Narrow Results

Association schemes are combinatorial objects that allow us solve problems in several branches of mathematics. They have been used in the study of permutation groups and graphs and also in the design of experiments, coding theory, partition designs etc. In this paper we show some techniques for computing properties of association schemes. The main framework arises from the fact that we can characterize completely the Bose–Mesner algebra in terms of a zero-dimensional ideal. A Gröbner basis of this ideal can be easily derived without the use of Buchberger algorithm in an efficient way. From this statement, some nice relations arise between the treatment of zero-dimensional ideals by reordering techniques (FGLM techniques) and some properties of the schemes such as P-polynomiality, and minimal generators of the algebra.

In this paper, we will investigate thin coherent configurations. We will show that thin coherent configurations are equivalent to finite connected groupoids. Also, we will investigate their representations.

This article discusses the representation theory of noncommutative algebras reality-based algebras with positive degree map over their field of definition. When the standard basis contains exactly two nonreal elements, the main result expresses the noncommutative simple component as a generalized quaternion algebra over its field of definition. The field of real numbers will always be a splitting field for this algebra, but there are noncommutative table algebras of dimension $6$ with rational field of definition for which it is a division algebra. The approach has other applications, one of which shows noncommutative association scheme of rank $7$ must have at least three symmetric relations.

A new method for the construction of the binary quantum stabilizer codes is provided, where the construction is based on Abelian and non-Abelian groups association schemes. The association schemes based on non-Abelian groups are constructed by bases for the regular representation from U_6n, T_4n, V_8n and dihedral D_2n groups. By using Abelian group association schemes followed by cyclic groups and non-Abelian group association schemes a list of binary stabilizer codes up to 40 qubits is given in tables 4, 5 and 10. Moreover, several binary stabilizer codes of minimum distances 5, 7 and 8 with good quantum parameters is presented. The preference of this method specially for Abelian group association schemes is that one can construct any binary quantum stabilizer code with any distance by using the commutative structure of association schemes.

Table Algebras (A,B) with distinct non-real basis elements a, b∈ B satisfying , (m, n∈ ℝ⁺) are studied in this paper. The two cases where either a is real and b is non-real or a, b are both real were completely classified by Arad and Blau. Here, we study the remaining case where a and b are non-real.

Table Algebras (A, B) with distinct non-trivial basis elements a, b ∈ B satisfying either or for some m, n ∈ ℝ⁺ are studied in this paper. The case was partially classified by Arad and Blau, and the case was partially classified by Arad and Cohen.

In this paper we prove the Krull-Schmidt type theorems for direct products of association schemes and tensor products of table algebras.

In this paper we give a brief overview of the theory of Majorana representations of finite groups.

We present an overview of results concerning partial periodic correlation of pseudorandom sequences, ranging from classical results on binary m–sequences to recent results on the first two partial period correlation moments of the sequences belonging to families A, B and C defined over Galois Rings. The use of Association Schemes provides us with a new uniform technique for analyzing the sequence families A, B and C.

These correlation moments have applications in synchronisation performance of CDMA communication systems using phase shift keying.