An Introduction to Algebraic Geometry codes

We present an introduction to the theory of algebraic geometry codes. Starting from evaluation codes and codes from order and weight functions, special attention is given to one-point codes and, in particular, to the family of Castle codes

On the evaluation codes given by simple d-sequences

Plane valuations at infinity are classified in five types. Valuations in one of them determine weight functions which take values on semigroups of Z2. These semigroups are generated by δ-sequences in Z2. We introduce simple δ-sequences in Z2 and study the evaluation codes of maximal length that they define. These codes are geometric and come from order domains. We give a bound on their minimum distance which improves the Andersen–Geil one. We also give coset bounds for the involved codes

A Review of Lightweight Thread Approaches for High Performance Computing

High-level, directive-based solutions are becoming the programming models (PMs) of the multi/many-core architectures. Several solutions relying on operating system (OS) threads perfectly work with a moderate number of cores. However, exascale systems will spawn hundreds of thousands of threads in order to exploit their massive parallel architectures and thus conventional OS threads are too heavy for that purpose. Several lightweight thread (LWT) libraries have recently appeared offering lighter mechanisms to tackle massive concurrency. In order to examine the suitability of LWTs in high-level runtimes, we develop a set of microbenchmarks consisting of commonly-found patterns in current parallel codes. Moreover, we study the semantics offered by some LWT libraries in order to expose the similarities between different LWT application programming interfaces. This study reveals that a reduced set of LWT functions can be sufficient to cover the common parallel code patterns andthat those LWT libraries perform better than OS threads-based solutions in cases where task and nested parallelism are becoming more popular with new architectures.The researchers from the Universitat Jaume I de Castelló were supported by project TIN2014-53495-R of the MINECO, the Generalitat Valenciana fellowship programme Vali+d 2015, and FEDER. This work was partially supported by the U.S. Dept. of Energy, Office of Science, Office of Advanced Scientific Computing Research (SC-21), under contract DEAC02-06CH11357. We gratefully acknowledge the computing resources provided and operated by the Joint Laboratory for System Evaluation (JLSE) at Argonne National Laboratory.Peer ReviewedPostprint (author's final draft

Codes and Sequences for Information Retrieval and Stream Ciphers

Given a self-similar structure in codes and de Bruijn sequences, recursive techniques may be used to analyze and construct them. Batch codes partition the indices of code words into m buckets, where recovery of t symbols is accomplished by accessing at most tau in each bucket. This finds use in the retrieval of information spread over several devices. We introduce the concept of optimal batch codes, showing that binary Hamming codes and first order Reed-Muller codes are optimal. Then we study batch properties of binary Reed-Muller codes which have order less than half their length. Cartesian codes are defined by the evaluation of polynomials at a subset of points in F_q. We partition F_q into buckets defined by the quotient with a subspace V. Several properties equivalent to (V intersect ) = {0} for all i,j between 1 and mu are explored. With this framework, a code in F_q^(mu-1) capable of reconstructing mu indices is expanded to one in F_q^(mu) capable of reconstructing mu+1 indices. Using a base case in F_q^3, we are able to prove batch properties for codes in F_q. We generalize this to Cartesian Codes with a limit on the degree mu of the polynomials. De Bruijn sequences are cyclic sequences of length q^n that contain every q-ary word of length n exactly once. The pseudorandom properties of such sequences make them useful for stream ciphers. Under a particular homomorphism, the preimages of a binary de Bruijn sequence form two cycles. We examine a method for identifying points where these sequences may be joined to make a de Bruijn sequence of order n. Using the recursive structure of this construction, we are able to calculate sums of subsequences in O(n^4 log(n)) time, and the location of a word in O(n^5 log(n)) time. Together, these functions allow us to check the validity of any potential toggle point, which provides a method for efficiently generating a recursive specification. Each successful step takes O(k^5 log(k)), for k from 3 to n

Evaluation codes defined by finite families of plane valuations at infinity

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A class of explicit high-order exponentially-fitted two-step methods for solving oscillatory IVPs

The derivation of new exponentially fitted (EF) modified two-step hybrid (MTSH) methods for the numerical integration of oscillatory second-order IVPs is analyzed. These methods are modifications of classical two-step hybrid methods so that they integrate exactly differential systems whose solutions can be expressed as linear combinations of the set of functions {exp(¿t), exp(-¿t)}, ¿¿C, or equivalently {sin(¿t), cos(¿t)} when ¿=i¿, ¿¿R, where ¿ represents an approximation of the main frequency of the problem. The EF conditions and the conditions for this class of EF schemes to have algebraic order p (with p=8) are derived. With the help of these conditions we construct explicit EFMTSH methods with algebraic orders seven and eight which require five and six function evaluation per step, respectively. These new EFMTSH schemes are optimal among the two-step hybrid methods in the sense that they reach a certain order of accuracy with minimal computational cost per step. In order to show the efficiency of the new high order explicit EFMTSH methods in comparison to other EF and standard two-step hybrid codes from the literature some numerical experiments with several orbital and oscillatory problems are presented