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Balanced sliced orthogonal arrays (BSOAs) are useful for designing sliced computer experiments with qualitative and quantitative factors or multiple models and nested computer experiments with two codes of different levels of accuracy. In this article, we propose a systematic method to construct BSOAs by making use of the A + B design structure and its modifications. A general method for construction of asymmetric BSOAs is also presented based on an orthogonal array with orthogonal partition. These methods are flexible for strength and factors. Numerous new symmetric and asymmetric BSOAs including a lot of ones with high strength can thus be constructed. Compared with the existing BSOAs, the constructed BSOAs have more number of columns. Moreover, these methods can also be used to construct BSOAs whose strength is different from the strength of their slices. Some selective BSOAs are tabulated for practical uses. [ABSTRACT FROM AUTHOR]

Using recent results of Keevash et al. [10] and Eberhard et al. [8] together with further new detailed techniques in combinatorics, we present constructions of two concrete families of generalized Maiorana-McFarland bent functions. Our constructions improve the lower bounds on the number of bent functions in n variables over a finite field F p if p is odd and n is odd in the limit as n tends to infinity. Moreover we obtain the asymptotically exact number of two dimensional vectorial Maiorana-McFarland bent functions in n variables over F 2 as n tends to infinity. [ABSTRACT FROM AUTHOR]

We describe the Terwilliger algebras of the Latin square association schemes arising from Cayley tables of Bol loops. These association schemes have four non-trivial classes. We give some necessary conditions involving Terwilliger algebras for a quasigroup to be a Bol loop. [ABSTRACT FROM AUTHOR]

We describe the computer-aided classification of equitable partitions of the 12-cube with quotient matrix [[2, 10], [6, 6]], or, equivalently, simple orthogonal arrays OA(1536, 12, 2, 7), or order-7 correlation-immune Boolean functions in 12 arguments with 1536 ones (which completes the classification of unbalanced order-7 correlation-immune Boolean functions in 12 arguments and, as derived objects, unbalanced order-6 correlation-immune Boolean functions in 11 arguments). We find that there are 103 equivalence classes of the considered objects, and there are only two almost-OA(1536, 12, 2, 8) among them. Additionally, we find that there are 40 equivalence classes of pairs of disjoint simple OA(1536, 12, 2, 7) (equivalently, equitable partitions of the 12-cube with quotient matrix [[2, 6, 4], [6, 2, 4], [6, 6, 0]]) and discuss the existence of a non-simple OA(1536, 12, 2, 7). [ABSTRACT FROM AUTHOR]

In this paper, we propose a new method for constructing 1-perfect mixed codes in the Cartesian product F n × F q n , where F n and F q are finite fields of orders n = q m and q. We consider generalized Reed-Muller codes of length n = q m and order (q - 1) m - 2 . Codes whose parameters are the same as the parameters of generalized Reed-Muller codes are called Reed-Muller-like codes. The construction we propose is based on partitions of distance-2 MDS codes into Reed-Muller-like codes of order (q - 1) m - 2 . We construct a set of q q cn nonequivalent 1-perfect mixed codes in the Cartesian product F n × F q n , where the constant c satisfies c < 1 , n = q m and m is a sufficiently large positive integer. We also prove that each 1-perfect mixed code in the Cartesian product F n × F q n corresponds to a certain partition of a distance-2 MDS code into Reed-Muller-like codes of order (q - 1) m - 2 . [ABSTRACT FROM AUTHOR]

Two resolutions of the same 3-design are said to be orthogonal when each parallel class of one resolution has at most one block in common with each parallel class of the other resolution. If a Steiner quadruple system has two mutually orthogonal resolutions, the design is called doubly resolvable and denoted by DRSQS. In this paper, we define almost doubly resolvable candelabra quadruple system and then to get a recursive construction of DRSQS, i.e., for n ≥ 16 , if there is a DRSQS(n), then there exists a DRSQS (14 n - 12) and a DRSQS (16 n - 12) . [ABSTRACT FROM AUTHOR]

