In this paper, the notion of 2-tuples distribution matrices of the rotation symmetric orbits is proposed, by using the properties of the 2-tuples distribution matrix, a new characterization of 2-resilient rotation symmetric Boolean functions is demonstrated. Based on the new characterization of 2-resilient rotation symmetric Boolean functions, constructions of 2-resilient rotation symmetric Boolean functions (RSBFs) are further studied, and new 2-resilient rotation symmetric Boolean functions with prime variables are constructed.

Variable strength orthogonal array, as a special form of variable strength covering array, plays an important role in computer software testing and cryptography. In this paper, we study the construction of variable strength orthogonal arrays with strength two containing strength greater than two by Galois field and construct some variable strength orthogonal arrays with strength l containing strength greater than l by Fan-construction.

Quantum combinatorial designs are gaining popularity in quantum information theory. Quantum Latin squares can be used to construct mutually unbiased maximally entangled bases and unitary error bases. Here we present a general method for constructing quantum Latin arrangements from irredundant orthogonal arrays. As an application of the method, many new quantum Latin arrangements are obtained. We also find a sufficient condition such that the improved quantum orthogonal arrays [10] are equivalent to quantum Latin arrangements. We further prove that an improved quantum orthogonal array can produce a quantum uniform state.

In this paper, by using the properties of the cyclic Hadamard matrices of order 4t, an infinite class of (4t-1)-variable 2-resilient rotation symmetric Boolean functions is constructed, and the nonlinearity of the constructed functions are also studied. To the best of our knowledge, this is the first class of direct constructions of 2-resilient rotation symmetric Boolean functions. The spirit of this method is different from the known methods depending on the solutions of an equation system proposed by Du Jiao, et al. Several situations are examined, as the direct corollaries, three classes of (4t-1)-variable 2-resilient rotation symmetric Boolean functions are proposed based on the corresponding sequences, such as m sequences, Legendre sequences, and twin primes sequences respectively.

In this work, we introduce notions of quantum frequency arrangements consisting of quantum frequency squares, cubes, hypercubes and a notion of orthogonality between them. We also propose a notion of quantum mixed orthogonal array (QMOA). By using irredundant mixed orthogonal array proposed by Goyeneche et al. we can obtain k-uniform states of heterogeneous systems from quantum frequency arrangements and QMOAs. Furthermore, some examples are presented to illustrate our method.

In the theory of orthogonal arrays, an orthogonal array (OA) is called schematic if its rows form an association scheme with respect to Hamming distances. In this paper, we study the Hamming distances of any two rows in an OA, construct some schematic OAs of strength two and give the positive solution to the open problem for classifying all schematic OAs. Some examples are given to illustrate our methods.

In this study, we developed a new orthogonal partition concept for asymmetric orthogonal arrays and used it for the construction of orthogonal arrays for the first time. Permutation matrices and the Kronecker product were also successfully and skillfully used as our main tools. Hence, a new general iterative construction method for asymmetric orthogonal arrays of high strength was developed, and some new infinite families of orthogonal arrays of strength 3 and several new orthogonal arrays of strength 4, 5, and 6 were obtained.

Orthogonal arrays and orthogonal partitions have great significance in communications and coding theory. In this letter, by using a generalized orthogonal partition, Latin squares and orthogonal Latin squares, we present an iterative construction method of orthogonal arrays of strength t and orthogonal partitions. As an application of the method, more orthogonal arrays of strength t and orthogonal partitions than the existing methods can be constructed.

The orthogonal array is an important object in combinatorial design theory, and it is applied to many fields, such as computer science, coding theory and cryptography etc. This paper mainly studies the existence of the mixed orthogonal arrays of strength two with seven factors and presents some new constructions. Consequently, a few new mixed orthogonal arrays are obtained.

Orthogonal arrays have important applications in statistics and computer science, as well as in coding theory. In this letter, a new construction method of symmetric orthogonal arrays of strength t is proposed, which is a concatenation of two orthogonal partitions according to a latin square. As far as we know, this is a new construction of symmetric orthogonal arrays of strength t, where t is a given integer. Based on the different latin squares, we also study the enumeration problem of orthogonal partitions, and a lower bound on the count of orthogonal partitions is derived.

Orthogonal Arrays (OAs) have been playing important roles in the field of experimental design. It has been known that OAs are closely related to error-correcting codes. Therefore, many OAs can be constructed from error-correcting codes. But these OAs are suitable for only cases that equal interaction effects can be assumed, for example, all two-factor interaction effects. Since these cases are rare in experimental design, we cannot say that OAs from error-correcting codes are practical. In this paper, we define OAs with unequal strength. In terms of our terminology, OAs from error-correcting codes are OAs with equal strength. We show that OAs with unequal strength are closer to practical OAs than OAs with equal strength. And we clarify the relation between OAs with unequal strength and unequal error-correcting codes. Finally, we propose some construction methods of OAs with unequal strength from unequal error-correcting codes.

In this paper, a definitce relation between the TSP's optimal solution and the attracting region in the parameters space of TSP's energy function is discovered. An many attracting region relating to the global optimal solution for TSP is founded. Then a neural network algorithm with the optimized parameters by using Orthogonal Array Table Method is proposed and used to solve the Travelling Salesman Problem (TSP) for 30, 31 and 300 cities and Map-coloring Problem (MCP). These results are very satisfactory.