The existence of almost difference families

Almost difference families (ADFs) were introduced by Ding and Yin as a useful generalization of almost difference sets (ADSs), and a number of infinite classes of almost difference families had been constructed. Suppose q is a prime power. To construct combinatorial designs in GF(q), one often needs to find an element xGF(q){0}, such that some polynomials in GF(q)[x] of degree one or two satisfying certain conditions. Weil's theorem on character sum estimates is very useful to do this. In this paper, a general bound for finding such x is given. By using this bound and computer searching, some known results on almost difference families by Ding and Yin are improved.     

On Whiteman's and Storer's difference sets

In this paper, we prove that all Storer's difference sets are cyclic. We also prove that for p<1.8×10²⁵, Whiteman's difference sets exist if and only if (p,q)=(17,53) and (46817,140453).     

Optimal designs,supplementary difference sets and multipliers

We investigate multipliers of 2 - {v; q², q²; λ} supplementary difference sets where cyclotomy has been used to construct D-optimal designs.     

Diffuse difference triangle sets

A particular kind of difference triangle sets (DTSs) called diffuse DTS (DDTS) are considered. Their combinatorial structure is underlying the construction of all known types of self-orthogonal diffuse codes. A number of constructions of DDTS are described, and lower and upper bounds on the maximal element of an optimal DDTS are given. The asymptotic behaviour of the maximal element is studied. Finally, tables of DDTS and of their minimal possible maximal elements are presented.     

A general construction of E(fNOD)-optimal multi-level supersaturated designs

The present paper deals with E(f_NOD)-optimal multi-level supersaturated designs. We present a new technique for the construction of supplementary difference sets. Based on the new supplementary difference sets, we also provide E(f_NOD)-optimal multi-level supersaturated designs with a large number of columns when compared with other designs. Moreover, these designs retain the equal occurrence property.     

Classification of difference matrices over cyclic groups

The existence of difference matrices over small cyclic groups is investigated in this computer-aided work. The maximum values of the parameters for which difference matrices exist as well as the number of inequivalent difference matrices in each case is determined up to the computational limit. Several new difference matrices have been found in this manner. The maximum number of rows is 9 for an r ×15 difference matrix over Z₃, 8 for an r ×15 difference matrix over Z₅, and 6 for an r ×12 difference matrix over Z₆; the number of inequivalent matrices with these parameters is 5, 2, and 7, respectively.     

On disjoint (3t, 3, 1) cyclic difference families

We give a new and easier proof of the existence of a disjoint (3t  , 3, 1) cyclic difference family for every t>3$t>3$, first proved by Dinitz and Shalaby (2002). Our purely theoretical construction is still elementary but simpler and does not need to be checked by computer.     

Weight hierarchies of linear codes of dimension 3

The weight hierarchy of a linear [n,k;q] code C over GF(q) is the sequence (d₁,d₂,…,d_k), where d_r is the smallest support of an r-dimensional subcode of C. The weight hierarchies of [n,3;q] codes are studied. In particular, for q⩽5 the possible weight hierarchies of [n,3;q] codes are determined.     

Minimal 2-coverings of a finite affine space based on GF(2)

Let EG(m, 2) denote the m-dimensional finite Euclidean space (or geometry) based on GF(2), the finite field with elements 0 and 1. Let T be a set of points in this space, then T is said to form a q-covering (where q is an integer satisfying 1?q?m) of EG(m, 2) if and only if T has a nonempty intersection with every (m-q)-flat of EG(m, 2). This problem first arose in the statistical context of factorial search designs where it is known to have very important and wide ranging applications. Evidently, it is also useful to study this from the purely combinatorial point of view. In this paper, certain fundamental studies have been made for the case when q=2. Let N denote the size of the set T. Given N, we study the maximal value of m.     

Concerning the vector v(m,t)

Let q = mt + 1 be a prime power, and let v(m, t) be the (m + 1)-vector (b₁, b₂, …, b_{m + 1}) of elements of GF(q) such that for each k, 1 ⩽ k ⩽ m + 1, the set {b_i−b_j:i∈{1,2,…m+1} − {m + 2 − k}, j  i + k(mod m + 2) and 1⩽j⩽m+1} forms a system of representatives for the cyclotomic classes of index m in GF(q). In this paper, we investigate the existence of such vectors. An upper bound on t for the existence of a v(m, t) is given for each fixed m unless both m and t are even, in which case there is no such a vector. Some special cases are also considered.     