A Boolean function f on n variables is said to be a bent function if the absolute value of all its Walsh coefficients is 2 n / 2 . Our main result is a new asymptotic lower bound on the number of Boolean bent functions. It is based on a modification of the Maiorana–McFarland family of bent functions and recent progress in the estimation of the number of transversals in latin squares and hypercubes. By-products of our proofs are the asymptotics of the logarithm of the numbers of partitions of the Boolean hypercube into 2-dimensional affine and linear subspaces. [ABSTRACT FROM AUTHOR]

Most of the approaches published in the literature to construct S-boxes via Cellular Automata (CA) work by either iterating a finite CA for several time steps, or by a one-shot application of the global rule. The main characteristic that brings together these works is that they employ a single CA rule to define the vectorial Boolean function of the S-box. In this work, we explore a different direction for the design of S-boxes that leverages on Orthogonal CA (OCA), i.e. pairs of CA rules giving rise to orthogonal Latin squares. The motivation stands on the facts that an OCA pair already defines a bijective transformation, and moreover the orthogonality property of the resulting Latin squares ensures a minimum amount of diffusion. We exhaustively enumerate all S-boxes generated by OCA pairs of diameter 4 ≤ d ≤ 6 , and measure their nonlinearity. Interestingly, we observe that for d = 4 and d = 5 all S-boxes are linear, despite the underlying CA local rules being nonlinear. The smallest nonlinear S-boxes emerges for d = 6 , but their nonlinearity is still too low to be used in practice. Nonetheless, we unearth an interesting structure of linear OCA S-boxes, proving that their Linear Components Space is itself the image of a linear CA, or equivalently a polynomial code. We finally classify all linear OCA S-boxes in terms of their generator polynomials. [ABSTRACT FROM AUTHOR]

A lower bound is presented for the minimal number of filled cells in a maximal partial Latin hypercube of dimension d and order n. The result generalises and extends previous results for d = 2 (Latin squares) and d = 3 (Latin cubes). Explicit constructions show that this bound is near-optimal for large n > d . For d > n , a connection with Hamming codes shows that this lower bound gives a related upper bound for the same quantity. The results can be interpreted in terms of independent dominating sets in certain graphs, and in terms of codes that have covering radius 1 and minimum distance at least 2. [ABSTRACT FROM AUTHOR]

We introduce a graph attached to mutually orthogonal Sudoku Latin squares. The spectra of the graphs obtained from finite fields are explicitly determined. As a corollary, we then use the eigenvalues to distinguish non-isomorphic Sudoku Latin squares. [ABSTRACT FROM AUTHOR]

This work brings together several types of combinatorial designs: difference matrices, difference covering arrays and difference schemes by defining the concept of covering scheme of strength t over an abelian additive group. Connections of covering schemes with orthogonal arrays and covering arrays are also established. We show general results of covering schemes of strength t using a method based on the factorization of a group and some refinements for particular classes. We apply the previous results to investigate covering schemes having three, four and five factors. Finally, a reformulation of covering schemes in terms of graph theory is established. [ABSTRACT FROM AUTHOR]

Strong orthogonal arrays (SOAs) have better space-filling properties than ordinary orthogonal arrays for computer experiments. SOAs of strengths two plus and two star can improve the space-filling properties in low dimensions, and column orthogonality plays a vital role in computer experiments. In this paper, we use matchings of the bipartite graphs and Hall's theorem to present several constructions of column-orthogonal SOAs of strengths two plus and two star. We also give a new direct construction of SOAs of strength two plus. Our constructions can provide larger numbers of factors of SOAs and column-orthogonal SOAs of strengths two plus and two star than those in the existing literature, having flexible run sizes. The constructions are convenient and flexible, and the resulting designs are good for computer experiments. [ABSTRACT FROM AUTHOR]

With the help of Generalized Estimating Equations, we identify locally D-optimal crossover designs for generalized linear models. We adopt the variance of parameters of interest as the objective function, which is minimized using constrained optimization to obtain optimal crossover designs. In this case, the traditional general equivalence theorem could not be used directly to check the optimality of obtained designs. In this manuscript, we derive a corresponding general equivalence theorem for crossover designs under generalized linear models. [ABSTRACT FROM AUTHOR]

In this paper, we use the intransitive starter-adder method and the standard starter-adder method to construct some new frames and doubly resolvable group divisible designs. Some infinite classes of frames and doubly resolvable group divisible designs are obtained by recursive constructions. On this basis, we almost establish the existence of frames and doubly resolvable group divisible designs with block size three and index two. [ABSTRACT FROM AUTHOR]

