Latin Squares

A Sudoku grid is a constrained Latin square. In this paper a reduced Sudoku grid is described, the properties of which differ, through necessity, from that of a reduced Latin square. The Sudoku symmetry group is presented and applied to determine a mathematical relationship between the number of reduced Sudoku grids and the total number of Sudoku grids for any size. This relationship simplifies the enumeration of Sudoku grids and an example of the use of this method is given.

A Sudoku grid is a 9 × 9 Latin square further constrained to have nine non-overlapping 3 × 3 mini-grids each of which contains the values 1-9. In ∆-Quasi-Magic Sudoku a further constraint is imposed such that every row, column and diagonal in each mini-grid sums to an integer in the interval [15-∆, 15+∆]. The problem of proving certain (computationally known) results for ∆ = 2 concerning mini-grids and bands (rows of mini-grids) was posed at the British Combinatorial Conference in 2007. These proofs are presented and extensions of these provide a full combinatorial enumeration for the total number of completed 2-Quasi-Magic Sudoku grids. It is also shown that there are 40 isomorphism classes of completed 2-Quasi-Magic Sudoku grids.

In , the author provided a Gray code for the set of n-length permutations with a given number of left-to-right minima in inversion array representation. In this paper, we give the first Gray code for the set of n-length permutations with a given number of left-to-right minima in one-line representation. In this code, each permutation is transformed into its successor by a product with a transposition or a cycle of length three. Also a generating algorithm for this code is given.

We give a Gray code and constant average time generating algorithm for derangements, i.e., permutations with no ÿxed points. In our Gray code, each derangement is transformed into its successor either via one or two transpositions or a rotation of three elements. We generalize these results to permutations with number of ÿxed points bounded between two constants.

We give a Gray code and constant average time generating algorithm for derangements, i.e., permutations with no ÿxed points. In our Gray code, each derangement is transformed into its successor either via one or two transpositions or a rotation of three elements. We generalize these results to permutations with number of ÿxed points bounded between two constants.

This volume is a C++ Programming Library for the study of the Orbit Theory of Natural Numbers. Over 100 C++ programs were written during the research that led to Orbit Theory in various categories from fundamentals to applications. This has been uploaded in PDF format for easy download and copying of program code to run on any device that handles C++ code. This file will be updated periodically as new programs are added. In this printed form, the code has been copyrighted. However, the code can be downloaded and used as is without charge provided that the author of the program has been credited appropriately.

Abstract: Prescott et al.(Technometrics 78: 268-276, 1993) proposed D-optimal orthogonally blocked designs in two blocks for Scheffé's quadratic mixture model with four components. Chan and Sandhu (J. Appl. Statist. 26 (1): 19-34, 1999) discussed the properties of D-, A-...

Given a one-factorization $\mathcal{F}$ of the complete bipartite graph $K_{n,n}$, let ${\sf pf}(\mathcal{F})$ denote the number of Hamiltonian cycles obtained by taking pairwise unions of perfect matchings in $\mathcal{F}$. Let ${\sf pf}(n)$ be the maximum of ${\sf pf}(\mathcal{F})$ over all one-factorizations $\mathcal{F}$ of $K_{n,n}$. In this work we prove that ${\sf pf}(n)\geq n^2/4$, for all $n\geq 2$.

In this paper, we constructed a type of parity-check matrices of QC-LDPC codes based on the progressive edge-growth (PEG) algorithm with suggested unreliable factors incorporated. The proposed algorithm can eliminate short cycles efficiently and gain low errorfloors. With this method, we can avoid the unreliable information transferred in decoding as far as possible. The proposed QC-LDPC codes are hardware-friendly due to the benefit from the quasi-cyclic structure of such codes. Simulation results demonstrate that the codes constructed with the proposed algorithm have better performance than those constructed with the PEG algorithm over the additive white Gaussian noise (AWGN) channel and the PR4-equalized magnetic recording channel (MRC).

