On the structure of generalized toric codes

Toric codes are obtained by evaluating rational functions of a nonsingular toric variety at the algebraic torus. One can extend toric codes to the so called generalized toric codes. This extension consists on evaluating elements of an arbitrary polynomial algebra at the algebraic torus instead of a linear combination of monomials whose exponents are rational points of a convex polytope. We study their multicyclic and metric structure, and we use them to express their dual and to estimate their minimum distance

Generalized Color Codes Supporting Non-Abelian Anyons

We propose a generalization of the color codes based on finite groups $G$. For non-abelian groups, the resulting model supports non-abelian anyonic quasiparticles and topological order. We examine the properties of these models such as their relationship to Kitaev quantum double models, quasiparticle spectrum, and boundary structure.Comment: 17 pages, 8 figures; references added, typos remove

Lattice polytopes in coding theory

In this paper we discuss combinatorial questions about lattice polytopes motivated by recent results on minimum distance estimation for toric codes. We also prove a new inductive bound for the minimum distance of generalized toric codes. As an application, we give new formulas for the minimum distance of generalized toric codes for special lattice point configurations.Comment: 11 pages, 3 figure

Secret Sharing Schemes with a large number of players from Toric Varieties

A general theory for constructing linear secret sharing schemes over a finite field \Fq from toric varieties is introduced. The number of players can be as large as $(q-1)^r-1$ for $r\geq 1$. We present general methods for obtaining the reconstruction and privacy thresholds as well as conditions for multiplication on the associated secret sharing schemes. In particular we apply the method on certain toric surfaces. The main results are ideal linear secret sharing schemes where the number of players can be as large as $(q-1)^2-1$. We determine bounds for the reconstruction and privacy thresholds and conditions for strong multiplication using the cohomology and the intersection theory on toric surfaces.Comment: 15 pages, 4 figures. arXiv admin note: text overlap with arXiv:1203.454

Graver Bases and Universal Gr\"obner Bases for Linear Codes

Two correspondences have been provided that associate any linear code over a finite field with a binomial ideal. In this paper, algorithms for computing their Graver bases and universal Gr\"obner bases are given. To this end, a connection between these binomial ideals and toric ideals will be established.Comment: 18 page

Subfield-Subcodes of Generalized Toric codes

We study subfield-subcodes of Generalized Toric (GT) codes over $\mathbb{F}_{p^s}$. These are the multidimensional analogues of BCH codes, which may be seen as subfield-subcodes of generalized Reed-Solomon codes. We identify polynomial generators for subfield-subcodes of GT codes which allows us to determine the dimensions and obtain bounds for the minimum distance. We give several examples of binary and ternary subfield-subcodes of GT codes that are the best known codes of a given dimension and length.Comment: Submitted to 2010 IEEE International Symposium on Information Theory (ISIT 2010

Foliated Field Theory and String-Membrane-Net Condensation Picture of Fracton Order

Foliated fracton order is a qualitatively new kind of phase of matter. It is similar to topological order, but with the fundamental difference that a layered structure, referred to as a foliation, plays an essential role and determines the mobility restrictions of the topological excitations. In this work, we introduce a new kind of field theory to describe these phases: a foliated field theory. We also introduce a new lattice model and string-membrane-net condensation picture of these phases, which is analogous to the string-net condensation picture of topological order.Comment: 22+15 pages, 8 figures; v3 added a summary of our model near the end of the introductio