom can be achieved by the difference set. Concrete examples are illustrated.

Since Horadam and de Launey introduced the cocyclic framework on combinatorial designs in the 1990s, it has revealed itself as a powerful technique for looking for (cocyclic) Hadamard matrices. Ten years later, the series of papers by Kotsireas, Koukouvinos and Seberry about Hadamard matrices with one or two circulant cores introduced a different structured approach to the Hadamard conjecture. This paper is built on both strengths, so that Hadamard matrices with cocyclic cores are introduced and studied. They are proved to strictly include usual Hadamard matrices with one and two circulant cores, and therefore provide a wiser uniform approach to a structured Hadamard conjecture. Junta de Andalucía FQM-016

We provide some applications of a polynomial criterion for difference sets. These include counting the difference sets with specified parameters in terms of Hilbert functions, in particular a count of bent functions. We also consider the question about the bentness of certain Boolean functions introduced by Carlet when the $\mathcal{C}$-condition introduced by him doesn't hold., Comment: 15 pages

The concept of a partial geometric difference set (or 1 1 2 -difference set) was introduced by Olmez in 2014. Recently, Nowak, Olmez and Song introduced the notion of a partial geometric difference family, which generalizes both the classical difference family and the partial geometric difference set. It was shown that partial geometric difference sets and partial difference families give rise to partial geometric designs. In this paper, a number of new infinite families of partial geometric difference sets and partial geometric difference families are constructed. From these partial geometric difference sets and difference families, we generate a list of infinite families of partial geometric designs.

In quantum information theory, symmetric informationally complete positive operator-valued measures (SIC-POVMs) are related to quantum state tomography (Caves et al., 2004), quantum cryptography (Fuchs and Sasaki, 2003) [ 1 ], and foundational studies (Fuchs, 2002) [ 2 ]. However, constructing SIC-POVMs is notoriously hard. Although some SIC-POVMs have been constructed numerically, there does not exist an infinite class of them. In this paper, we propose two constructions of approximately SIC-POVMs, where a small deviation from uniformity of the inner products is allowed. We employ difference sets to present the first construction and the dimension of the approximately SIC-POVMs is q + 1 , where q is a prime power. Notably, the dimension of this framework is new. The second construction is based on partial geometric difference sets and works whenever the dimension of the framework is a prime power.

Let N be the set of positive integers. For a nonempty set A of integers and every integer u, denote by d A ( u ) the number of ( a , a ′ ) with a , a ′ ∈ A such that u = a − a ′ . For a sequence S of positive integers, let S ( x ) be the counting function of S. The set A ⊆ N is called a perfect difference set if d A ( u ) = 1 for every positive integer u. In 2008, Cilleruelo and Nathanson (2008) [4] constructed dense perfect difference sets from dense Sidon sets. In this paper, as a main result, we prove that: let f : N → N be an increasing function satisfying f ( n ) ≥ 2 for any positive integer n, then for every Sidon set B and every function ω ( x ) → ∞ , there exists a set A ⊆ N such that d A ( u ) = f ( u ) for every positive integer u and B ( x / 3 ) − ω ( x ) ≤ A ( x ) ≤ B ( x / 3 ) + ω ( x ) for all x ≥ C f , B , ω .

In this paper, we construct a novel coarray named as the difference and sum (diff–sum) coarray by exploiting an improved Conjugate Augmented MUSIC (CAM) estimator, which utilizes both the temporal information and the spatial information. The diff–sum coarray is the union of the difference coarray and the sum coarray. When taking the coprime array as the array model, we find that the elements of the sum coarray can fill up all the holes in the difference coarray. Besides, the sum coarray contains bonus uniform linear array (ULA) segments which extend the consecutive range of the difference coarray. As a result, the consecutive lags of the diff–sum coarray are much more than those of the difference coarray. For analysis, we derive the hole locations and consecutive ranges of the difference set and the sum set, discuss the complementarity of the two sets, and provide the analytical expression of the diff–sum virtual aperture. Simulations verify the effectivity of the improved method and show the high DOF of the diff–sum coarray.

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.

We present a new construction of almost difference sets. The construction occurs in nonabelian groups of order 4N with a subgroup H of order N so that H has an (N,\(\frac{N-1}{2}\),\(\frac{N-3}{4}\)) difference set (and hence N must be an integer that is 3 (mod 4)).

By further exploring some ideas in Buratti (1999), Chang and Ding (2006) several new infinite classes of difference families from finite fields are obtained and some results of Chang and Ding (2006) are extended.

In this paper we extend previous results regarding SL2(k) over any finite field k by investigating the structure of the symmetric spaces for the family of special linear groups SLn(k) for any integer n>2. Specifically, we discuss the generalized and extended symmetric spaces of SLn(k) for all conjugacy classes of involutions over a finite field of odd or even characteristic. We characterize the structure of these spaces and provide an explicit difference set in cases where the two spaces are not equal.