Cellular automata (CA) are an interesting computational model for designing pseudorandom number generators (PRNG), due to the complex dynamical behavior they can exhibit depending on the underlying local rule. Most of the CA-based PRNGs proposed in the literature, however, suffer from poor diffusion since a change in a single cell can propagate only within its neighborhood during a single time step. This might pose a problem especially when such PRNGs are used for cryptographic purposes. In this paper, we consider an alternative approach to generate pseudorandom sequences through orthogonal CA (OCA), which guarantees a better amount of diffusion. After defining the related PRNG, we perform an empirical investigation of the maximal cycles in OCA pairs up to diameter d = 8 . Next, we focus on OCA induced by linear rules, giving a characterization of their cycle structure based on the rational canonical form of the associated Sylvester matrix. Finally, we devise an algorithm to enumerate all linear OCA pairs characterized by a single maximal cycle, and apply it up to diameter d = 16 and d = 13 for OCA respectively over the binary and ternary alphabets. [ABSTRACT FROM AUTHOR]

We give a theoretical proof of the fact that the orthomorphism graph of group Z 2 × Z 4 has maximal clique 2 by determining the structure of the graph. [ABSTRACT FROM AUTHOR]

Orthogonal array is one of the important research subjects in combinatorial design theory and experimental design theory, and it is widely applied to statistics, computer science, coding theory and cryptography. There are many constructions and results for orthogonal array of strength 2, however the results on orthogonal array of strength t ≥ 3 are rare. In this paper, we first present two new effective constructions for orthogonal arrays of strength t ≥ 3 based on large sets of orthogonal arrays. Second, many infinite families of large sets of orthogonal arrays are obtained and then some new series of orthogonal array of strength t ≥ 3 are produced. [ABSTRACT FROM AUTHOR]

An order-m Latin square is a factor pair Latin square if, for each ordered pair (a, b) of positive integers satisfying a b = m , there is no repetition of symbols in any a × b tiling region. If p is prime and n is a natural number, we introduce a new characterization of order- p n linear factor pair Latin squares. We use this characterization to show that the Latin squares constructed by Mariot et al. (Des Codes Cryptogr 88:391-411, 2020) corresponding to F q - linear bipermutive cellular automata are linear factor pair Latin squares when q is prime. These linear factor pair Latin squares are called cellular factor pair Latin squares. We then apply the results and methods of Mariot et al. (2020) to construct maximally sized families of pairwise mutually orthogonal cellular factor pair Latin squares of order p n , and likewise for diagonal cellular factor pair Latin squares of order p n . The size of each of these two families is asymptotic to p n / n . [ABSTRACT FROM AUTHOR]

A relation on a k -net (n) (or, equivalently, a set of k - 2 mutually orthogonal Latin squares of order n) is an F 2 linear dependence within the incidence matrix of the net. Dukes and Howard (2014) showed that any 6 -net (10) satisfies at least two non-trivial relations, and classified the relations that could appear in such a net. We find that, up to equivalence, there are 18 526 320 pairs of MOLS satisfying at least one non-trivial relation. None of these pairs extend to a triple. We also rule out one other relation on a set of 3-MOLS from Dukes and Howard's classification. [ABSTRACT FROM AUTHOR]

The fundamental problem in the orthogonal design theory is the design isomorphism, which involves two classes of methods in the statistical literature. One is to identify the isomorphic designs by costly computation, another is only to detect the non-isomorphic designs as a feasible alternative. In this paper we explore the design structure to propose the degree of isomorphism, as a novel criterion showing the similarity between orthogonal designs. A column-wise framework is proposed to accommodate different issues of the design isomorphism, including the detection of non-isomorphism, identification of isomorphism and determination of subclasses for symmetric orthogonal designs. Our framework shows surprisingly high efficiency, where the average time of identifying the isomorphism between two designs in selected classes is all down to about one second. By applying the hierarchical clustering on the average linkage, a novel classification is also presented for non-isomorphic orthogonal designs in a combinatorial view. [ABSTRACT FROM AUTHOR]

