Drue Coles

Efficient filters for the simulated evolution of small sorting networks

A sorting network is a mathematical model of an oblivious sorting algorithm - that is, a sorting algorithm in which all comparisons take place in a fixed order at predetermined positions in the list. The search for sorting networks of optimal size is an old problem. Asymptotically optimal networks of size O(n log n) are known, where n is the number of inputs, but the hidden constants are enormous. Genetic algorithms have been used to tackle this problem in a number of studies since the early 1990s, often focusing on the historically interesting case of 16 inputs. In this special case, the best known bound of 60 comparisons has been attained, but often through the use of a particular, highly structured, initial sequence of comparisons that narrows the search space by filtering out all but a small core of input sequences. We make explicit the concept of a filter - a fixed sequence of comparisons to be extended to a sorting network through a stochastic process - and present a new construction for any perfect square number of inputs. In the case of 9 inputs, we extend the filter to a sorting network of size 25, attaining the best known bound. For 16 and 25 inputs, we present a simple GA variant that extends the filter to produce small sorting networks with highly regular structure.

Vector bundles and codes on the Hermitian curve

The construction of algebraic-geometry (AG) codes can be seen as a distinctly geometric process, and yet decoding procedures tend to rely on algebraic ideas that have no direct geometric interpretation. Recently, however, Trygve Johnsen observed that decoding can be viewed in abstract terms of a class of vector bundles on the underlying curve. The present paper describes these objects at a concrete computational level for the Hermitian codes COmega(D,mPinfin) defined over Fq 2 (q a power of 2). The construction of explicit representations of the vector bundles by transition matrices involves finding functions on the curve that satisfy a certain property in their power series expansions around Pinfin, computing the image of the corresponding global sections under Serre duality, and finding a suitable open cover of the curve. The cover enables any rational point to be expressed as a line bundle by a simple kind of transition function. A special case is considered in which these functions can be realized as ratios of linear forms

A geometric view of decoding AG codes

We investigate the use of vector bundles over finite fields to obtain a geometric view of decoding algebraic-geometric codes. Building on ideas of Trygve Johnsen, who revealed a connection between the errors in a received word and certain vector bundles on the underlying curve, we give explicit constructions of the relevant geometric objects and efficient algorithms for some general computations needed in the constructions. The use of vector bundles to understand decoding as a geometric process is the first application of these objects to coding theory.

On constructing AG codes without basis functions for riemann-roch spaces

The standard construction of linear error-correcting codes on algebraic curves requires determining a basis for the Riemann-Roch space

Decoding by rank-2 bundles over plane quartics

\mathcal{L}

Predicting the types of file fragments

(G) associated to a given divisor G, often a hard problem. Here we consider the problem of constructing the code without any knowledge of such a basis. We interpret the columns of a generator matrix as points on an embedded copy of the curve, and show that in certain cases these points can be realized in principle as the images of a set of vector bundles under a standard map to a class of repartitions.

Goppa codes and Tschirnhausen modules

Motivated by error-correcting coding theory, we pose some hard questions regarding moduli spaces of rank-2 vector bundles over algebraic curves. We propose a new approach to the role of rank-2 bundles in coding theory, using recent results over the complex numbers, namely restriction of vector bundles from the projective space where the curve is embedded. We specialize our analysis to plane quartic curves which, if smooth, are canonical curves of genus three, and remark that all the bundles in question are restrictions. Using the vector-bundle approach, we work out explicit equations for the error divisors viewed as points of a multisecant variety. We specialize canonical quartics even more, to Kleins curve, and finite fields of characteristic two, a situation in which bundles can be neatly trivialized and codes have been produced. We give explicit equations, work out counting results for curves, Jacobians, and varieties of bundles, revealing several surprising features.