A T -user UD k-ary codes for MAAC is proposed. First, a coding scheme for a Tf+g-user UD k-ary code with code length f + g is proposed that is constructed from a Tf-user UD k-ary code and a Tg-user UD (2k − 1)-ary difference set. In fact, the Tg-user UD (2k − 1)-ary difference set is associated with Tg-user UD (2k − 1)-ary code. Second, originally from the multi-user UD (2i(k − 1) + 1)-ary (i = 0, 1, 2,…, m) codes with unitary code length, by recursively employing the coding scheme, 2m+1-user k-ary code with code length 2m is obtained. Finally, by recursively employing the coding scheme, 2n-user UD k-ary code with arbitrary code length n from the codes with length 2i (i = 0, 1,…, ⌊log 2 n⌋) is given. Since introducing the high-order multiary difference sets, the total rates of the proposed codes are higher those of conventional codes.

It is shown that, if D is a nontrivial 2-(v,k,λ) symmetric design, with (k,λ)=1, admitting a flag-transitive automorphism group G of affine type, then v=pd, p an odd prime, and G is a point-primitive, block-primitive subgroup of AΓL1(pd). Moreover, O(G) acts flag-transitively, point-primitively on D, and D is isomorphic to the development of a difference set whose parameters and structure are also provided.

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).

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, 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 $2\ensuremath{\lceil}\sqrt{d}\ensuremath{\rceil}+\ensuremath{\lceil}\frac{\sqrt{d}}{4}\ensuremath{\rceil}+1$.

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 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 $\mathrm{GR}(2^2,2^{2s})$.

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.

This paper addresses the question 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 also give an alternative and more direct version of the construction, leaving out the intermediate step via Youden squares. This is used when we analyse 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. Construction methods for triple arrays

We present a novel transform for periodic complementary sets (PCSs) over two-valued alphabets from a large set of difference families. This is achieved by generalizing Golomb's idea in 1992, which was for transformed perfect sequences with zero autocorrelations only. Based on the properties of difference family, a sufficient condition for such two-valued PCSs is derived. Systematic constructions of two-valued periodic complementary pairs are presented. It is shown that many lengths for which binary PCSs do not exist become admissible for our proposed two-valued PCSs. NRF (Natl Research Foundation, S’pore) Accepted version

We show that the assumption n"1>@l in the Second Multiplier Theorem can be replaced by a divisibility condition weaker than the condition in McFarland's multiplier theorem, thus obtaining significant progress towards the multiplier conjecture.

Difference systems of sets (DSS) are combinatorial configurations that arise in connection with code synchronization. This paper proposes a new method to construct DSSs, which uses known DSSs to partition some of the cosets of Zν relative to subgroup of order k, where ν = km is a composite number. As applications, we obtain some new optimal DSSs.

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.

In this letter, we present a new proof of the ideal two-level correlation binary sequence pairs introduced by Peng et al. We also provide a new construction of ideal two-level correlation binary sequence pair. As a co-product, new classes of difference set pairs can be derived.

Almost difference sets are an interesting subject of combinatorics, and have applications in many areas of engineering such as CDMA communications, error correcting codes and cryptography. The objective of this paper is to present some new constructions of almost difference sets, together with several results on the equivalence relation.

Analog coding decouples the tasks of protecting against erasures and noise. For erasure correction, it creates an "analog redundancy" by means of band-limited discrete Fourier transform (DFT) interpolation, or more generally, by an over-complete expansion based on a frame. We examine the analog coding paradigm for the dual setup of a source with "erasure" side-information (SI) at the encoder. The excess rate of analog coding above the rate-distortion function (RDF) is associated with the energy of the inverse of submatrices of the frame, where each submatrix corresponds to a possible erasure pattern. We give a partial theoretical as well as numerical evidence that a variety of structured frames, in particular DFT frames with difference-set spectrum and more general equiangular tight frames (ETFs), with a common MANOVA limiting spectrum, minimize the excess rate over all possible frames. However, they do not achieve the RDF even in the limit as the dimension goes to infinity.

A pair of binary sequences is generalized from the concept of a two-level autocorrelation function of a single binary sequence. In this paper, new families of binary sequence pairs with period N = np, where gcd(n,p) = 1, and optimal correlation values -1 are constructed, as well as new families of binary sequence pairs with three-level correlation and period N≡0(mod 4). For the new binary sequence pairs with optimal correlation values, their corresponding new difference set pairs and almost difference set pairs are also derived.

