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Cyclic codes are among the most important families of codes in coding theory for both theoretical and practical reasons. Despite their prominence and intensive research on cyclic codes for over a half century, there are still open problems…
Multi-Higgs models equipped with global symmetry groups, either exact or softly broken, offer a rich framework for constructions beyond the Standard Model and lead to remarkable phenomenological consequences. Knowing all the symmetry…
The study of symmetric configurations $v_3$ with block size 3 has a long and rich history. In this paper we consider two colouring problems which arise naturally in the study of these structures. The first of these is weak colouring, in…
A group is properly 3-realizable if it is the fundamental group of a compact polyhedron whose universal covering is proper homotopically equivalent to some 3-manifold. We prove that when such a group is also quasi-simply filtered then it…
The projective special linear group $\PSL_2(n)$ is $2$-transitive for all primes $n$ and $3$-homogeneous for $n \equiv 3 \pmod{4}$ on the set $\{0,1, \cdots, n-1, \infty\}$. It is known that the extended odd-like quadratic residue codes are…
Let $p$ be an odd prime and let $\mathbf{B}$ be a $p$-block of a finite group, such that $\mathbf{B}$ has cyclic defect groups. We describe the self-dual indecomposable $\mathbf{B}$-modules and for each such module determine whether it is…
Constacyclic codes over finite fields are of theoretical importance as they are closely related to a number of areas of mathematics such as algebra, algebraic geometry, graph theory, combinatorial designs and number theory. However, the…
In this paper, we further investigate the constructions on three-dimensional $(u\times v\times w,k,1)$ optical orthogonal codes with the at most one optical pulse per wavelength/time plane restriction (briefly AM-OPP $3$-D $(u\times v\times…
This paper is an attempt to compute the decomposition numbers of the blocks of the symmetric group which have "small defect"; that is, blocks of weight smaller than the characteristic. We present various methods for computing such…
Large sets of combinatorial designs has always been a fascinating topic in design theory. These designs form a partition of the whole space into combinatorial designs with the same parameters. In particular, a large set of block designs,…
This paper is concerned with the limit cycles for planar semi-quasi-homogeneous polynomial systems. We give some explicit criteria for the nonexistence and existence of periodic orbits. Let $N=N(p,q,m,n)$ be the maximum number of limit…
In this work it is shown that there are symmetries beyond the Euclidean group $E\left(3\right)$ in 3-body problem, and by extension in many-body problem, with inverse squared distance inter particle force. The symmetries in 3-body problem…
Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems and communication systems as they have efficient encoding and decoding algorithms. In this paper, we settle an open problem…
It has recently been suggested that an exactly solvable problem characterized by a new quantum number may underlie the electronic shell structure observed in the mass spectra of medium-sized sodium clusters. We investigate whether the…
We provide a method to construct $t$-designs from weighing matrices and association schemes. One instance of our method can produce a $3$-design from any (symmetric or skew-symmetric) conference matrix, thereby providing a partial answer to…
An integer feasibility problem is a fundamental problem in many areas, such as operations research, number theory, and statistics. To study a family of systems with no nonnegative integer solution, we focus on a commutative semigroup…
Motivated by the application in geometric orthogonal codes (GOCs), Wang et al. introduced the concept of generalized perfect difference families (PDFs), and established the equivalence between GOCs and a certain type of generalized PDFs…
We point out that the moduli spaces of all known 3d $\mathcal{N}=$ 8 and $\mathcal{N}=$ 6 SCFTs, after suitable gaugings of finite symmetry groups, have the form $\mathbb{C}^{4r}/\Gamma$ where $\Gamma$ is a real or complex reflection group…
Quasipatterns (two-dimensional patterns that are quasiperiodic in any spatial direction) remain one of the outstanding problems of pattern formation. As with problems involving quasiperiodicity, there is a small divisor problem. In this…
It is well known that the problem of determining the weight distributions of families of cyclic codes is, in general, notoriously difficult. An even harder problem is to find characterizations of families of cyclic codes in terms of their…