We present a construction of partial spread bent functions using subspaces generated by linear recurring sequences (LRS). We first show that the kernels of the linear mappings defined by two LRS have a trivial intersection if and only if their feedback polynomials are relatively prime. Then, we characterize the appropriate parameters for a family of pairwise coprime polynomials to generate a partial spread required for the support of a bent function, showing that such families exist if and only if the degrees of the underlying polynomials are either 1 or 2. We then count the resulting sets of polynomials and prove that, for degree 1, our LRS construction coincides with the Desarguesian partial spread. Finally, we perform a computer search of all PS - and PS + bent functions of n = 8 variables generated by our construction and compute their 2-ranks. The results show that many of these functions defined by polynomials of degree d = 2 are not EA-equivalent to any Maiorana–McFarland or Desarguesian partial spread function. [ABSTRACT FROM AUTHOR]

Regard T = { (g , ... , g) ∈ F q n : g ∈ F q } as an additive subgroup of the vector space F q n over a finite field F q . A code C ⊂ F q n is T-invariant if T acts on C induced by the addition of F q n . Using linear algebra and finite fields results, we investigate which important families of codes are T-invariant, including perfect codes, first-order Reed–Muller codes, and some Reed–Solomon codes. In particular, we characterize which perfect linear codes are T-invariant. A version of the Gilbert–Varshamov theorem for T-invariant codes is presented. We characterize which Hadamard codes are T-invariant for nonlinear codes. As an application, constructions of difference schemes are obtained using our results. Finally, difference schemes of strength 2 over a finite field are characterized using quasigroups endowed with a distributive law. [ABSTRACT FROM AUTHOR]

In an earlier paper by three of the present authors and Csaba Schneider, it was shown that, for m ≥ 2 , a set of m + 1 partitions of a set Ω , any m of which are the minimal non-trivial elements of a Cartesian lattice, either form a Latin square (if m = 2 ), or generate a join-semilattice of dimension m associated with a diagonal group over a base group G. In this paper we investigate what happens if we have m + r partitions with r ≥ 2 , any m of which are minimal elements of a Cartesian lattice. If m = 2 , this is just a set of mutually orthogonal Latin squares. We consider the case where all these squares are isotopic to Cayley tables of groups, and give an example to show the groups need not be all isomorphic. For m > 2 , things are more restricted. Any m + 1 of the partitions generate a join-semilattice admitting a diagonal group over a group G. It may be that the groups are all isomorphic, though we cannot prove this. Under an extra hypothesis, we show that G must be abelian and must have three fixed-point-free automorphisms whose product is the identity. (We describe explicitly all abelian groups having such automorphisms.) Under this hypothesis, the structure gives an orthogonal array, and conversely in some cases. If the group is cyclic of prime order p, then the structure corresponds exactly to an arc of cardinality m + r in the (m - 1) -dimensional projective space over the field with p elements, so all known results about arcs are applicable. More generally, arcs over a finite field of order q give examples where G is the elementary abelian group of order q. These examples can be lifted to non-elementary abelian groups using p-adic techniques. [ABSTRACT FROM AUTHOR]

In this paper a new class of orthogonal arrays (OAs), i.e., OAs without interaction columns, are proposed which are applicable in factor screening, interaction detection and other cases. With the tools of di erence matrices, we present some general recursive methods for constructing OAs of such type. Several families of OAs with high percent saturation are constructed. In particular, for any integer λ ≥ 3, such a two-level OA of run 4λ can always be obtained if the corresponding Hadamard matrix exists. [ABSTRACT FROM AUTHOR]

Let Fd be the finite field with d elements. Extraordinary subgroups in Fd × Fd play an important role in the field of quantum information theory, especially for the study of mutually unbiased bases. Recently, Ghiu et al. introduced the concept of supersquare of order d which is related to extraordinary subgroups. They have given a method of construction of the mutually orthogonal supersquares, and determined all the complete sets of mutually orthogonal extraordinary supersquares of order 4. In this article, we present the construction of a complete set of mutually orthogonal extraordinary supersquares of order pn where p is a prime. We also determine all the complete sets of mutually orthogonal extraordinary supersquares of order 9. [ABSTRACT FROM AUTHOR]