Abstract: Prescott et al.(Technometrics 78: 268-276, 1993) proposed D-optimal orthogonally blocked designs in two blocks for Scheffé's quadratic mixture model with four components. Chan and Sandhu (J. Appl. Statist. 26 (1): 19-34, 1999) discussed the properties of D-, A-...

Once an application steps out of the bounds of a single-computer box, its external communication is immediately exposed to a multitude of outside observers with various intentions, good or bad. In order to protect sensitive data while these are en route, applications invoke different methods. In today's world, most of the means of secure data and code storage and distribution rely on using cryptographic schemes, such as certificates or encryption keys. Thus, cryptography mechanisms form a foundation upon which many important aspects of a solid security system are built. Cryptography is the science of writing in secret code and is an ancient art. Some experts argue that cryptography appeared spontaneously sometime after writing was invented, with applications ranging from diplomatic missives to wartime battle plans. It is no surprise, then; those new forms of cryptography came soon after the widespread development of computer communications. There are two basic types of cryptography: Symmetric Key and Asymmetric Key. Symmetric key algorithms are the quickest and most commonly used type of encryption. Here, a single key is used for both encryption and decryption. There are few well-known symmetric key algorithms i.e. DES, RC2, RC4, IDEA etc. This paper describes cryptography and its types and then proposes a new symmetric key algorithm X-S cryptosystem based on stream cipher. Algorithms for both encryption and decryption are provided here. The advantages of this new algorithm over the others are also explained.

It is well known that given a Steiner triple system (STS) one can define a binary operation * upon its base set by assigning x * x = x for all x and x * y = z, where z is the third point in the block containing the pair {x, y}. The same can be done for Mendelsohn triple systems (MTSs) as well as hybrid triple systems (HTSs), where (x, y) is considered to be ordered. In the case of STSs and MTSs, the operation is a quasigroup, however this is not necessarily the case for HTSs. In this paper we study the binary operation induced by HTSs. It turns out that each such operation * satisfies y ∈ {x * (x * y), (x * y) * x} and y ∈ {(y * x) * x, x * (y * x)} for all x and y from the base set. We call every binary operation that fulfils this condition hybridly symmetric. Not all idempotent hybridly symmetric operations can be obtained from HTSs. We show that these operations correspond to decompositions of a complete digraph into certain digraphs on three vertices. However, an idempotent hybridly symmetric quasigroup always comes from an HTS. The corresponding HTS is then called a latin HTS (LHTS). The core of this paper is the characterization of LHTSs and the description of their existence spectrum.

We present the number of totally symmetric quasigroups (equivalently, totally symmetric Latin squares) of order 16, as well as the number of isomorphism classes of such objects. Totally symmetric quasigroups of orders up to and including 16 that are (respectively) medial, idempotent, and unipotent are also enumerated.

We give new constructions for regular group divisible designs, pairwise balanced designs, generalized Bhaskar Rao supplementary difference sets and generalized weighing matrices. In particular if p is a prime power and q divides p-1 we show the following exist: (i) GDD(2(p2 +p+ 1), 2(p2 +p+ 1), rp2, 2p2, λ1 = p2λ, λ2 = (p2-p)r, m = p2 + p+ 1, n = 2), r = 1,2; (ii) GDD(q(p+ 1), q(p+ 1), p(q-1), p(q-1), λ1 = (q-1)(q-2), λ2 = (p-1)(q-1)2/q, m = q, n = p+1); (iii) PBD(21, 10; K), K = {3, 6, 7} and PBD(78, 38; K), K = {6, 9, 45}; (iv) GW(vk, k2; EA(k)) whenever a (v, k, λ)-difference set exists and k is a prime power; (v) PBIBD(vk2, vk2, k2, k2; λ1 = 0, λ2 = λ, λ3 = k) whenever a (v, k, λ)-difference set exists and k is a prime power; (vi) we give a GW(21; 9; Z3).

