Related papers: New code upper bounds for the folded n-cube
We generalize upper bounds for constant dimension codes containing a lifted maximum rank distance code first studied by Etzion and Silberstein. The proof allows to construct several improved codes.
Linear codes in the projective space $\mathbb{P}_q(n)$, the set of all subspaces of the vector space $\mathbb{F}_q^n$, were first considered by Braun, Etzion and Vardy. The Grassmannian $\mathbb{G}_q(n,k)$ is the collection of all subspaces…
We prove that every connected cubic graph with $n$ vertices has a maximal matching of size at most $\frac{5}{12} n+ \frac{1}{2}$. This confirms the cubic case of a conjecture of Baste, F\"urst, Henning, Mohr and Rautenbach (2019) on regular…
In the context of constant--dimension subspace codes, an important problem is to determine the largest possible size $A_q(n, d; k)$ of codes whose codewords are $k$-subspaces of $\mathbb{F}_q^n$ with minimum subspace distance $d$. Here in…
A classic result by Cook, Gerards, Schrijver, and Tardos provides an upper bound of $n \Delta$ on the proximity of optimal solutions of an Integer Linear Programming problem and its standard linear relaxation. In this bound, $n$ is the…
We obtain some new upper bounds on the maximum number $f(n)$ of edges in $n$-vertex graphs without containing cycles of length four. This leads to an asymptotically optimal bound on $f(n)$ for a broad range of integers $n$ as well as a…
Let $n_q(M,d)$ be the minimum length of a $q$-ary code of size $M$ and minimum distance $d$. Bounding $n_q(M,d)$ is a fundamental problem that lies at the heart of coding theory. This work considers a generalization $n^\bx_q(M,d)$ of…
We address the maximum size of binary codes and binary constant weight codes with few distances. Previous works established a number of bounds for these quantities as well as the exact values for a range of small code lengths. As our main…
We consider linear codes associated to Schubert varieties in Grassmannians. A formula for the minimum distance of these codes was conjectured in 2000 and after having been established in various special cases, it was proved in 2008 by…
We present a new algorithm for improving lower bounds on $ex(n;\{C_3,C_4\})$, the maximum size (number of edges) of an $n$-vertex graph of girth at least 5. The core of our algorithm is a variant of a hill-climbing heuristic introduced by…
An additive quaternary $[n,k,d]$-code (length $n,$ quaternary dimension $k,$ minimum distance $d$) is a $2k$-dimensional F_2-vector space of $n$-tuples with entries in $Z_2\times Z_2$ (the $2$-dimensional vector space over F_2) with minimum…
An open question about Gabidulin codes is whether polynomial-time list decoding beyond half the minimum distance is possible or not. In this contribution, we give a lower and an upper bound on the list size, i.e., the number of codewords in…
The dimension of a graph $G$ is the smallest $d$ for which its vertices can be embedded in $d$-dimensional Euclidean space in the sense that the distances between endpoints of edges equal $1$ (but there may be other unit distances).…
In this short note we give a new upper bound for the size of a set family with a single Hamming distance. Our proof is an application of the linear algebra bound method.
We investigate the problem of the maximum number of cubic subwords (of the form $www$) in a given word. We also consider square subwords (of the form $ww$). The problem of the maximum number of squares in a word is not well understood.…
Determine the size of $r$-graphs with given graph parameters is an interesting problem. Chv\'atal and Hanson (JCTB, 1976) gave a tight upper bound of the size of 2-graphs with restricted maximum degree and matching number; Khare (DM, 2014)…
We show that, for a constant-degree algebraic curve $\gamma$ in $\mathbb{R}^D$, every set of $n$ points on $\gamma$ spans at least $\Omega(n^{4/3})$ distinct distances, unless $\gamma$ is an {\it algebraic helix} (see Definition 1.1). This…
We study the maximum dimension $d=d(n,p)$ for which an Erd\H{o}s-R\'enyi $G(n,p)$ random graph is $d$-rigid. Our main results reveal two different regimes of rigidity in $G(n,p)$ separated at $p_c=C_*\log n/n,~C_*=2/(1-\log 2)$ -- the point…
A Maximum Distance Separable code over an alphabet $F$ is defined via an encoding function $C:F^k \rightarrow F^n$ that allows to retrieve a message $m \in F^k$ from the codeword $C(m)$ even after erasing any $n-k$ of its symbols. The…
We improve the best known upper bound on the number of edges in a unit-distance graph on $n$ vertices for each $n\in\{16,\ldots,30\}$. When $n\leq 21$, our bounds match the best known lower bounds, and we fully enumerate the densest…