A characterization of some {3v1 + v3, 3v0 + v2; 3, 3}-minihypers and its applications to error-correcting codes

Recently, Hamada et al. (Des. Codes Cryptogr. 2(1992), 225–229) showed that there exists an [87,5,57;3]-code meeting the Grismer bound. But it is unknown whether or not an [87,5,57;3]-code is unique up to equivalence. In order to characterize all [87,5,57;3]-codes, it is sufficient to characterize all {2v₂ + 2v₃,2v₁ + 2v₂; 4, 3}-minihypers and in order to characterize all {2v₂ + 2v₃,2v₁ + 2v₂;4,3}-minihypers, it is necessary to characterize all {3v₁ + v₃,3v₀ + v₂;3,3}-minihypers, where v₀ = 0, v₁ = 1, v₂ = 4 and v₃ = 13 (cf. Hamada and Helleseth, J. Statist. Plann. Inference, 56 (1996)). The purpose of this paper is to characterize all {3v₁ + v₃,3v₀ + v₂;3,3}-minihypers.     

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.     

Self-dual codes which are principal ideals of the group algebra F2[{F2m, + }]

This paper follows up Camion's contribution on self-dual codes which are principal ideals of the algebra F₂[{F_2^m, + }], the so-called H-codes. Our main result is that this class of codes does not meet the Gilbert-Varshamov bound. We obtain this result by giving an upper bound on the minimal distance of any H-code. We characterize extremal H-codes and link up their generators with certain difference sets.     

On some joint distributions in fluctuation theory

For non-negative integral valued interchangeable random variables v₁, v₂,…,v_n, Takács (1967, 70) has derived the distributions of the statistics ?_n' ?¹_n' ?^(c)_n and ?^(-c)_n concerning the partial sums N_r = v₁ + v₂ + ··· + v_rr = 1,…,n. This paper deals with the joint distributions of some other statistics viz., (α^(c)_n, δ^(c)_n, Z_n), (β^(c)_n, Z_n) and (β^(-c)_n, Z_n) concerning the partial sums N_r = ε₁ + ··· + ε_rr = 1,2,…,n, of geometric random variables ε₁, ε₂,…,ε_n.     

On the spectrum of [v,k, t] trades

A [v, k, t] trade of volume m consists of two disjoint collections T₁ and T₂, each of m k-subsets of a v-set V, such that each t-subset of V is contained in the same number of blocks of T₁ and T₂, and each element of V is contained in at least one block of T₁. We study [v, k, t] trades, and investigate their spectrum (i.e., the collections of allowable volumes), using both theoretical techniques and computer-based searches.     

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.     

Asymptotic distribution of hotelling's trace for two unequal covariance matrices and robustness study of test of equality of mean vectors

Asymptotic expansions for the percentiles and c.d.f., up to terms of order $\text{1}\text{n}{}^2$ of the statistic $\text{T =m}\text{T}\text{r}\text{S}{}_1\text{S}{}^{-1}{}_2$, where mS₁ and nS₂ independently distributed W(m, p, Σ₁) and W(n, p, Σ₂) respectively, are obtained using methods similar to those of Ito [4], Chattopadhyay and Pillai [2]. These expansions hold when $\text{Σ}{}_1\text{Σ}{}^{-1}{}_2\text{=}\text{I}\text{+}\text{F}$ and$\text{|}\text{Ch}{}_{\mathrm{i}}\text{(}\text{F}\text{)| < 1}$. Tables of powers of T for p = 3 and p = 4 for m = 4 and various values of n are given and comparison made with the exact powers for p = 3. These powers are useful for the study of (i) the test of equality of covariance matrices in two p-variate normal populations and (ii) robustness of test of equality of mean vectors of l normal populations against the violation of the assumption of equality of covariance matrices.     

Partial differential equations for hypergeometric functions 3F2 of matrix argument

Many multivariate non-null distributions and moment formulas can be expressed in terms of hypergeometric functions _pF_q of matrix arqument. Muirhead [6] and Constantine and Muirhead [2] gave partial differential equations for the functions of ₂F₁ of one argument matrix and two argument matrices, respectively. Such differential equations have been used to obtain asymptotic expansions of the functions (Muirhead [7], [8], [9], Sugiura [10]). The purpose of this paper is to derive partial differential equations for the functions ₃F₂ (a₁ a₂, a₃; b₁, b₂, R) and ₃F₂ (a₁, a₂, a₃; b₁, b₂; R, S). Differential equations for ₂F₂ are also obtained.     

Resolvable systems of 8-tuples

Let R be a family of k-element blocks of a v-element set V such that any two elements of V are contained in λ blocks of R and R=R₁∪…∪R_v?1, R_i∩ R_j=? (i≠j) and ?{B_i∈R_j∣i=1,…,v?k}=V (B_i a block in R_j), i.e. R is a resolvable block design RB(v, k, λ). In this paper it will be shown that a sufficient condition for the existence of an RB(v, 8, 7) is that v≡0 (mod 8) and v is nondivisible by 3, 5, 7.     

Complete designs with blocks of maximal multiplicity

The set of distinct blocks of a block design is known as its support. We construct complete designs with parameters v(?7), k=3, λ=v ? 2 which contain a block of maximal multiplicity and with support size $\text{b}{}^1\text{= (}{}^{\mathrm{v}}{}_3\text{) ? 4(v ? 2)}$. Any complete design which contains such a block, and has parameters v, k, λ as above, must be supported on at most (^v₃) ? 4(v ? 2) blocks. Attention is given to complete designs because of their direct relationship to simple random sampling.     