A perfect code in a graph Γ = (V, E) is a subset C of V such that no two vertices in C are adjacent and every vertex in V \ C is adjacent to exactly one vertex in C. A subgroup H of a group G is called a subgroup perfect code of G if there exists a Cayley graph of G which admits H as a perfect code. Equivalently, H is a subgroup perfect code of G if there exists an inverse-closed subset A of G containing the identity element such that (A, H) is a tiling of G in the sense that every element of G can be uniquely expressed as the product of an element of A and an element of H. In this paper we obtain multiple results on subgroup perfect codes of finite groups, including a few necessary and sufficient conditions for a subgroup of a finite group to be a subgroup perfect code, a few results involving 2-subgroups in the study of subgroup perfect codes, and several results on subgroup perfect codes of metabelian groups, generalized dihedral groups, nilpotent groups and 2-groups.

A partial Latin square is premature if it has no completion, but it admits a completion after removing any of its symbols. This type of partial Latin square has been introduced by Brankovic, Honik, Miller and Rosa [Ars Combinatoria, to appear] where the authors showed that the number of empty cells in an n x n premature latin square is at least 371-4. vVe improve this lower bound to 7n/2-0(71,).

This paper describes the construction and enumeration of mixed orthogonal arrays (MOA) to produce optimal experimental designs. A MOA is a multiset whose rows are the different combinations of factor levels, discrete values of the variable under study, having very well defined features such as symmetry and strength three (all main interactions are taken in consideration). The applied methodology blends the fields of combinatorics and group theory by applying the ideas of orbits, stabilizers and isomorphisms to array generation and enumeration. Integer linear programming was used in order to exploit the symmetry property of the arrays under study. The backtrack search algorithm was used to find suitable arrays in the underlying space of possible solutions. To test the performance of the MOAs, an engineered system was used as a case study within the stage of parameter design. The analysis showed how the MOAs were capable of meeting the fundamental engineering design axioms and principles, creating optimal experimental designs within the desired context.

This paper proposes a decoding algorithm for nonbinary low-density parity-check (NB-LDPC) codes, aiming to improve the error rate performance for NAND flash memory. Several NB-LDPC decoding methods for NAND flash memory have been studied. Some approaches rely on hard decisions, and these are relatively simple but do not have a good error rate performance. Others are based on soft decisions that require multiple reads for each flash memory cell, leading to significant memory throughput degradation. To improve the error rate performance without suffering performance degradation owing to multiple reads, an iterative pseudo-soft-reliability-based decoding algorithm is proposed. Using Galois field addition to calculate the Hamming distance at the initialization, the proposed algorithm not only improves the error rate performance but also reduces the average number of iterations compared with those of conventional hard-decision-based decoding algorithms. INDEX TERMS Error correction codes, Hamming distance, hard decision, iterative hard-reliability-based majority-logic decoding algorithm, NAND flash memory, nonbinary low-density parity-check codes.

It is well known that all n×n partial Latin squares with at most n−1 entries are completable. Our intent is to extend this well known statement to partial Latin cubes. We show that if an n×n×n partial Latin cube contains at most n − 1 entries, no two of which occupy the same row, then the partial Latin cube is completable. Also included in this paper is the problem of completing 2×n×n partial Latin boxes with at most n − 1 entries. Given certain sufficient conditions, we show when such partial Latin boxes are completable and then extendable to a deeper Latin box.

The first known families of cages arised from the incidence graphs of generalized polygons of order q, q a prime power. In particular, (q + 1, 6)-cages have been obtained from the projective planes of order q. Morever, infinite families of small regular graphs of girth 5 have been constructed performing algebraic operations on F q. In this paper, we introduce some combinatorial operations to construct new infinite families of small regular graphs of girth 7 from the (q + 1, 8)-cages arising from the generalized quadrangles of order q, q a prime power.

In our research, we present a method to build a private key cryptosystem, this method is based on the dynamic priority coloured petri net to produced complex long key sequence. In our research, there are two parts, the first one produced random number based on four colored petri nets which have the same design and one ordinary petri net , all petri nets is combined in such a way that generate long nonlinear random key that can be used in encryption process. The same petri nets model is used in both encryption/decryption algorithm and received the same input vector to produce the same output, this input vector must be shared by secured channel between sender and receiver. The key generated is more complex and tested by cryptography randomness tests. In the second part, the plaintext has been converted to gray code and permutated in such a way to increase the complexity of attack it. Finally, the permutated plaintext is combined with the key by exclusive or operation to produce cipher text.

