Related papers: The Geometry of Homogeneous Two-Weight Codes
It is shown in this paper that, if $R$ is a Frobenius ring, then the quaternion ring $\mathcal{H}_{a,b}(R)$ is a Frobenius ring for all units $a,b \in R$. In particular, if $q$ is an odd prime power then $\mathcal{H}_{a,b}(\mathbb{F}_q)$ is…
Inspired by the Z2Z4-additive codes, linear codes over Z2^r x(Z2+uZ2)^s have been introduced by Aydogdu et al. more recently. Although these family of codes are similar to each other, linear codes over Z2^r x(Z2+uZ2)^s have some advantages…
We study the class of hyponormal 2-variable weighted shifts with two consecutive equal weights in the weight sequence of one of the coordinate operators. We show that under natural assumptions on the coordinate operators, the presence of…
In this work, we define a modification of a bordered construction for self-dual codes which utilises $\lambda$-circulant matrices. We provide the necessary conditions for the construction to produce self-dual codes over finite commutative…
We show that regular homogeneous two-weight $\mathbb{Z}_{p^k}$-codes where $p$ is odd and $k\geqslant 2$ with dual Hamming distance at least four do not exist. The proof relies on existence conditions for the strongly regular graph built on…
We follow up on previous work which found that commonly used graph evolution moves lead to conserved quantities that can be expressed in terms of the braiding of the graph in its embedding space. We study non-embedded graphs under three…
In this article we mainly study linear codes over $\mathbb{F}_{2^n}$ and their binary subfield codes. We construct linear codes over $\mathbb{F}_{2^n}$ whose defining sets are the certain subsets of $\mathbb{F}_{2^n}^m$ obtained from…
Every second flat Reidemeister move of knot projections can be decomposed into two types thorough an inverse or direct self-tangency modification, respectively called strong or weak, when orientations of the knot projections are arbitrarily…
We construct an infinite family of two-Lee-weight and three-Lee-weight codes over the chain ring $\mathbb{F}_p+u\mathbb{F}_p.$ They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using Gauss…
The standard and fractional projections are extended from binary two-mode networks to weighted two-mode networks. Some interesting properties of the extended projections are proved.
We consider $q$-ary (linear and nonlinear) block codes with exactly two distances: $d$ and $d+\delta$. Several combinatorial constructions of optimal such codes are given. In the linear (but not necessary projective) case, we prove that…
The parametric geometry of numbers has allowed to visualize the simultaneous approximation properties of a collection of real numbers through the combined graph of the related successive minima functions. Several inequalities among…
A weighted Hamming metric is introduced in [4] and it showed that the binary generalized Goppa code is a perfect code in some weighted Hamming metric. In this paper, we study the weight structures which admit the binary Hamming code and the…
Self-orthogonal codes are of interest as they have important applications in quantum codes, lattices and many areas. In this paper, based on the weakly regular plateaued functions or plateaued Boolean functions, we construct a family of…
We consider codes defined over an affine algebra $\mathcal A=R[X_1,\dots,X_r]/\left\langle t_1(X_1),\dots,t_r(X_r)\right\rangle$, where $t_i(X_i)$ is a monic univariate polynomial over a finite commutative chain ring $R$. Namely, we study…
We study pseudo-geometric strongly regular graphs whose second subconstituent with respect to a vertex is a cover of a strongly regular graph or a complete graph. By studying the structure of such graphs, we characterize all graphs…
Recently, the weight distributions of the duals of the cyclic codes with two zeros have been obtained for several cases. In this paper we use the method developed before to solve one more special case. We make extensive use of standard…
We study the interplay between the classical theory of linear series on curves, and the recent theory of linear series on graphs. We prove that every d-gonal (weighted) graph of Hurwitz type is the dual graph of a d-gonal curve. Conversely…
We study random 2-dimensional complexes in the Linial - Meshulam model and find torsion in their fundamental groups at various regimes. We find a simple algorithmically testable criterion for a subcomplex of a random 2-complex to be…
Studying the generalized Hamming weights of linear codes is a significant research area within coding theory, as it provides valuable structural information about the codes and plays a crucial role in determining their performance in…