Perfect octagon quadrangle systems with upper C4-systems

An octagon quadrangle is the graph consisting of an 8-cycle (x₁, x₂,…, x₈) with two additional chords: the edges {x₁, x₄} and {x₅, x₈}. An octagon quadrangle system of order v and index ρ [OQS] is a pair (X,H), where X is a finite set of v vertices and H is a collection of edge disjoint octagon quadrangles (called blocks) which partition the edge set of ρK_v defined on X. An octagon quadrangle systemΣ=(X,H) of order v and index λ is said to be upper C₄-perfect if the collection of all of the upper4-cycles contained in the octagon quadrangles form a μ-fold 4-cycle system of order v; it is said to be upper strongly perfect, if the collection of all of the upper4-cycles contained in the octagon quadrangles form a μ-fold 4-cycle system of order v and also the collection of all of the outside8-cycles contained in the octagon quadrangles form a ?-fold 8-cycle system of order v. In this paper, the authors determine the spectrum for these systems.     

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.     

E-optimal incomplete block designs with two distinct block sizes

Sufficient conditions are derived for the determination of E-optimal designs in the class D(v,b₁,b₂,k₁,k₂) of incomplete block designs for v treatments in b₁ blocks of size k₁ each and b₂ blocks of size k₂ each. Some constructions for E-optimal designs that satisfy the sufficient conditions obtained here are given. In particular, it is shown that E-optimal designs in D(v,b₁,b₂,k₁,k₂) can be constructed by augmenting b₂ blocks, with k₂ − k₁ extra plots each, of a BIBD(v,b = b₁ + b₂,k₁,λ) and GDD(v,b = b₁ + b₂,k₁,λ₁,λ₂). It is also shown that equireplicate E-optimal designs in D(v,b₁,b₂,k₁,k₂) can be constructed by combining disjoint blocks of BIBD(v,b,k₁,λ) and GDD(v,b,k₁,λ₁,λ₂) into larger blocks. As applications of the construction techniques, several infinite series of E-optimal designs with small block sizes differing by at most two are given. Lower bounds for the A-efficiency are derived and it is found that A-efficiency exceeds 99% for v ⩾ 10, and at least 97.5% for 5 ⩽v < 10.     

Comparison of rotatable designs for regression on balls,I (quadratic)

Designs for quadratic and cubic regression are considered when the possible choices of the controlable variable are points x=( x₁,x₂,…,x_q) in the q-dimensional. Full of radius R, B_q(R) ={x:Σ⁴_ix²_i?R²}. The designs that are optimum among rotatable designs with respect to the D-, A-, and E-optimality criteria are compared in their performance relative to these and other criteria, including extrapolation. Additionally, the performance of a design optimum for one value of R, when it is implemented for a different value of R, is investigated. Some of the results are developed algebraically; others, numerically. For example, in quadratic regression the A-optimum design appears to be fairly robust in its efficiency, under variation of criterion.     

The intersection problem for K4 - e designs

A G-design of order n is a pair (P,B) where P is the vertex set of the complete graph K_n and B is an edge-disjoint decomposition of K_n into copies of the simple graph G. Following design terminology, we call these copies “blocks”. Here K₄ - e denotes the complete graph K₄ with one edge removed. It is well-known that a K₄ - e design of order n exists if and only if n ≡ 0 or 1 (mod 5), n ⩾ 6. The intersection problem here asks for which k is it possible to find two K₄ - e designs (P,B₁) and (P,B₂) of order n, with |B₁ ∩ B₂| = k, that is, with precisely k common blocks. Here we completely solve this intersection problem for K₄ - e designs.     

A construction of group divisible designs

Given any affine design with parameters v, b, r, k, λ and μ = k²/v and any design with parameters v′, b′, r′, k′, λ′ where r′ = tr for some natural number `t and k′?r, we construct a group divisible design with parameters v′' = vv′, m = v′, n = v, b′' = vb′, k′' = kk′, r′'= kr′, λ₁ = tkλ and λ₂ = μλ′. This is applied to some series of designs. As a lemma, we also show that any 0-1-matrix with row sums tr and column sums ?r may be written as the sum of r 0-1-matrices with row sums t and column sums ?1.     

Existence of holey LSSOM of type 2n with application to G7-packings of Kv

In this paper, we show that the set of all integers n⩾4 is the spectrum of two mutually orthogonal holey Latin squares, of type 2ⁿ, in which one square is symmetric. As an application, we determine the maximum number of pairwise edge disjoint copies of G₇ in K_v for each v, and all possible graphs that arise from the unused edges.     

Many (22, 44, 14, 7, 4) — and (15, 42, 14, 5, 4)-balanced incomplete block designs

A Balanced Incomplete Block Design (BIBD) is a pair (V, B) where V is a v-set and B is a collection of b k-subsets of V, called blocks, such that every element of V occurs in exactly r of the k-subsets and every 2-subset of V occurs in exactly λ of the blocks. The number of non-isomorphic designs of a BIBD (22, 44, 14, 7, 4) whose automorphism group is divisible by 7 or 11 are investigated. From this work, results are obtained on the number of non-isomorphic BIBDs (15, 42, 14, 5, 4).