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Let $m$ be a positive integer such that $p$ does not divide $m$ where $p$ is prime. In this paper we find the number of conjugacy classes of completely reducible cyclic subgroups in GL$(2, q)$ of order $m$, where $q$ is a power of $p$.
we construct infinitely many non-isotrivial families of abelian varieties of $GL_2$-type over four punctured projective lines with bad reduction of type-$(1/2)_\infty$ via $p$-adic Hodge theory and Langlands correspondence. They lead to…
We study a two-boundary extension of the Temperley-Lieb algebra which has recently arisen in statistical mechanics. This algebra lies in a quotient of the affine Hecke algebra of type C and has a natural diagrammatic representation. The…
We prove that the parameter $x$ of a tight set $\mathcal{T}$ of a hyperbolic quadric $\mathsf{Q}^+(2n+1,q)$ of an odd rank $n+1$ satisfies ${x\choose 2}+w(w-x)\equiv 0\mod q+1$, where $w$ is the number of points of $\mathcal{T}$ in any…
We prove that the set of CM points on the Shimura curve associated to an Eichler order inside an indefinite quaternion $\mathbb{Q}$-algebra, is in bijection with the set of certain classes of $p$-adic binary quadratic forms, where $p$ is a…
In a recent paper, it is shown that functions of the form $L_1(x^3)+L_2(x^9)$, where $L_1$ and $L_2$ are linear, are a good source for construction of new infinite families of APN functions. In the present work we study necessary and…
The interplay between coding theory and $t$-designs has attracted a lot of attention for both directions. It is well known that the supports of all codewords with a fixed weight in a code may hold a $t$-design. In this paper, by determining…
We introduce a new class of possibly infinite dimensional Lie algebras and study their structural properties. Examples of this new class of Lie algebras are finite dimensional simple Lie algebras containing a nonzero split torus, affine and…
We describe some recent computer investigations with the `Constraint Logic Programming over Finite Domains' -- CLP(FD) -- library in the Prolog programming environment to search for new simple Lie algebras over the field $\GF(2)$ of $2$…
For any integer $\rho \geq 1$ and for any prime power q, the explicit construction of a infinite family of completely regular (and completely transitive) q-ary codes with d=3 and with covering radius $\rho$ is given. The intersection array…
Projective two-weight linear codes are closely related to finite projective spaces and strongly regular graphs. In this paper, a family of $q$-ary projective two-weight linear codes is presented, where $q$ is a power of 2. The parameters of…
In this paper, we present an infinite dimensional associative diagram algebra that satisfies the relations of the generalized Temperley--Lieb algebra having a basis indexed by the fully commutative elements (in the sense of Stembridge) of…
In this paper, using group actions, we introduce a new method for constructing partial geometric designs (sometimes referred to as $1\frac{1}{2}$-designs). Using this new method, we construct several infinite families of partial geometric…
Let $n$ be a prime power, $r$ be a prime with $r\mid n-1$, and $\varepsilon\in (0,1/2)$. Using the theory of multiplicative character sums and superelliptic curves, we construct new codes over $\mathbb F_r$ having length $n$, relative…
This paper is mainly devoted to constructions of \(q\)-analogs of group divisible designs and their applications. We give a complete description of the action of \(G=\GL(m,q^l)\) on \(\Omega_k^{k-1}\), where $3\leq k\leq \min\left\lbrace…
We present family of automatic sequences that define algebraic continued fractions in characteristic 2. This family is constructed from ultimately period words and contains the period-doubling sequence.
If C is a binary linear code, let C^2 be the linear code spanned by intersections of pairs of codewords of C. We construct an asymptotically good family of binary linear codes such that, for C ranging in this family, the C^2 also form an…
We compute the characters of real irreducible representations of SL(2,q), the special linear group on q letters, for an odd prime $q$. Moreover, we give the dimensions of these irreducible representations under the actions of cyclic…
The primary focus of this paper is overpartitions, a type of partition that plays a significant role in $q$-series theory. In 2006, Treneer discovered an explicit infinite family of congruences of overpartitions modulo $5$. In our research,…
We construct an infinite family of imaginary quadratic number fields with 2-class groups of type (2,2,2) whose Hilbert 2-class fields are finite.