On the existence of retransmission permutation arrays

We investigate retransmission permutation arrays (RPAs) that are motivated by applications in overlapping channel transmissions. An RPA is an $n\times n$ array in which each row is a permutation of $\{1,\dots ,n\}$, and for $1?i?n$, all $n$ symbols occur in each $i\times ?\frac{n}{i}?$ rectangle in specified corners of the array. The array has types 1, 2, 3 and 4 if the stated property holds in the top left, top right, bottom left and bottom right corners, respectively. It is called latin if it is a latin square. We show that for all positive integers $n$, there exists a type-1, 2, 3, 4 $\mathrm{RPA}\left(n\right)$ and a type-1, 2 latin $\mathrm{RPA}\left(n\right)$.     

Constructions of permutation arrays for certain scheduling cost measures

Constructions of permutation arrays are presented that are optimal or nearly-optimal with respect to two cost measures: the so-called longest-jump measure and the longest-monotone-greedy-subsequence measure. These measures arise in the context of scheduling problems in asynchronous, shared memory, multiprocessor machines.     

Frequency permutation arrays

Motivated by recent interest in permutation arrays, we introduce and investigate the more general concept of frequency permutation arrays (FPAs). An FPA of length n = mλ and distance d is a set T of multipermutations on a multiset of m symbols, each repeated with frequency λ, such that the Hamming distance between any distinct x,y ∈ T is at least d. Such arrays have potential applications in powerline communication. In this article, we establish basic properties of FPAs, and provide direct constructions for FPAs using a range of combinatorial objects, including polynomials over finite fields, combinatorial designs, and codes. We also provide recursive constructions, and give bounds for the maximum size of such arrays. © 2006 Wiley Periodicals, Inc. J Combin Designs 14: 463–478, 2006     

Constructions of complete permutation polynomials

Based on the Feistel and MISTY structures, this paper presents several new constructions of complete permutation polynomials (CPPs) of the finite field \({\mathbb {F}}_{2^{n}}^2\) for a positive integer n and three constructions of CPPs over \({\mathbb {F}}_{p^{n}}^m\) for any prime p and positive integer \(m\ge 2\). In addition, we investigate the upper bound on the algebraic degree of these CPPs and show that some of them can have nearly optimal algebraic degree.     

Constructing permutation arrays from groups

Let M(n, d) be the maximum size of a permutation array on n symbols with pairwise Hamming distance at least d. We use various combinatorial, algebraic, and computational methods to improve lower bounds for M(n, d). We compute the Hamming distances of affine semilinear groups and projective semilinear groups, and unions of cosets of AGL(1, q) and PGL(2, q) with Frobenius maps to obtain new, improved lower bounds for M(n, d). We give new randomized algorithms. We give better lower bounds for M(n, d) also using new theorems concerning the contraction operation. For example, we prove a quadratic lower bound for \(M(n,n-2)\) for all \(n\equiv 2 \pmod 3\) such that \(n+1\) is a prime power.     

Constructions of augmented orthogonal arrays

Augmented orthogonal arrays (AOAs) were introduced by Stinson, who showed the equivalence between ideal ramp schemes and AOAs (Discrete Math. 341 (2018), 299–307). In this paper, we show that there is an AOA if and only if there is an OA which can be partitioned into subarrays, each being an OA, and that there is a linear AOA if and only if there is a linear maximum distance separable (MDS) code of length and dimension over , which contains a linear MDS subcode of length and dimension over . Some constructions for AOAs and some new infinite classes of AOAs are also given.     

Decoding permutation arrays with ternary vectors

It is well known that the MacWilliams transform of the weight enumerator of some code having integer coefficients is equivalent to a set of congruences having integer solutions. In this paper, we prove an equivalent condition of Ward’s bound on dimension of divisible codes, which is part of this set of congruences having integer solutions. This new interpretation makes the generalization of Ward’s bound an explicit one.     

A construction for equidistant permutation arrays of index one

An equidistant permutation array (EPA) which we denote by A(r, λ; ν) is a ν × r array such that every row is a permutation of the integers 1, 2,…, r and such that every pair of distinct rows has precisely λ columns in common. R(r, λ) is the maximum ν such that there exists an A(r, λ; ν). In this paper we show that R(n² + n + 2, 1) ? 2n² + n where n is a prime power.     