The number of solutions to the alternate matrix equation over a finite field and a q-identity

Let F_q be a finite field with q elements, where q is a power of a prime. In this paper, we first correct a counting error for the formula N(K_2ν,0^(m)) occurring in Carlitz (1954. Arch. Math. V, 19–31). Next, using the geometry of symplectic group over F_q, we have given the numbers of solutions X of rank k and solutions X to equation XAX′=B over F_q, where A and B are alternate matrices of order n, rank 2ν and order m, rank 2s, respectively. Finally, an elementary q-identity is obtained from N(K_2ν,0⁽⁰⁾), and the explicit results for N(K_n,2ν,K_m,2s) is represented by terminating q-hypergeometric series.     

Some necessary conditions on the parameters of partial difference sets

In this paper, we study some necessary conditions on the parameters of nontrivial regular (υ, κ, λ, μ)-partial difference sets in abelian groups. In particular, we settle some undecided cases in Ma's table [Designs, Codes Cryptography, 4 (1994)]. Also, the case when λ ⩽ 1 is studied. Nonexistence results are obtained when λ = 0 and a complete characterization is given when λ = 1. Finally, parameters of partial difference sets with an odd μ are determined.     

Construction of orthogonal Latin hypercube designs with flexible run sizes

Latin hypercube designs (LHDs) have recently found wide applications in computer experiments. A number of methods have been proposed to construct LHDs with orthogonality among main-effects. When second-order effects are present, it is desirable that an orthogonal LHD satisfies the property that the sum of elementwise products of any three columns (whether distinct or not) is 0. The orthogonal LHDs constructed by Ye (1998), Cioppa and Lucas (2007), Sun et al. (2009) and Georgiou (2009) all have this property. However, the run size n of any design in the former three references must be a power of two (n=2^c) or a power of two plus one (n=2^c+1), which is a rather severe restriction. In this paper, we construct orthogonal LHDs with more flexible run sizes which also have the property that the sum of elementwise product of any three columns is 0. Further, we compare the proposed designs with some existing orthogonal LHDs, and prove that any orthogonal LHD with this property, including the proposed orthogonal LHD, is optimal in the sense of having the minimum values of ave(|t|), t_max, ave(|q|) and q_max.     

Nested Hadamard difference sets

A Hadamard difference set (HDS) has the parameters (4N², 2N² − N, N² − N). In the abelian case it is equivalent to a perfect binary array, which is a multidimensional matrix with elements ±1 such that all out-of-phase periodic autocorrelation coefficients are zero. We show that if a group of the form H × Z²_p^r contains a (hp^2r, √hp^r(2√hp^r − 1), √hp^r(√hp^r − 1)) HDS (HDS), p a prime not dividing |H| = h and p^j ≡ −1 (mod exp(H)) for some j, then H × Z²_p^t has a (hp^2t, √hp^t(2√hp^t − 1), √hp^t(√hp^t − 1)) HDS for every 0⩽t⩽r. Thus, if these families do not exist, we simply need to show that H × Z²_p does not support a HDS. We give two examples of families that are ruled out by this procedure.     

An application of difference sets to a problem concerning graphical codes

By using difference sets, we give an answer to the following problem concerning graphical codes: When is the binary code generated by the complete graph K_n contained in some binary Hamming code? It turns out that this holds if and only if n is one of the numbers 2, 3 and 6.     

Existence of certain class of duadic group algebra codes

Duadic codes are defined in terms of idempotents of a group algebra GF(q)G, where G is a finite group and gcd(q,|G|)=1. Under the conditions of (1) q=2^m, and (2) the idempotents are taken to be central and (3) the splitting is μ₋₁, we show that such duadic codes exist if and only if q has odd-order modulo |G|.     