We present a construction for hash-based one-time group signature schemes, and develop a traceable post-quantum multi-time group signature upon it. A group signature scheme allows group members to anonymously sign a message on behalf of the entire group. The signatures are unforgeable, and the scheme enables authorized openers to trace the signature back to the original signer when needed. Our construction utilizes three nested layers to build the group signature scheme. The first layer performs the key-management task; it deploys a transversal design to assign keys to the group members and the openers, establishing anonymity and providing the construction with traceability. The second layer utilizes sets of hash values, hash pools, to build the group public verification key and to connect group members together. The final layer uses a post-quantum hash-based signature scheme, that adds unforgeability to our construction. We extend our scheme to multi-time signatures using Merkle trees and show that this process maintains the scalability property of Merkle-based signatures, while it supports the group members signing any number of messages. [ABSTRACT FROM AUTHOR]

An SQS(v) is said to be doubly resolvable if it has two orthogonal resolutions and denoted by DRSQS(v). In this paper, we give two new constructions, i.e., doubling construction and doubly resolvable MPCQS construction. We also give some new results of doubly resolvable SQS(v). [ABSTRACT FROM AUTHOR]

Covering arrays (CA) of strength t, mixed level or fixed level, have been applied to software testing to aim for a minimum coverage of all t-way interactions among components. The size of CA increases with the increase of strength interaction t, which increase the cost of software testing. However, it is quite often that some certain components have strong interactions, while others may have fewer or none. Hence, a better way to test software system is to identify the subsets of components which are involved in stronger interactions and apply high strength interaction testing only on these subsets. For this, in 2003, the notion of variable strength covering arrays was proposed by Cohen et al. to satisfy the need to vary the size of t in an individual test suite. In this paper, an effective deterministic construction of variable strength covering arrays is presented. Based on the construction, some series of variable strength covering arrays are then obtained, which are all optimal in the sense of their sizes. In the procedure, two classes of new difference matrices of strength 3 are also mentioned. [ABSTRACT FROM AUTHOR]

A frequency square is a square matrix in which each row and column is a permutation of the same multiset of symbols. A frequency square is of type (n ; λ) if it contains n / λ symbols, each of which occurs λ times per row and λ times per column. In the case when λ = n / 2 we refer to the frequency square as binary. A set of k-MOFS (n ; λ) is a set of k frequency squares of type (n ; λ) such that when any two of the frequency squares are superimposed, each possible ordered pair occurs equally often. A set of k-maxMOFS (n ; λ) is a set of k-MOFS (n ; λ) that is not contained in any set of (k + 1) -MOFS (n ; λ) . For even n, let μ (n) be the smallest k such that there exists a set of k-maxMOFS(n; n/2). It was shown in Britz et al. (Electron. J. Combin. 27(3):#P3.7, 26 pp, 2020) that μ (n) = 1 if n/2 is odd and μ (n) > 1 if n/2 is even. Extending this result, we show that if n/2 is even, then μ (n) > 2 . Also, we show that whenever n is divisible by a particular function of k, there does not exist a set of k ′ -maxMOFS(n; n/2) for any k ′ ⩽ k . In particular, this means that lim sup μ (n) is unbounded. Nevertheless we can construct infinite families of maximal binary MOFS of fixed cardinality. More generally, let q = p u be a prime power and let p v be the highest power of p that divides n. If 0 ⩽ v - u h < u / 2 for h ⩾ 1 then we show that there exists a set of (q h - 1) 2 / (q - 1) -maxMOFS(n; n/q). [ABSTRACT FROM AUTHOR]

An n × n matrix A consisting of nonnegative integers is a general magic square of order n if the sum of elements in each row, column, and main diagonal is the same. A general magic square A of order n is called a magic square, denoted by MS(n), if the entries of A are distinct. A magic square A of order n is normal if the entries of A are n2 consecutive integers. Let A*d denote the matrix obtained by raising each element of A to the d-th power. The matrix A is a d-multimagic square, denoted by MS(n,d), if A*e is an MS(n) for 1 ≤ e ≤ d. In this paper we investigate the existence of normal bimagic squares of order 2u and prove that there exists a normal bimagic square of order 2u, where u and 6 are coprime and u ≥ 5. [ABSTRACT FROM AUTHOR]

