Cyclic and BCH Codes whose Minimum Distance Equals their Maximum BCH bound

In this paper we study the family of cyclic codes such that its minimum distance reaches the maximum of its BCH bounds. We also show a way to construct cyclic codes with that property by means of computations of some divisors of a polynomial of the form X^n-1. We apply our results to the study of those BCH codes C, with designed distance delta, that have minimum distance d(C)= delta. Finally, we present some examples of new binary BCH codes satisfying that condition. To do this, we make use of two related tools: the discrete Fourier transform and the notion of apparent distance of a code, originally defined for multivariate abelian codes

Constructions of Abelian Codes multiplying dimension of cyclic codes

In this note, we apply some techniques developed in [1]-[3] to give a particular construction of bivariate Abelian Codes from cyclic codes, multiplying their dimension and preserving their apparent distance. We show that, in the case of cyclic codes whose maximum BCH bound equals its minimum distance the obtained abelian code verifies the same property; that is, the strong apparent distance and the minimum distance coincide. We finally use this construction to multiply Reed-Solomon codes to abelian codesComment: arXiv admin note: text overlap with arXiv:2402.0393

Numerical cubature using error-correcting codes

We present a construction for improving numerical cubature formulas with equal weights and a convolution structure, in particular equal-weight product formulas, using linear error-correcting codes. The construction is most effective in low degree with extended BCH codes. Using it, we obtain several sequences of explicit, positive, interior cubature formulas with good asymptotics for each fixed degree $t$ as the dimension $n \to \infty$. Using a special quadrature formula for the interval [arXiv:math.PR/0408360], we obtain an equal-weight $t$-cubature formula on the $n$-cube with O(n^{\floor{t/2}}) points, which is within a constant of the Stroud lower bound. We also obtain $t$-cubature formulas on the $n$-sphere, $n$-ball, and Gaussian $\R^n$ with $O(n^{t-2})$ points when $t$ is odd. When $\mu$ is spherically symmetric and $t=5$, we obtain $O(n^2)$ points. For each $t \ge 4$, we also obtain explicit, positive, interior formulas for the $n$-simplex with $O(n^{t-1})$ points; for $t=3$, we obtain O(n) points. These constructions asymptotically improve the non-constructive Tchakaloff bound. Some related results were recently found independently by Victoir, who also noted that the basic construction more directly uses orthogonal arrays.Comment: Dedicated to Wlodzimierz and Krystyna Kuperberg on the occasion of their 40th anniversary. This version has a major improvement for the n-cub

A new approach to the Berlekamp-Massey-Sakata Algorithm. Improving Locator Decoding

We study the problem of the computation of Groebner basis for the ideal of linear recurring relations of a doubly periodic array. We find a set of indexes such that, along with some conditions, guarantees that the set of polynomials obtained at the last iteration in the Berlekamp-Massey-Sakata algorithm is exactly a Groebner basis for the mentioned ideal. Then, we apply these results to improve locator decoding in abelian codes.Comment: 21 page

Symmetries in algebraic Property Testing

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 94-100).Modern computational tasks often involve large amounts of data, and efficiency is a very desirable feature of such algorithms. Local algorithms are especially attractive, since they can imply global properties by only inspecting a small window into the data. In Property Testing, a local algorithm should perform the task of distinguishing objects satisfying a given property from objects that require many modifications in order to satisfy the property. A special place in Property Testing is held by algebraic properties: they are some of the first properties to be tested, and have been heavily used in the PCP and LTC literature. We focus on conditions under which algebraic properties are testable, following the general goal of providing a more unified treatment of these properties. In particular, we explore the notion of symmetry in relation to testing, a direction initiated by Kaufman and Sudan. We investigate the interplay between local testing, symmetry and dual structure in linear codes, by showing both positive and negative results. On the negative side, we exhibit a counterexample to a conjecture proposed by Alon, Kaufman, Krivelevich, Litsyn, and Ron aimed at providing general sufficient conditions for testing. We show that a single codeword of small weight in the dual family together with the property of being invariant under a 2-transitive group of permutations do not necessarily imply testing. On the positive side, we exhibit a large class of codes whose duals possess a strong structural property ('the single orbit property'). Namely, they can be specified by a single codeword of small weight and the group of invariances of the code. Hence we show that sparsity and invariance under the affine group of permutations are sufficient conditions for a notion of very structured testing. These findings also reveal a new characterization of the extensively studied BCH codes. As a by-product, we obtain a more explicit description of structured tests for the special family of BCH codes of design distance 5.by Elena Grigorescu.Ph.D

Correcting errors and erasures via the syndrome variety

AbstractWe propose a new syndrome variety, which can be used to decode cyclic codes. We present also a generalization to erasure and error decoding. We can exhibit a polynomial whose roots give the error locations, once it has been specialized to a given syndrome. This polynomial has degree t in the variable corresponding to the error locations and its coefficients are polynomials in the syndromes

Algebraic Codes For Error Correction In Digital Communication Systems

Access to the full-text thesis is no longer available at the author's request, due to 3rd party copyright restrictions. Access removed on 29.11.2016 by CS (TIS).Metadata merged with duplicate record (http://hdl.handle.net/10026.1/899) on 20.12.2016 by CS (TIS).C. Shannon presented theoretical conditions under which communication was possible error-free in the presence of noise. Subsequently the notion of using error correcting codes to mitigate the effects of noise in digital transmission was introduced by R. Hamming. Algebraic codes, codes described using powerful tools from algebra took to the fore early on in the search for good error correcting codes. Many classes of algebraic codes now exist and are known to have the best properties of any known classes of codes. An error correcting code can be described by three of its most important properties length, dimension and minimum distance. Given codes with the same length and dimension, one with the largest minimum distance will provide better error correction. As a result the research focuses on finding improved codes with better minimum distances than any known codes. Algebraic geometry codes are obtained from curves. They are a culmination of years of research into algebraic codes and generalise most known algebraic codes. Additionally they have exceptional distance properties as their lengths become arbitrarily large. Algebraic geometry codes are studied in great detail with special attention given to their construction and decoding. The practical performance of these codes is evaluated and compared with previously known codes in different communication channels. Furthermore many new codes that have better minimum distance to the best known codes with the same length and dimension are presented from a generalised construction of algebraic geometry codes. Goppa codes are also an important class of algebraic codes. A construction of binary extended Goppa codes is generalised to codes with nonbinary alphabets and as a result many new codes are found. This construction is shown as an efficient way to extend another well known class of algebraic codes, BCH codes. A generic method of shortening codes whilst increasing the minimum distance is generalised. An analysis of this method reveals a close relationship with methods of extending codes. Some new codes from Goppa codes are found by exploiting this relationship. Finally an extension method for BCH codes is presented and this method is shown be as good as a well known method of extension in certain cases