Literature shows that there are several ways of generating Latin squares, but there is not enough implementation about Supersymmetric Latin squares. This paper proposes a mathematical algorithm to construct Super-symmetric Latin squares of order 2 ୬ by substituting blocks of order 2 n which has the basic properties of a recursive algorithm. The proposed algorithm was tested for a large number of orders and the results proved that the algorithm could be generalized for any input order n where n is a positive integer.

For every positive integer n greater than 4 there is a set of Latin squares of order n such that every permutation of the numbers 1,. .. , n appears exactly once as a row, a column, a reverse row or a reverse column of one of the given Latin squares. If n is greater than 4 and not of the form p or 2p for some prime number p congruent to 3 modulo 4, then there always exists a Latin square of order n in which the rows, columns, reverse rows and reverse columns are all distinct permutations of 1,. .. , n, and which constitute a permutation group of order 4n. If n is prime congruent to 1 modulo 4, then a set of (n − 1)/4 mutually orthogonal Latin squares of order n can also be constructed by a classical method of linear algebra in such a way, that the rows, columns, reverse rows and reverse columns are all distinct and constitute a permutation group of order n(n − 1).

Sudoku problems are some of the most known and enjoyed pastimes, with a never diminishing popularity, but, for the last few years those problems have gone from an entertainment to an interesting research area, a twofold interesting area, in fact. On the one side Sudoku problems, being a variant of Gerechte Designs and Latin Squares, are being actively used for experimental design, as in [8, 44, 39, 9]. On the other hand, Sudoku problems, as simple as they seem, are really hard structured combinatorial search problems, and thanks to their characteristics and behavior, they can be used as benchmark problems for refining and testing solving algorithms and approaches. Also, thanks to their high inner structure, their study can contribute more than studies of random problems to our goal of solving real-world problems and applications and understanding problem characteristics that make them hard to solve. In this work we use two techniques for solving and modeling Sudoku problems, namely, Constraint Satisfaction Problem (CSP) and Satisfiability Problem (SAT) approaches. To this effect we define the Generalized Sudoku Problem (GSP), where regions can be of rectangular shape, problems can be of any order, and solution existence is not guaranteed. With respect to the worst-case complexity, we prove that GSP with block regions of m rows and n columns with m = n is NP-complete. For studying the empirical hardness of GSP, we define a series of instance generators, that differ in the balancing level they guarantee between the constraints of the problem, by finely controlling how the holes are distributed in the cells of the GSP. Experimentally, we show that the more balanced are the constraints, the higher the complexity of solving the GSP instances, and that

Shapeless quasigroups are needed for cryptography purposes. In this paper, we construct shapeless quasigroups by the diagonal method from orthomorphisms over abelian groups. We use generalizations of Feistel networks as orthomorphisms. We introduce parameters into several types of Extended Feistel networks and Generalized Feistel-non linear feedback shift registers and, by suitable choice of the parameter values, different shapeless quasigroup can be used in every application.

Designing new cryptosystems and their cryptanalysis is the basic cycle of advancement in the field of cryptography. In this paper we introduce a block cipher based on the quasigroup transformations, which are defined by the matrix presentation of the quasigroup operations. This type of quasigroup presentation is suitable for constructing a block cipher since it doesn't require too much memory space to store all the necessary data, so it can be used even for lightweight cryptographic purposes. For now, we are considering only the quasigroups of order 4. Constructions with quasigroups of higer order and examination of the strengths and weaknesses of this design will be considered in next papers. Index Terms-block cipher, quasigroup, matrix form of quasigroup

Diagonal method and orthomorphisms Algebraic properties of the quasigroup (G , •) Different generalizations of the Feistel Network as orthomorphisms Some constructions of shapeless quasigroups Conclusions Complete mappings and orthomorphisms Definition (Mann (1949), Johnson et al (1961), Evans (1989)) A complete mapping of a quasigroup (group) (G , +) is a permutation φ : G → G such that the mapping θ : G → G defined by θ(x) = x + φ(x) (θ = I + φ, where I is the identity mapping) is again a permutation of G. The mapping θ is the orthomorphism associated to the complete mapping φ. A quasigroup (group) G is admissible if there is a complete mapping φ : G → G .