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Two n × n Latin squares L 1 , L 2 are said to be orthogonal if, for every ordered pair (x, y) of symbols, there are coordinates (i, j) such that L 1 (i , j) = x and L 2 (i , j) = y . A k-MOLS is a sequence of k pairwise-orthogonal Latin squares, and the existence and enumeration of these objects has attracted a great deal of attention. Recent work of Keevash and Luria provides, for all fixed k, log-asymptotically tight bounds on the number of k-MOLS. To study the situation when k grows with n, we bound the number of ways a k-MOLS can be extended to a (k + 1) -MOLS. These bounds are again tight for constant k, and allow us to deduce upper bounds on the total number of k-MOLS for all k. These bounds are close to tight even for k linear in n, and readily generalise to the broader class of gerechte designs, which include Sudoku squares. [ABSTRACT FROM AUTHOR]

Let t , k , λ , s and v be nonnegative integers, and let X be a set of v symbols. A generalized Howell design, denoted t-GHD k (s , v ; λ) , is an s × s array, each cell of which is either empty or contains a k-set of symbols from X, called a block, such that: (i) each symbol appears exactly once in each row and in each column (i.e. each row and column is a resolution of X); (ii) no t-subset of elements from X appears in more than λ cells. A generalized Howell design is a class of doubly resolvable designs , which generalize a number of well-known objects. Particular instances of the parameters correspond to generalized Howell designs are doubly resolvable group divisible designs (DRGDDs). In this paper, we concentrate on the case that t = 2 , k = 3 and λ = 2 , and simply write GHD(s, v; 2). The spectrum of GHD (3 n - 3 , 3 n ; 2) 's and GHD (6 n - 6 , 6 n ; 2) 's is completely established by solving the existence of (3, 2)-DRGDDs of types 3 n and 6 n . At the same time, we also survey rummage the existence of GHD 4 (n , 4 n ; 1) 's. As their applications, several new classes of multiply constant-weight codes are obtained. [ABSTRACT FROM AUTHOR]

A division sudoku is a latin square whose all six conjugates are sudoku squares. We enumerate division sudokus up to a suitable equivalence, introduce powerful invariants of division sudokus, and also study latin squares that are division sudokus with respect to multiple partitions at the same time. We use nearfields and affine geometry to construct division sudokus of prime power rank that are rich in sudoku partitions. [ABSTRACT FROM AUTHOR]

The concept of detecting arrays was developed to locate and detect interaction faults arising between the factors in a component-based system during software testing. In this paper, we propose a family of consecutive detecting arrays (CDAs) in which the interactions between factors are considered to be ordered. CDAs can be used to generate test suites for locating and detecting interaction faults between neighboring factors. We establish a general criterion for measuring the optimality of CDAs in terms of their size. Based on this optimality criterion, the equivalence between optimum CDAs and consecutive orthogonal arrays with prescribed properties is explored. Using the advantages of this equivalence, a great number of optimum CDAs are presented. In particular, the existence of optimum CDAs with few factors is completely determined. [ABSTRACT FROM AUTHOR]

We show that for all $m,k,r\in \mathbb{N}$ , there is an $n\in \mathbb{N}$ such that whenever $L$ is a Latin square of order $m$ and the Cartesian product $L^{n}$ of $n$ copies of $L$ is $r$ -coloured, there is a monochrome Latin subsquare of $L^{n}$ , isotopic to $L^{k}$. In particular, for every prime $p$ and for all $k,r\in \mathbb{N}$ , there is an $n\in \mathbb{N}$ such that whenever the multiplication table $L({\mathbb{Z}_{p}}^{n})$ of the group ${\mathbb{Z}_{p}}^{n}$ is $r$ -coloured, there is a monochrome Latin subsquare of order $p^{k}$. On the other hand, we show that for every group $G$ of order $\leq 15$ , there is a 2-colouring of $L(G)$ without a nontrivial monochrome Latin subsquare. [ABSTRACT FROM AUTHOR]

We complete the classification of tight 4-designs in Hamming association schemes H(n,q), i.e., that of tight orthogonal arrays of strength 4, which had been open since a result by Noda (1979). To do so, we construct an association scheme attached to a tight 4-design in H(n,q) and analyze its triple intersection numbers to conclude the non-existence in all open cases. [ABSTRACT FROM AUTHOR]

