Related papers: Explicit Abelian Lifts and Quantum LDPC Codes

A $k$-lift of an $n$-vertex base graph $G$ is a graph $H$ on $n\times k$ vertices, where each vertex $v$ of $G$ is replaced by $k$ vertices $v_1,\cdots{},v_k$ and each edge $(u,v)$ in $G$ is replaced by a matching representing a bijection…

We present a new explicit construction for expander graphs with nearly optimal spectral gap. The construction is based on a series of 2-lift operations. Let $G$ be a graph on $n$ vertices. A 2-lift of $G$ is a graph $H$ on $2n$ vertices,…

An $\ell$-lift of a graph $G$ is any graph obtained by replacing every vertex of $G$ with an independent set of size $\ell$, and connecting every pair of two such independent sets that correspond to an edge in $G$ by a matching of size…

We give the first construction of explicit constant-degree lossless vertex expanders. Specifically, for any $\varepsilon > 0$ and sufficiently large $d$, we give an explicit construction of an infinite family of $d$-regular graphs where…

Let $G$ be a graph on $n$ vertices and $(H,+)$ be an abelian group. What is the minimum size ${\sf S}_H(G)$ of the set of all sums $A(u)+A(v)$ over all injections $A:V(G)\to H$? In 2012, the first author, Angel, the second author, and…

The power graph of a group $G$, denoted as $P(G)$, constitutes a simple undirected graph characterized by its vertex set $G$. Specifically, vertices $a,b$ exhibit adjacency exclusively if $a$ belongs to the cyclic subgroup generated by $b$…

We construct a new family of explicit codes that are list decodable to capacity and achieve an optimal list size of $O(\frac{1}{\epsilon})$. In contrast to existing explicit constructions of codes achieving list decoding capacity, our…

Given an epimorphism between topological groups $f:G\to H$, when can a generating set of $H$ be lifted to a generating set of $G$? We show that for connected Lie groups the problem is fundamentally abelian: generators can be lifted if and…

In this article, we provide a general set-up for arbitrary linear Lie groups $H\leq \mathrm{GL}(n,\mathbb{R})$ which allows to characterise the almost Abelian Lie algebras admitting a torsion-free $H$-structure. In more concrete terms,…

We present a general framework for constructing high rate error correcting codes that are locally correctable (and hence locally decodable if linear) with a sublinear number of queries, based on lifting codes with respect to functions on…

An abelian lattice-ordered group, or abelian $\ell$-group, is an abelian group equipped with a compatible lattice ordering. In this paper, we introduce two multi-sorted extensions of abelian lattice-ordered groups inspired by the zero-set…

A random $n$-lift of a base graph $G$ is its cover graph $H$ on the vertices $[n]\times V(G)$, where for each edge $u v$ in $G$ there is an independent uniform bijection $\pi$, and $H$ has all edges of the form $(i,u),(\pi(i),v)$. A main…

A gain graph is a triple (G,h,H), where G is a connected graph with an arbitrary, but fixed, orientation of edges, H is a group, and h is a homomorphism from the free group on the edges of G to H. A gain graph is called balanced if the…

The \textbf{Co-Prime Order Graph} $\Theta (G)$ of a given finite group is a simple undirected graph whose vertex set is the group $G$ itself, and any two vertexes x,y in $\Theta (G)$ are adjacent if and only if $gcd(o(x),o(y))=1$ or prime.…

We present near-linear time list decoding algorithms (in the block-length $n$) for expander-based code constructions. More precisely, we show that (i) For every $\delta \in (0,1)$ and $\epsilon > 0$, there is an explicit family of good…

Deterministic constructions of expander graphs have been an important topic of research in computer science and mathematics, with many well-studied constructions of infinite families of expanders. In some applications, though, an infinite…

This article proposes an effective criterion for lifting automorphisms along regular coverings of graphs, with the covering transformation group being any finite abelian group.

Amit and Linial showed that a random lift of a graph with minimum degree $\delta\ge3$ is asymptotically almost surely $\delta$-connected, and mentioned the problem of estimating this probability as a function of the degree of the lift. We…

The enhanced power graph of a group $G$ is a simple undirected graph with vertex set $G$ and two vertices are adjacent if they belong to same cyclic subgroup. In this paper, we study distant properties and detour distant properties such as…

We canonically identify the groups of isometries and dilations of local fields and their rings of integers with subgroups of the automorphism group of the $(d+1)$-regular tree $\widetilde T_{d+1}$, where $d$ is the residual degree. Then we…