Random sampling and randomized experimentation are inextricably linked. Beginning with their common origins in the work of Fisher and Neyman from the 1920s and the 1930s, one can trace the development of parallel concepts and structures in the two areas (see Fienberg and Tanur [Bull. Int. Stat. Inst. 51 (1985) Art. ID 10.1; Int. Stat. Rev. 55 (1987) 75-96]). One of the more important lessons to be learned from the parallel concepts and structures is that they can profitably be linked and intertwined, with sampling embedded in experiments and formal experimental structures embedded in sampling designs. In this paper, we trace some of parallels between sampling theory and theory of experimental design. We then explore some of the ways that experimental and sampling structures have been combined in statistical practice and the principles that underlie their combination; we also make some suggestions toward the improvement of practice.

Salah satu faktor yang mengakibatkan kehilangan hasil pada produk pertanian tanaman pangan khususnya padi adalah proses penggilingan. Mesin penggilingan yang sudah tua merupakan salah satu penyebab terjadinya kehilangan hasil dan kurang bagusnya produk beras yang dihasilkan. Untuk mengatasi masalah tersebut, pemerintah melalui dinas pertanian memberikan bantuan mesin penggiling padi satu fase pada kelompok-kelompok tani tertentu. Namun demikian, sebagian dari mesin ini tidak difungsikan karena kapasitas mesin yang terlalu tinggi dan hasil kurang memuaskan. Tujuan dari kegiatan ini adalah melakukan pengujian kinerja mesin penggiling padi satu fase khususnya pada kualitas produk beras yang dihasilkan. Proses evaluasi dilakukan pada mesin yang bekerja pada kondisi putaran poros utama rata-rata 949 rpm serta dilakukan dengan dua kali ulangan. Bahan uji adalah gabah dengan kadar air rata-rata 13,5% yang terdiri atas 92% beras bernas, 6,65% beras kapang dan 1,05% kotoran. Hasil evaluasi yang dilakukan menunjukkan kapasitas input mesin adalah 1432,14 kg/jam dengan rendemen hasil rata-rata 63%. Analisis yang dilakukan pada beras pecah kulit (BPK) yang keluar dari husker adalah 82,86% BPK, 14,29% masih berupa gabah dan 2,8% sisanya adalah sekam. Kualitas beras hasil proses penggilingan terdiri atas 76% beras kepala, 13,44% beras patah dan 10,56% beras menir. Hasil ini menunjukkan bahwa kerja mesin penggiling padi ini sudah cukup bagus meskipun kapasitas mesin kurang sesuai untuk digunakan oleh petani dengan hasil yang masih kecil. Kata kunci: Mesin penggiling padi satu fase, kualitas beras

We give some constructions of new infinite families of group divisible designs, GDD(n, 2, 4; 1 , 2), including one which uses the existence of Bhaskar Rao designs. We show the necessary conditions are sufficient for 3 n 8. For n = 10 there is one missing critical design. If 1 > 2 , then the necessary conditions are sufficient for n ≡ 4, 5, 8 (mod 12). For each of n=10, 15, 16, 17, 18, 19, and 20 we indicate a small minimal set of critical designs which, if they exist, would allow construction of all possible designs for that n. The indices of each of these designs are also among those critical indices for every n in the same congruence class mod 12.

