On Dual Toric Complete Intersection Codes

In this paper we study duality for evaluation codes on intersections of ℓ hypersurfaces with given ℓ-dimensional Newton polytopes, so called toric complete intersection codes. In particular, we give a condition for such a code to be quasi-self-dual. In the case of it reduces to a combinatorial condition on the Newton polygons. This allows us to give an explicit construction of dual and quasi-self-dual toric complete intersection codes. We provide a list of examples over and an algorithm for producing them

Quantum Codes from Toric Surfaces

A theory for constructing quantum error correcting codes from Toric surfaces by the Calderbank-Shor-Steane method is presented. In particular we study the method on toric Hirzebruch surfaces. The results are obtained by constructing a dualizing differential form for the toric surface and by using the cohomology and the intersection theory of toric varieties. In earlier work the author developed methods to construct linear error correcting codes from toric varieties and derive the code parameters using the cohomology and the intersection theory on toric varieties. This method is generalized in section to construct linear codes suitable for constructing quantum codes by the Calderbank-Shor-Steane method. Essential for the theory is the existence and the application of a dualizing differential form on the toric surface. A.R. Calderbank, P.W. Shor and A.M. Steane produced stabilizer codes from linear codes containing their dual codes. These two constructions are merged to obtain results for toric surfaces. Similar merging has been done for algebraic curves with different methods by A. Ashikhmin, S. Litsyn and M.A. Tsfasman.Comment: IEEE copyrigh

AG Codes from Polyhedral Divisors

A description of complete normal varieties with lower dimensional torus action has been given by Altmann, Hausen, and Suess, generalizing the theory of toric varieties. Considering the case where the acting torus T has codimension one, we describe T-invariant Weil and Cartier divisors and provide formulae for calculating global sections, intersection numbers, and Euler characteristics. As an application, we use divisors on these so-called T-varieties to define new evaluation codes called T-codes. We find estimates on their minimum distance using intersection theory. This generalizes the theory of toric codes and combines it with AG codes on curves. As the simplest application of our general techniques we look at codes on ruled surfaces coming from decomposable vector bundles. Already this construction gives codes that are better than the related product code. Further examples show that we can improve these codes by constructing more sophisticated T-varieties. These results suggest to look further for good codes on T-varieties.Comment: 30 pages, 9 figures; v2: replaced fansy cycles with fansy divisor

Fracton topological order via coupled layers

In this work, we develop a coupled layer construction of fracton topological orders in $d=3$ spatial dimensions. These topological phases have sub-extensive topological ground-state degeneracy and possess excitations whose movement is restricted in interesting ways. Our coupled layer approach is used to construct several different fracton topological phases, both from stacked layers of simple $d=2$ topological phases and from stacks of $d=3$ fracton topological phases. This perspective allows us to shed light on the physics of the X-cube model recently introduced by Vijay, Haah, and Fu, which we demonstrate can be obtained as the strong-coupling limit of a coupled three-dimensional stack of toric codes. We also construct two new models of fracton topological order: a semionic generalization of the X-cube model, and a model obtained by coupling together four interpenetrating X-cube models, which we dub the "Four Color Cube model." The couplings considered lead to fracton topological orders via mechanisms we dub "p-string condensation" and "p-membrane condensation," in which strings or membranes built from particle excitations are driven to condense. This allows the fusion properties, braiding statistics, and ground-state degeneracy of the phases we construct to be easily studied in terms of more familiar degrees of freedom. Our work raises the possibility of studying fracton topological phases from within the framework of topological quantum field theory, which may be useful for obtaining a more complete understanding of such phases.Comment: 20 pages, 18 figures, published versio