Geometry of vector bundle extensions and applications to a generalised theta divisor

Let E and F be vector bundles over a complex projective smooth curve X, and suppose that 0 -> E -> W -> F -> 0 is a nontrivial extension. Let G be a subbundle of F, and D an effective divisor on X. We give a criterion for the subsheaf G(-D) \subset F to lift to W, in terms of the geometry of a scroll in the extension space \PP H^1 (X, Hom(F, E)). We use this criterion to describe the tangent cone to the generalised theta divisor on the moduli space of semistable bundles of rank r and slope g-1 over X, at a stable point. This gives a generalisation of a case of the Riemann-Kempf singularity theorem for line bundles over X. In the same vein, we generalise the geometric Riemann-Roch theorem to vector bundles of slope g-1 and arbitrary rank.Comment: Main theorem slightly weakened; statement and proof significantly more compac

New Classes of Partial Geometries and Their Associated LDPC Codes

The use of partial geometries to construct parity-check matrices for LDPC codes has resulted in the design of successful codes with a probability of error close to the Shannon capacity at bit error rates down to $10^{-15}$. Such considerations have motivated this further investigation. A new and simple construction of a type of partial geometries with quasi-cyclic structure is given and their properties are investigated. The trapping sets of the partial geometry codes were considered previously using the geometric aspects of the underlying structure to derive information on the size of allowable trapping sets. This topic is further considered here. Finally, there is a natural relationship between partial geometries and strongly regular graphs. The eigenvalues of the adjacency matrices of such graphs are well known and it is of interest to determine if any of the Tanner graphs derived from the partial geometries are good expanders for certain parameter sets, since it can be argued that codes with good geometric and expansion properties might perform well under message-passing decoding.Comment: 34 pages with single column, 6 figure

Parameters of AG codes from vector bundles

AbstractWe investigate the parameters of the algebraic–geometric codes constructed from vector bundles on a projective variety defined over a finite field. In the case of curves we give a method of constructing weakly stable bundles using restriction of vector bundles on algebraic surfaces and illustrate the result by some examples

Error Correcting Codes on Algebraic Surfaces

Error correcting codes are defined and important parameters for a code are explained. Parameters of new codes constructed on algebraic surfaces are studied. In particular, codes resulting from blowing up points in \proj^2 are briefly studied, then codes resulting from ruled surfaces are covered. Codes resulting from ruled surfaces over curves of genus 0 are completely analyzed, and some codes are discovered that are better than direct product Reed Solomon codes of similar length. Ruled surfaces over genus 1 curves are also studied, but not all classes are completely analyzed. However, in this case a family of codes are found that are comparable in performance to the direct product code of a Reed Solomon code and a Goppa code. Some further work is done on surfaces from higher genus curves, but there remains much work to be done in this direction to understand fully the resulting codes. Codes resulting from blowing points on surfaces are also studied, obtaining necessary parameters for constructing infinite families of such codes. Also included is a paper giving explicit formulas for curves with more \field{q}-rational points than were previously known for certain combinations of field size and genus. Some upper bounds are now known to be optimal from these examples.Comment: This is Chris Lomont's PhD thesis about error correcting codes from algebriac surface

Codes as fractals and noncommutative spaces

We consider the CSS algorithm relating self-orthogonal classical linear codes to q-ary quantum stabilizer codes and we show that to such a pair of a classical and a quantum code one can associate geometric spaces constructed using methods from noncommutative geometry, arising from rational noncommutative tori and finite abelian group actions on Cuntz algebras and fractals associated to the classical codes.Comment: 18 pages LaTeX, one png figur

Quantum Low-Density Parity-Check Codes

Quantum error correction is an indispensable ingredient for scalable quantum computing. In this Perspective we discuss a particular class of quantum codes called “quantum low-density parity-check (LDPC) codes.” The codes we discuss are alternatives to the surface code, which is currently the leading candidate to implement quantum fault tolerance. We introduce the zoo of quantum LDPC codes and discuss their potential for making quantum computers robust with regard to noise. In particular, we explain recent advances in the theory of quantum LDPC codes related to certain product constructions and discuss open problems in the field