W e study critical sets of F-squares, latin squares and Youden squares. W e prove that some subsets of F-squares, latin squares and Youden squares are critical sets. W e consider minimal critical sets. W e show that some critical sets for certain types of F-squares and a certain class of Youden squares are minimal. W e study subsets of critical sets and their importance in the completion of a critical set to a full design. W e study the nest, influence, power and strongbox of these subsets, and their ability to hold information about the critical set. W e calculate the influence of elements and subsets of a critical set of a back circulant latin square, as well as the strongbox of a critical set of a back circulant latin square. W e solve the spectrum of the type Fin; l,n-1) of F-squares and give partial spectrums for type P(n;2,n-2). Connections to finite topology and graph theory are considered. W e show that certain collections of subsets of critical sets m a y form the basis of some topology on the critical set. T h e problem of ranking elements in a critical set is looked at, with some examples of possible rankings given. W e look very briefly at the notion of secret sharing schemes based on critical sets, and show how an access structure m a y be formed based on the properties of the influence of a set in a critical set. v vii In this paper the concepts of nest, influence, power and strongbox of a subset are introduced and studied. Chapters 3 and 5 are partly based on this paper. This author contributed about 75 percent to this paper.

Composite Circular hollow Steel tubes with and without GFRP infill for three different grades of Light weight concrete are tested for ultimate load capacity and axial shortening , under Cyclic loading. Steel tubes are compared for different lengths, cross sections and thickness. Specimens were tested separately after adopting Taguchi's L9 (Latin Squares) Orthogonal array in order to save the initial experimental cost on number of specimens and experimental duration. Analysis was carried out using ANN (Artificial Neural Network) technique with the assistance of Mini Tab-a statistical soft tool. Comparison for predicted, experimental & ANN output is obtained from linear regression plots. From this research study, it can be concluded that *Cross sectional area of steel tube has most significant effect on ultimate load carrying capacity, *as length of steel tube increased-load carrying capacity decreased & *ANN modeling predicted acceptable results. Thus ANN tool can be utilized for predicting ultimate load carrying capacity for composite columns.

Generalised Quasi-Cyclic Root-Check LDPC codes based on Progressive Edge Growth (PEG) techniques for blockfading channels are proposed. The proposed Root-Check LDPC codes are suitable for channels under = 3, 4 independent fadings per codeword which is a scenario that has not been previously considered. A generalised Quasi-Cyclic Root-Check structure is devised for = 3, 4 independent fadings. The proposed PEG-based algorithm for generalised Quasi-Cyclic Root-Check LDPC codes takes into account the specific structure, and designs a parity-check matrix with a reduced number of cycles. The performance of the new codes is investigated in terms of the Frame Error Rate (FER). Numerical results show that the codes constructed by the proposed algorithm perform about 0.5dB from the theoretical limit.

Wireless body area network (WBAN) is a promising network aiming at enhancing the communication in medical applications. It is adopted by medical organizations due to its flexibility in remotely monitoring patient health status. WBANs suffer from many limitations due to excessive channel impairments. Low density parity check (LDPC) codes are proposed to mitigate WBAN's impairments concerning the bit error rate, complexity, and dissipated energy. In this paper, a comprehensive performance analysis of various LDPC decoding algorithms is used to improve communication and reduce complexity in implant to implant WBAN channel. Moreover, a novel low complex LDPC decoding algorithm, which has a performance close to soft decision and a decoding time close to hard decision algorithms is proposed to minimize the dissipated energy. The proposed algorithm can be classified as a hybrid decision algorithm. The results demonstrate extensive analysis and comparisons between hard, soft, and hybrid decision algorithms. INDEX TERMS Low density parity check codes, wireless body area network, implementation efficient reliability ratio weighted bit flipping, forward error correction, turbo codes, hybrid decoding.

In this survey recent results about q-analogues of some classical theorems in extremal set theory are collected. They are related to determining the chromatic number of the q-analogues of Kneser graphs. For the proof one needs results on the number of 0-secant subspaces of point sets, so in the second part of the paper recent results on the structure of point sets having few 0-secant subspaces are discussed. Our attention is focussed on the planar case, where various stability results are given.

Orthogonal block designs for Scheffé's quadratic model have been considered previously by Draper et al. (1993), John (1984), Lewis et al. (1994) and Prescott, Draper, Dean, and Lewis (1993). Prescott and Draper (2004) obtained mixture component-amount designs via projections of standard mixture designs, viz., the simplex-lattice, the simplex-centroid and the orthogonally blocked mixture designs based on latin squares. Aggarwal, Singh, Sarin, and Husain (2009) considered the case of components assuming equal volume fractions and obtained mixture designs in orthogonal blocks using F-squares. In this paper, we construct orthogonal blocks of two and three mixture component-amount blends by projecting the class of four component mixture designs presented by Aggarwal et al. (2009).