Let n > m ⩾ 2 be positive integers and n = (m + 1)l + r, where ⩽ r ⩽ m. Let C be a subset of {0, 1, …, n}. We prove that if $$|C| > \left\{ {\matrix{{\left\lfloor {n/2} \right\rfloor + 1{\rm{ if }}m{\rm{ is old,}}} \cr {ml/2 + \delta {\rm{ if }}m{\rm{ is even,}}} \cr } } \right.$$ , where └x┘ denotes the largest integer less than or equal to x and δ denotes the cardinality of even numbers in the interval [0, min{r,m − 2}], then C − C contains a power of m. We also show that these lower bounds are best possible.

In this article we consider -difference set in . Several classes of such difference sets are known. We classify these classes into two typical types and characterize them.

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.

In this article, we propose two successive search methods for solving a canonical DC programming problem constrained by the difference set between two compact convex sets in the case where the dimension number is greater than or equal to three. In order to find feasible solutions, the algorithms generate the directions based on a branch and bound procedure, successively. By exploring the provisional solutions throughout the intersection of the boundaries of two compact convex sets, both algorithms calculate an approximate solution.

Difference systems of sets (DSSs) are combinatorial structures introduced by Levenshtein in connection with code synchronization. In this paper, some recursive constructions of DSSs obtained from finite projective geometry are presented. As a consequence, new infinite families of optimal DSSs are obtained.

We construct an infinite family of (2 ns , 2 ns/2 -1(2 ns/2?1), 2 ns/2 -1(2 ns/2 -1 ?1)) difference sets over a Galois ring GR(2 n , s) with characteristic an even power n of 2 and an odd extension degree s. It makes a chain of difference sets preserving the structures when n increases and s is fixed. We introduce a new operation into GR(2 n , s). The Gauss sum associated with the multiplicative character defined by the subgroup with respect to the new operation plays an important role in the construction.

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.

Let K ⊆ R 2 be a compact set such that K + Z 2 = R 2 and let K − K = { a − b | a , b ∈ K } . We prove, via Algebraic Topology, that the integer points of the difference set of K, ( K − K ) ∩ Z 2 , is not contained on the coordinate axes, Z × { 0 } ∪ { 0 } × Z . This result implies the negative answer to the inverse problem posed by M.B. Nathanson (2010) [5] .

Let p = ef + 1 be an odd prime for some e and e and let f, be the finite field with Fp elements. In this paper, we explicitly describe the trace representations of the binary characteristic sequences (of period p) of all the cyclic difference sets D which are some union of cosets of eth powers He in Fp* (=Δ Fp\{0}) for e ≤ 12. For this, we define eth power residue sequences of period p, which include all the binary characteristic sequences mentioned above as special cases, and reduce the problem of determining their trace representations to that of determining the values of the generating polynomials of cosets of He in Fρ* at some primitive pth root of unity, and some properties of these values are investigated. Based on these properties, the trace representation and linear complexity not only of the characteristic sequences of all the known eth residue difference sets, but of all the sixth power residue sequences are determined. Furthermore, we have determined the linear complexity of a nonconstant eth power residue sequence for any e to be either p - 1 or p whenever (e, (p-1)/n) = 1, where n is the order of 2 mod p.

Binary sequences with optimal autocorrelation are needed in many applications. Two constructions of binary sequences with optimal autocorrelation of period N ≡ 0 (mod 4) are investigated. The two constructions are powerful and generic in the sense that many classes of binary sequences with optimal autocorrelation could be obtained from binary sequences with ideal autocorrelation. General results on the minimal polynomials of these binary sequences are derived. Based on the results, both the linear complexities and the minimal polynomials are determined.

A partial difference set with parameters (v,v-12,v-54,v-14) is said to be of Paley type. In this paper, we give a recursive theorem that for all odd n>1 constructs Paley partial difference sets in certain groups of order n^4 and 9n^4. We are also able to construct Paley-Hadamard difference sets of the Stanton-Sprott family in groups of order n^4(n^4+/-2) when n^4+/-2 is a prime power and 9n^4(9n^4+/-2) when 9n^4+/-2 is a prime power. Many of these are new parameters for such difference sets, and also give new Hadamard designs and matrices.

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

Difference systems of sets (DSSs) are combinatorial configurations which were introduced in 1971 by Levenstein for the construction of codes for synchronization. In this paper, we present two kinds of constructions of difference systems of sets by using disjoint difference families and a special type of difference sets, respectively. As a consequence, new infinite classes of optimal DSSs are obtained.

Doubly Regular Asymmetric Digraphs (DRAD) with rank 4 automorphism groups were previously thought to be rare. We exhibit difference sets in Galois Rings that can be used to construct an infinite family of DRADs with rank 4 automorphism groups. In addition, we construct difference sets in groups Z2r2 for all r⩾2 that can be used to construct DRADs and nonsymmetric 3-class imprimitive association schemes. Finally, we prove a new product construction for difference sets so that the resulting difference sets can be used to build nonsymmetric 3-class imprimitive association schemes.