Related papers: Explicit Abelian Lifts and Quantum LDPC Codes

The semisimple part of d-dimensional Galilean conformal algebra g^(d) is given by h^(d)=O(2,1)+O(d), which after adding via semidirect sum the 3d-dimensional Abelian algebra t^(d) of translations, Galilean boosts and constant accelerations…

We study the problem of constructing explicit codes whose rate and distance match the Gilbert-Varshamov bound in the low-rate, high-distance regime. In 2017, Ta-Shma gave an explicit family of codes where every pair of codewords has…

We show that any pointed, preordered module map $\mathfrak{BF}_{\mathrm{gr}}(E) \to \mathfrak{BF}_{\mathrm{gr}}(F)$ between Bowen-Franks modules of finite graphs can be lifted to a unital, graded, diagonal preserving $\ast$-homomorphism…

We demonstrate the emergence of non-Abelian fusion rules for excitations of a two dimensional lattice model built out of Abelian degrees of freedom. It can be considered as an extension of the usual toric code model on a two dimensional…

We derive two families of EA-QC quantum LDPC (EA-QC-QLDPC) codes by tiling permutation matrices of prime and composite orders. The unassisted portion of the Tanner graphs corresponding to these codes, constructed from two distinct classical…

Let $G$ be the semidirect product $N \rtimes \mathbb{R}$, where $N$ is a stratified Lie group and $\mathbb{R}$ acts on $N$ via automorphic dilations. Homogeneous left-invariant sub-Laplacians on $N$ and $\mathbb{R}$ can be lifted to $G$,…

For an Abelian surface $A$ with a symplectic action by a finite group $G$, one can define the partition function for $G$-invariant Hilbert schemes \[Z_{A, G}(q) = \sum_{d=0}^{\infty} e(\text{Hilb}^{d}(A)^{G})q^{d}.\] We prove the reciprocal…

We consider the task of locally correcting, and locally list-correcting, multivariate linear functions over the domain $\{0,1\}^n$ over arbitrary fields and more generally Abelian groups. Such functions form error-correcting codes of…

We consider quantum two-block group algebra (2BGA) codes, a previously unstudied family of smallest lifted-product (LP) codes. These codes are related to generalized-bicycle (GB) codes, except a cyclic group is replaced with an arbitrary…

In this paper we are interested in lifting a prescribed group of automorphisms of a finite graph via regular covering projections. Here we describe with an example the problems we address and refer to the introductory section for the…

Let $H$ be a finite abelian (commutative) group of order $n \geq 2$, and $m >1$ be an integer. We define the $m$-graph of $H$, denoted by $m-G(H)$, as a simple undirected graph with vertex set $H$, and two distinct vertices, $a, b \in H$,…

In this work, we introduce constructions for non-Abelian qLDPC codes obtained by gauging transversal Clifford gates using measurement and feedback. In particular, we identify two qualitatively different approaches to gauging qLDPC codes to…

We introduce lattice gauge theories which describe three-dimensional, gapped quantum phases exhibiting the phenomenology of both conventional three-dimensional topological orders and fracton orders, starting from a finite group $G$, a…

We study the doubled edge-stage lift \[ \HL'_2(G)=L(G\otimes K_2), \] the line graph of the canonical bipartite double cover of a graph \(G\). The natural involution \((u,v)\leftrightarrow(v,u)\) has quotient isomorphic to \(L(G)\), and…

We construct a family of constant-rate highly-symmetric self-dual qLDPC codes on high dimensional expanders. This is the first self-dual code constructed on high dimensional expanders and also the first such code with a rich (e.g.…

A well-known result of Brylawski constructs an elementary lift of a matroid $M$ from a linear class of circuits of $M$. We generalize this result by showing how to construct a rank-$k$ lift of $M$ from a rank-$k$ matroid on the set of…

In order to compute with $l$--adic sheaves or crystals on a line over $\mathbb{F} _q$ a low-technology alternative to the traditional computation with the Hecke operators on the automorphic side could be helpful. A program which has evolved…

Gauging introduces gauge fields in order to localize an existing global symmetry, resulting in a dual global symmetry on the gauge fields that can be gauged again. By iterating the gauging process on spin chains with Abelian group…

The paper is devoted to the study of abelian (in the sense defined by Godsil and Hensel) distance-regular $r$-covers of the complete graphs $K_n$. According to the construction by Coutinho, Godsil, Shirazi, and Zhan (2016), each such cover…

We study the problem of constructing explicit sparse graphs that exhibit strong vertex expansion. Our main result is the first two-sided construction of imbalanced unique-neighbor expanders, meaning bipartite graphs where small sets…