An (n, k) sequence covering array is a set of permutations of [n] such that each sequence of k distinct elements of [n] is a subsequence of at least one of the permutations. An (n, k) sequence covering array is perfect if there is a positive integer λ such that each sequence of k distinct elements of [n] is a subsequence of precisely λ of the permutations. While relatively close upper and lower bounds for the minimum size of a sequence covering array are known, this is not the case for perfect sequence covering arrays. Here we present new nontrivial bounds for the latter. In particular, for k = 3 we obtain a linear lower bound and an almost linear upper bound. [ABSTRACT FROM AUTHOR]

Partial Latin squares constitute an interesting approach to improve the teaching of different subjects in the mathematics classroom. This paper delves into this topic by introducing the use of a Dynamic Geometry System to deal with these combinatorial structures as a source of loci of points in the Euclidean plane. It is assumed to this end that the distance matrix among all these points contains a zero-diagonal symmetric partial Latin square with at least one non-zero filled cell per row. [ABSTRACT FROM AUTHOR]

Akbari and Alipour [1] conjectured that any Latin array of order n with at least n2/2 symbols contains a transversal. For large n, we confirm this conjecture, and moreover, we show that n399/200 symbols suffice. [ABSTRACT FROM AUTHOR]

We investigate sets of mutually orthogonal latin squares (MOLS) generated by cellular automata (CA) over finite fields. After introducing how a CA defined by a bipermutive local rule of diameter d over an alphabet of q elements generates a Latin square of order q d - 1 , we study the conditions under which two CA generate a pair of orthogonal Latin squares. In particular, we prove that the Latin squares induced by two Linear Bipermutive CA (LBCA) over the finite field F q are orthogonal if and only if the polynomials associated to their local rules are relatively prime. Next, we enumerate all such pairs of orthogonal Latin squares by counting the pairs of coprime monic polynomials with nonzero constant term and degree n over F q . Finally, we present a construction for families of MOLS based on LBCA, and prove that their cardinality corresponds to the maximum number of pairwise coprime polynomials with nonzero constant term. Although our construction does not yield all such families of MOLS, we show that the resulting lower bound is asymptotically close to their actual number. [ABSTRACT FROM AUTHOR]

We give explicit constructions for incomplete pairwise balanced designs IPBD((v; w), K), or, equivalently, edge-decompositions of a difference of two cliques K v \ K w into cliques whose sizes belong to the set K. Our constructions produce such designs whenever v and w satisfy the usual divisibility conditions, have ratio v / w bounded away from the smallest value in K minus one, say v / w > k - 1 + ϵ , for k = min K and ϵ > 0 , and are sufficiently large (depending on K and ϵ ). As a consequence, some new results are obtained on many related designs, including class-uniformly resolvable designs, incomplete mutually orthogonal latin squares, and group divisible designs. We also include several other applications that illustrate the power of using IPBDs as 'templates'. [ABSTRACT FROM AUTHOR]

A (t, s, v)-all-or-nothing transform (AONT) is a bijective mapping defined on s-tuples over an alphabet of size v, which satisfies that if any s - t of the s outputs are given, then the values of any t inputs are completely undetermined. When t and v are fixed, to determine the maximum integer s such that a (t, s, v)-AONT exists is the main research objective. In this paper, we solve three open problems proposed in Nasr Esfahani et al. (IEEE Trans Inf Theory 64:3136–3143, 2018) and show that there do exist linear (2, p, p)-AONTs. Then for the size of the alphabet being a prime power, we give the first infinite class of linear AONTs which is better than the linear AONTs defined by Cauchy matrices. Besides, we also present a recursive construction for general AONTs and a new relationship between AONTs and orthogonal arrays. [ABSTRACT FROM AUTHOR]

