Related papers: A Goppa-like bound on the trellis state complexity…
The classical Crossing Lemma by Ajtai et al.~and Leighton from 1982 gave an important lower bound of $c \frac{m^3}{n^2}$ for the number of crossings in any drawing of a given graph of $n$ vertices and $m$ edges. The original value was $c=…
In this work, a coding technique called cost constrained Geometric Huffman coding (ccGhc) is developed. ccGhc minimizes the Kullback-Leibler distance between a dyadic probability mass function (pmf) and a target pmf subject to an affine…
Linear codes over finite rings become one of hot topics in coding theory after Hommons et al.([4], 1994) discovered that several remarkable nonlinear binary codes with some linear-like properties are the images of Gray map of linear codes…
Separable codes were introduced to provide protection against illegal redistribution of copyrighted multimedia material. Let $\mathcal{C}$ be a code of length $n$ over an alphabet of $q$ letters. The descendant code ${\sf…
Algorithms working with linear algebraic groups often represent them via defining polynomial equations. One can always choose defining equations for an algebraic group to be of the degree at most the degree of the group as an algebraic…
The determination of the maximal length of maximum distance separable (MDS) codes arising from elliptic curves is a central problem in coding theory. For an elliptic curve $E$ over $\mathbb{F}_q$, let $\operatorname{MEC}(k,q)$ denote the…
For every integer $k \geq 3$ we construct a $k$-gonal curve $C$ along with a very ample divisor of degree $2g + k - 1$ (where $g$ is the genus of $C$) to which the vanishing statement from the Green-Lazarsfeld gonality conjecture does not…
The {\it toughness} $\tau(G)=\mathrm{min}\{\frac{|S|}{c(G-S)}: S~\mbox{is a vertex cut in}~G\}$ for $G\ncong K_n,$ which was initially proposed by Chv\'{a}tal in 1973. A graph $G$ is called {\it $t$-tough} if $\tau(G)\geq t.$ Let…
Various methods have been used to obtain improvements of the Goppa lower bound for the minimum distance of an algebraic geometric code. The main methods divide into two categories and all but a few of the known bounds are special cases of…
Let $q=p^m$ be a prime power, $e$ be an integer with $0\leq e\leq m-1$ and $s=\gcd(e,m)$. In this paper, for a vector $v$ and a $q$-ary linear code $C$, we give some necessary and sufficient conditions for the equivalent code $vC$ of $C$…
For a graph $G$, let $c_k(G)$ be the number of spanning trees of $G$ with maximum degree at most $k$. For $k \ge 3$, it is proved that every connected $n$-vertex $r$-regular graph $G$ with $r \ge \frac{n}{k+1}$ satisfies $$ c_k(G)^{1/n} \ge…
An identifying code is a subset of vertices of a graph such that each vertex is uniquely determined by its neighbourhood within the identifying code. If $\M(G)$ denotes the minimum size of an identifying code of a graph $G$, it was…
In the 1980's Daryl Cooper introduced the notion of a C-complex (or clasp-complex) bounded by a link and explained how to compute signatures and polynomial invariants using a C-complex. Since then this was extended by works of Cimasoni,…
Currently, the best upper bounds on the number of rational points on an absolutely irreducible, smooth, projective algebraic curve of genus g defined over a finite field F_q come either from Serre's refinement of the Weil bound if the genus…
For any complex number $c$, let $\sigma_c\colon\mathbb N\rightarrow\mathbb C$ denote the divisor function defined by $\sigma_c(n)=\displaystyle{\sum_{d|n}d^c}$ for all $n\in\mathbb N$, and define $R(c)=\{\sigma_c(n)\in\mathbb C\colon…
Consider the family S of irreducible plane curves of degree n with d nodes and k cusps as singularities. Let W be an irreducible component of S. We consider the natural rational map from W to the moduli space of curves of genus…
The node-averaged complexity of a problem captures the number of rounds nodes of a graph have to spend on average to solve the problem in the LOCAL model. A challenging line of research with regards to this new complexity measure is to…
We investigate the computational complexity of some problems in three-dimensional topology and geometry. We show that the problem of determining a bound on the genus of a knot in a 3-manifold, is NP-complete. Using similar ideas, we show…
We study search trees with 2-way comparisons (2WCST's), which involve separate less-than and equal-to tests in their nodes, each test having two possible outcomes, yes and no. These trees have a much subtler structure than standard search…
The study of subblock-constrained codes has recently gained attention due to their application in diverse fields. We present bounds on the size and asymptotic rate for two classes of subblock-constrained codes. The first class is binary…