Using permutation rational functions to obtain permutation arrays with large hamming distance

Designs, Codes and Cryptography - We consider permutation rational functions (PRFs), V(x)/U(x), where both V(x) and U(x) are polynomials over a finite field $$\mathbb {F}_q$$ . Permutation rational...     

Constructions of covering arrays of strength five

A covering array of size N, strength t, degree k and order v, or a CA(N; t, k, v) in short, is an N × k array on v symbols. In every N × t subarray, each t-tuple occurs in at least one row. Covering arrays have been studied for their significant applications to generating software test suites to cover all t-sets of component interactions. In this paper, we present two constructive methods to obtain covering arrays of strength 5 by using difference covering arrays and holey difference matrices with a prescribed property. As a consequence, some new upper bounds on the covering numbers are derived.     

Constructions of new orthogonal arrays and covering arrays of strength three

A covering array of size N, strength t, degree k, and order v, or a CA(N;t,k,v) in short, is a k×N array on v symbols. In every t×N subarray, each t-tuple column vector occurs at least once. When ‘at least’ is replaced by ‘exactly’, this defines an orthogonal array, OA(t,k,v). A difference covering array, or a DCA(k,n;v), over an abelian group G of order v is a k×n array (aij) (1?i?k, 1?j?n) with entries from G, such that, for any two distinct rows l and h of D (1?l<h?k), the difference list Δlh={dh1−dl1,dh2−dl2,…,dhn−dln} contains every element of G at least once.Covering arrays have important applications in statistics and computer science, as well as in drug screening. In this paper, we present two constructive methods to obtain orthogonal arrays and covering arrays of strength 3 by using DCAs. As a consequence, it is proved that there are an OA(3,5,v) for any integer v?4 and v?2 (mod 4), and an OA(3,6,v) for any positive integer v satisfying gcd(v,4)≠2 and gcd(v,18)≠3. It is also proved that the size CAN(3,k,v) of a CA(N;3,k,v) cannot exceed v³+v² when k=5 and v≡2 (mod 4), or k=6, v≡2 (mod 4) and gcd(v,18)≠3.     

Covering arrays of higher strength from permutation vectors

A covering array CA(N;t,k,v) is an N × k array such that every N × t sub‐array contains all t‐tuples from v symbols at least once, where t is the strength of the array. Covering arrays are used to generate software test suites to cover all t‐sets of component interactions. We introduce a combinatorial technique for their construction, focussing on covering arrays of strength 3 and 4. With a computer search, covering arrays with improved parameters have been found. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 202–213, 2006     

Diagonally neighbour transitive codes and frequency permutation arrays

Constant composition codes have been proposed as suitable coding schemes to solve the narrow band and impulse noise problems associated with powerline communication, while at the same time maintaining a constant power output. In particular, a certain class of constant composition codes called frequency permutation arrays have been suggested as ideal, in some sense, for these purposes. In this paper we characterise a family of neighbour transitive codes in Hamming graphs in which frequency permutation arrays play a central rode. We also classify all the permutation codes generated by groups in this family.     

Covering and radius-covering arrays: Constructions and classification

The minimum number of rows in covering arrays (equivalently, surjective codes) and radius-covering arrays (equivalently, surjective codes with a radius) has been determined precisely only in special cases. In this paper, explicit constructions for numerous best known covering arrays (upper bounds) are found by a combination of combinatorial and computational methods. For radius-covering arrays, explicit constructions from covering codes are developed. Lower bounds are improved upon using connections to orthogonal arrays, partition matrices, and covering codes, and in specific cases by computation. Consequently for some parameter sets the minimum size of a covering array is determined precisely. For some of these, a complete classification of all inequivalent covering arrays is determined, again using computational techniques. Existence tables for up to 10 columns, up to 8 symbols, and all possible strengths are presented to report the best current lower and upper bounds, and classifications of inequivalent arrays.     

Constructions of optimal orthogonal arrays with repeated rows

We construct orthogonal arrays ${\mathsf{OA}}_{\lambda }(k,n)$ (of strength two) having a row that is repeated $m$ times, where $m$ is as large as possible. In particular, we consider OAs where the ratio $m∕\lambda$ is as large as possible; these OAs are termed optimal. We provide constructions of optimal OAs for any $k\ge n+1$, albeit with large $\lambda$. We also study basic OAs; these are optimal OAs in which $gcd(m,\lambda )=1$. We construct a basic OA with $n=2$ and $k=4t+1$, provided that a Hadamard matrix of order $8t+4$ exists. This completely solves the problem of constructing basic OAs with $n=2$, modulo the Hadamard matrix conjecture.     