Theory of factorial designs of the parallel flats type. I: the coefficient matrix

Let GF(s) be the finite field with s elements.(Thus, when s=3, the elements of GF(s) are 0, 1 and 2.)Let A(r×n), of rank r, and c_i(i=1,…,f), (r×1), be matrices over GF(s). (Thus, for n=4, r=2, f=2, we could have A=[¹¹¹⁰₀₁₂₁], c₁=[¹₀], c₂=[⁰₂].) Let T_i (i=1,…,f) be the flat in EG(n, s) consisting of the set of all the s^n?r solutions of the equations At=c_i, wheret′=(t₁,…,t_n) is a vector of variables.(Thus, EG(4, 3) consists of the 3⁴=81 points of the form (t₁,t₂,t₃,t₄), where t's take the values 0,1,2 (in GF(3)). The number of solutions of the equations At=c_i is s^n?r, where r=Rank(A), and the set of such solutions is said to form an (n?r)-flat, i.e. a flat of (n?r) dimensions. In our example, both T₁ and T₂ are 2-flats consisting of 3^4?2=9 points each. The flats T₁,T₂,…,T_f are said to be parallel since, clearly, no two of them can have a common point. In the example, the points of T₁ are (1000), (0011), (2022), (0102), (2110), (1121), (2201), (1212) and (0220). Also, T₂ consists of (0002), (2010), (1021), (2101), (1112), (0120), (1200), (0211) and (2222).) Let T be the fractional design for a sⁿ symmetric factorial experiment obtained by taking T₁,T₂,…,T_f together. (Thus, in the example, 3⁴=81 treatments of the 3⁴ factorial experiment correspond one-one with the points of EG(4,3), and T will be the design (i.e. a subset of the 81 treatments) consisting of the 18 points of T₁ and T₂ enumerated above.)In this paper, we lay the foundation of the general theory of such ‘parallel’ types of designs. We define certain functions of A called the alias component matrices, and use these to partition the coefficient matrix X (n×v), occuring in the corresponding linear model, into components X._j(j=0,1,…,g), such that the information matrix X is the direct sum of the X′._jX._j. Here, v is the total number of parameters, which consist of (possibly μ), and a (general) set of (geometric) factorial effects (each carrying (s?1) degrees of freedom as usual). For j≠0, we show that the spectrum of X′._jX._j does not change if we change (in a certain important way) the usual definition of the effects. Assuming that such change has been adopted, we consider the partition of the X._j into the X_ij (i=1,…,f). Furthermore, the X_ij are in turn partitioned into smaller matrices (which we shall here call the) X_ijh. We show that each X_ijh can be factored into a product of 3 matrices J, ζ (not depending on i,j, and h) and Q(j,h,i)where both the Kronecker and ordinary product are used. We introduce a ring R using the additive groups of the rational field and GF(s), and show that the Q(j,h,i) belong to a ring isomorphic to R. When s is a prime number, we show that R is the cyclotomic field. Finally, we show that the study of the X._j and X′._jX._j can be done in a much simpler manner, in terms of certain relatively small sized matrices over R.     

Precise large deviations for the difference of two sums of END random variables with heavy tails

Assume that there are two types of insurance contracts in an insurance company, and the ith related claims are denoted by {X_ij, j ? 1}, i = 1, 2. In this article, the asymptotic behaviors of precise large deviations for non random difference ∑^n₁(t)_{j = 1}X_1j ? ∑^n₂(t)_{j = 1}X_2j and random difference ∑^N₁(t)_{j = 1}X_1j ? ∑^N₂(t)_{j = 1}X_2j are investigated, and under several assumptions, some corresponding asymptotic formulas are obtained.     

Geometric ergodicity of nonlinear autoregressive models with changing conditional variances

The authors give easy‐to‐check sufficient conditions for the geometric ergodicity and the finiteness of the moments of a random process x_t = ?(x_t‐1,…, x_t‐p) + ?_tσ(x_t‐1,…, x_t‐q) in which ?: R^p → R, σ R^q → R and (?_t) is a sequence of independent and identically distributed random variables. They deduce strong mixing properties for this class of nonlinear autoregressive models with changing conditional variances which includes, among others, the ARCH(p), the AR(p)‐ARCH(p), and the double‐threshold autoregressive models.