Let v ≡ 4 (mod 6) and v ≥ 4 . A canonical Kirkman packing design of order v, denoted by CKPD (v) , is a resolvable packing with r = (v - 4) / 2 parallel classes such that (i) each parallel class consists of a size 4 block and (v - 4) / 3 triples; (ii) the leave consists of the union of (v - 4) / 2 vertex-disjoint edges and a K 4 with no vertices in common with those edges. A canonical Kirkman packing design is said to be doubly resolvable if there exist a pair of orthogonal resolutions. A doubly resolvable packing design is the generalization of the Kirkman square, which is inextricably bound up with the existence of some constant weight codes such as constant composition codes and permutation codes, etc. In this paper, we establish the spectra of doubly resolvable CKPD (v) s with 36 possible exceptions for v. As its direct application, a class of permutation codes are obtained. [ABSTRACT FROM AUTHOR]

Every partial colouring of a Hamming graph is uniquely related to a partial Latin hyper-rectangle. In this paper we introduce the Θ-stabilized (a, b)-colouring game for Hamming graphs, a variant of the (a, b)-colouring game so that each move must respect a given autotopism Θ of the resulting partial Latin hyperrectangle. We examine the complexity of this variant by means of its chromatic number. We focus in particular on the bi-dimensional case, for which the game is played on the Cartesian product of two complete graphs, and also on the hypercube case. [ABSTRACT FROM AUTHOR]

Latin squares are used as scramblers on symmetric-key algorithms that generate pseudo-random sequences of the same length. The robustness and effectiveness of these algorithms are respectively based on the extremely large key space and the appropriate choice of the Latin square under consideration. It is also known the importance that isomorphism classes of Latin squares have to design an effective algorithm. In order to delve into this last aspect, we improve in this paper the efficiency of the known methods on computational algebraic geometry to enumerate and classify partial Latin squares. Particularly, we introduce the notion of affine algebraic set of a partial Latin square L = (lij) of order n over a field K as the set of zeros of the binomial ideal 〈 x i x j − x l ij : (i , j) is a non-empty cell in L 〉 ⊆ K [ x 1 , ... , x n ] . Since isomorphic partial Latin squares give rise to isomorphic affine algebraic sets, every isomorphism invariant of the latter constitutes an isomorphism invariant of the former. In particular, we deal computationally with the problem of deciding whether two given partial Latin squares have either the same or isomorphic affine algebraic sets. To this end, we introduce a new pair of equivalence relations among partial Latin squares: being partial transpose and being partial isotopic. [ABSTRACT FROM AUTHOR]

The paper provides a complete solution to the existence problem of two orthogonal one-factorizations of a complete balanced multipartite graph K p × q . In particular, new classes of Howell designs are constructed. [ABSTRACT FROM AUTHOR]

We define a cover of a Latin square to be a set of entries that includes at least one representative of each row, column and symbol. A cover is minimal if it does not contain any smaller cover. A partial transversal is a set of entries that includes at most one representative of each row, column and symbol. A partial transversal is maximal if it is not contained in any larger partial transversal. We explore the relationship between covers and partial transversals. We prove the following: (1) The minimum size of a cover in a Latin square of order n is n + a if and only if the maximum size of a partial transversal is either n - 2 a or n - 2 a + 1 . (2) A minimal cover in a Latin square of order n has size at most μ n = 3 (n + 1 / 2 - n + 1 / 4) . (3) There are infinitely many orders n for which there exists a Latin square having a minimal cover of every size from n to μ n . (4) Every Latin square of order n has a minimal cover of a size which is asymptotically equal to μ n . (5) If 1 ⩽ k ⩽ n / 2 and n ⩾ 5 then there is a Latin square of order n with a maximal partial transversal of size n - k . (6) For any ε > 0 , asymptotically almost all Latin squares have no maximal partial transversal of size less than n - n 2 / 3 + ε . [ABSTRACT FROM AUTHOR]

We consider designs for cancer trials which allow each medical centre to treat only a limited number of cancer types with only a limited number of drugs. We specify desirable properties of these designs, and prove some consequences. Then we give several different constructions. Finally we generalize this to three or more factors, such as biomarkers. [ABSTRACT FROM AUTHOR]

Yantrams have been used to generate mixture designs in the interior of a simplex. In this note, we show a connection between Parshvanath yantram and a particular partially balanced incomplete block design. This block design is rather special and somewhat unexpected due to the feature that sum of the treatment symbols in any block is constant. [ABSTRACT FROM AUTHOR]