Generalized orthogonal arrays: Constructions and related graphs

Generalized orthogonal arrays were first defined to provide a combinatorial characterization of (t, m, s)-nets. In this article we describe three new constructions for generalized orthogonal arrays. Two of these constructions are straightforward generalizations of constructions for orthogonal arrays and one employs new techniques. We construct families of generalized orthogonal arrays using orthogonal arrays and provide net parameters obtained from our constructions. In addition, we define a set of graphs associated with a generalized orthogonal array which provide information about the structure of the array. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 31–39, 1999     

Equidistant frequency permutation arrays and related constant composition codes

In this paper we study the special class of equidistant constant composition codes of type CCC(n, d, μ ^m ) (where n = m μ), which correspond to equidistant frequency permutation arrays; we also consider related codes with composition “close to” μ ^m . We establish various properties of these objects and give constructions for optimal families of codes.     

Covering arrays of strength three from extended permutation vectors

A covering array \(\text{ CA }(N;t,k,v)\) is an \(N\times k\) array such that in every \(N\times t\) subarray each possible t-tuple over a v-set appears as a row of the subarray at least once. The integers t and v are respectively the strength and the order of the covering array. Let v be a prime power and let \({\mathbb {F}}_v\) denote the finite field with v elements. In this work the original concept of permutation vectors generated by a \((t-1)\)-tuple over \({\mathbb {F}}_v\) is extended to vectors generated by a t-tuple over \({\mathbb {F}}_v\). We call these last vectors extended permutation vectors (EPVs). For every prime power v, a covering perfect hash family \(\text{ CPHF }(2;v^2-v+3,v^3,3)\) is constructed from EPVs given by subintervals of a linear feedback shift register sequence over \({\mathbb {F}}_v\). When \(v\in \{7,9,11,13,16,17,19,23,25\}\) the covering array \(\text{ CA }(2v^3-v;3,v^2-v+3,v)\) generated by \(\text{ CPHF }(2;v^2-v+3,v^3,3)\) has less rows than the best-known covering array with strength three, \(v^2-v+3\) columns, and order v. CPHFs formed by EPVs are also constructed using simulated annealing; in this case the results improve the size of eighteen covering arrays of strength three.     

Pairwise “Orthogonal generalized room squares” and equidistant permutation arrays

A generalized Room square S(r, λ; v) is an r × r array such that every cell in the array contains a subset of a v-set V. This subset could of course be the empty set. The array has the property that every element of V is contained precisely once in every row and column and that any two distinct elements of V are contained in precisely λ common cells. In this paper we define pairwise orthogonal generalized Room squares and give a construction for these using finite projective geometries. This is another generalization of the concept of pairwise orthogonal latin squares. We use these orthogonal arrays to construct permutations having a constant Hamming distance.     

On the existence of doubly resolvable Kirkman systems and equidistant permutation arrays

The purpose of this paper is twofold. A doubly resolvable Kirkman system is a (v, k, 1)-BIBD whose blocks can be resolved into two resolutions R and R' such that any resolution class from R has at most one block in common with any resolution class from R'. Room squares are examples of such systems in the case k = 2. It was not known whether such systems existed for k ? 3. In this paper, we construct infinite classes of such designs. In particular, we display several doubly resolvable Kirkman systems with v = 27 and k = 3.An equidistant permutation array (EPA) is a v × r array defined on an r-set V of symbols such that each row is a permutation of V and any two distinct rows have Hamming distance r ? 1. Such arrays are closely related to the problem described above and have attracted much interest recently. A central problem of EPA's is to determine the maximum value of v for a fixed value of r. Until now, there has been only one value of r for which a direct construction existed to produce an array having v=2r+1. The present paper provides a direct construction for these arrays which have v the order of $\text{r}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$. Hence, this establishes that the maximum value of v grows non-linearly with r. In the case of r = 13, the largest value of v previously known was 27. We display such an array with v = 36.