Related papers: The Projective General Linear Group $\mathrm{PGL}_…
We consider the ring of invariants of n points on the projective line. The space (P^1)^n // PGL_2 is perhaps the first nontrivial example of a Geometry Invariant Theory quotient. The construction depends on the weighting of the n points.…
Let X be a non-empty finite set, E be a finite dimensional euclidean vector space and G a finite subgroup of O(E), the orthognal group of E. Suppose GG={U_i | i in X} is a finite set of linear lines in E and an orbit of G on which its…
Invariant notions of a class of Segre varieties $\Segrem(2)$ of PG(2^m - 1, 2) that are direct products of $m$ copies of PG(1, 2), $m$ being any positive integer, are established and studied. We first demonstrate that there exists a…
The main result here is a characterisation of binary $2$-neighbour-transitive codes with minimum distance at least $5$ via their minimal subcodes, which are found to be generated by certain designs. The motivation for studying this class of…
We discuss quantum two-block codes, a large class of CSS codes constructed from two commuting square matrices.Interesting families of such codes are generalized-bicycle (GB) codes and two-block group-algebra (2BGA) codes, where a cyclic…
The dominant theme of this thesis is the construction of matrix representations of finite solvable groups using a suitable system of generators. For a finite solvable group $G$ of order $N = p_{1}p_{2}\dots p_{n}$, where $p_{i}$'s are…
The prime graph of a finite group $G$ is denoted by $\ga(G)$. Also $G$ is called recognizable by prime graph if and only if each finite group $H$ with $\ga(H)=\ga(G)$, is isomorphic to $G$. In this paper, we classify all finite groups with…
We consider the geometric problem of determining the maximum number $n_q(r,h,f;s)$ of $(h-1)$-spaces in the projective space $\operatorname{PG}(r-1,q)$ such that each subspace of codimension $f$ does contain at most $s$ elements. In coding…
Combinatorial $t$-designs have wide applications in coding theory, cryptography, communications and statistics. It is well known that the supports of all codewords with a fixed weight in a code may give a $t$-design. In this paper, we first…
Few-weight codes over finite chain rings are associated with combinatorial objects such as strongly regular graphs (SRGs), strongly walk-regular graphs (SWRGs) and finite geometries, and are also widely used in data storage systems and…
The $k$-Weisfeiler-Leman ($k$-WL) graph isomorphism test hierarchy is a common method for assessing the expressive power of graph neural networks (GNNs). Recently, GNNs whose expressive power is equivalent to the $2$-WL test were proven to…
Combinatorial designs are closely related to linear codes. In recent year, there are a lot of $t$-designs constructed from certain linear codes. In this paper, we aim to construct $2$-designs from binary three-weight codes. For any binary…
For a smooth affine algebraic group $G$ over an algebraically closed field, we consider several two-variables generalizations of the affine Grassmannian $G(\!(t)\!)/G[\![t]\!]$, given by quotients of the double loop group…
Codes are crucial in many areas of applications. Different types of codes are designed to meet specific needs, which makes them more effective and useful. Linear codes are extensively used in data storage systems. Identifying codes are…
The set of functions parameterized by a linear fully-connected neural network is a determinantal variety. We investigate the subvariety of functions that are equivariant or invariant under the action of a permutation group. Examples of such…
The nonsoluble length $\lambda(G)$ of a finite group $G$ is defined as the minimum number of nonsoluble factors in a normal series of $G$ each of whose quotients either is soluble or is a direct product of nonabelian simple groups. The…
We obtain certain algebraic invariants relevant to study codes on subgroups of weighted projective tori inside an $n$-dimensional weighted projective space. As application, we compute all the main parameters of generalized toric codes on…
In this paper, we examine a class of doubly resolvable combinatorial objects. Let $t, k, \lambda, s$ and $v$ be nonnegative integers, and let $X$ be a set of $v$ symbols. A generalized Howell design, denoted $t$-$GHD_{k}(s,v;\lambda)$, is…
The paper gives a succinct appraisal of the properties of the projective line defined over the direct product ring $R\_{\triangle} \equiv$ GF(2)$\otimesGF(2)\otimes$GF(2). The ring is remarkable in that except for unity, all the remaining…
The concept of Generalized Inverse based Decoding (GID) is introduced, as an algebraic framework for the syndrome decoding problem (SDP) and low weight codeword problem (LWP). The framework has ground on two characterizations by generalized…