It is well known that there does not exist a Boolean function f: Z_2^m ightarrow Z_2^n satisfying both basic cryptologic criteria, balancedness and perfect nonlinearity. In /9/ it was shown that, if we use as a domain quasigroup G instead of the group Z_2^n, one can find functions which are at the same time balanced and perfectly nonlinear. Such functions have completely flat difference table. We continue in our previous work, but we turn our attention to the worst case when all lines of Cayley table of G define so called linear structure of f (/5/). We solve this problem in both directions: We describe all such bijections f:G ightarrow Z_2^n, for a given quasigroup |G|=2^n, and describe such quasigroups for a given function f

In this paper we discuss pertinent questions closely related to well known RSA cryptosystem [5]. From practical point of view it is reasonable to use as a public exponent an integer s = 2 fc + 1, i.e., so called short exponent, with the lowest possible binary weight. The most common are for k = 1 and k = 2 4 , the two Fermat primes. In this paper we prove two theorems which give a percentage of acceptable public exponents s = 2 k + 1, 1 ^ fc ^ 1023 to two randomly selected primes of 512 bits each. In fact, our results are valid for arbitrary set of exponents s. We also present results of our experiments. In our simulation, for all such acceptable public exponents, the corresponding secret exponent t had a weight within the range of 451-567. Thus, although it is recommended in [8] not to use short public exponents, by our observation to use the attack based on continuos fractions is infeasible.

We use a free construction to prove the existence of perfect Steiner triple systems on a countably infinite point set. We use a specific countably infinite family of partial Steiner triple systems to start the construction, thus yielding 2 ℵ0 non-isomorphic perfect systems.

The Neyman-Fisher controversy considered here originated with the 1935 presentation of Jerzy Neyman's Statistical Problems in Agricultural Experimentation to the Royal Statistical Society. Neyman asserted that the standard ANOVA F-test for randomized complete block designs is valid, whereas the analogous test for Latin squares is invalid in the sense of detecting differentiation among the treatments, when none existed on average, more often than desired (i.e., having a higher Type I error than advertised). However, Neyman's expressions for the expected mean residual sum of squares, for both designs, are generally incorrect. Furthermore, Neyman's belief that the Type I error (when testing the null hypothesis of zero average treatment effects) is higher than desired, whenever the expected mean treatment sum of squares is greater than the expected mean residual sum of squares, is generally incorrect. Simple examples show that, without further assumptions on the potential outcomes, one cannot determine the Type I error of the F-test from expected sums of squares. Ultimately, we believe that the Neyman-Fisher controversy had a deleterious impact on the development of statistics, with a major consequence being that potential outcomes were ignored in favor of linear models and classical statistical procedures that are imprecise without applied contexts.

A famous conjecture of Caccetta and Häggkvist is that in a digraph on n vertices and minimum out-degree at least n r there is a directed cycle of length r or less. We consider the following generalization: in an undirected graph on n vertices, any collection of n disjoint sets of edges, each of size at least n r , has a rainbow cycle of length r or less. We focus on the case r = 3, and prove the existence of a rainbow triangle under somewhat stronger conditions than in the conjecture. For any fixed k and large enough n, we determine the maximum number of edges in an n-vertex edge-coloured graph where all colour classes have size at most k and there is no rainbow triangle. Moreover, we characterize the extremal graphs for this problem.

Orthogonal arrays of strength 3 permit estimation of all the main effects of the experimental factors free from confounding or contamination with 2-factor interactions. We introduce methods of using arithmetic formulations and Latin squares to construct mixed orthogonal arrays of strength 3. Although the methods could be well extended to computing larger arrays, we confine computing to at most 100 run orthogonal arrays for practical uses. We find new arrays with run sizes 80 and 96, each has many distinct factor levels.