Exponential Sums and Goppa Codes: I
نویسندگان
چکیده
A bound is obtained which generalizes the Carlitz-Uchiyama result, based on a theorem of Bombieri and Weil about exponential sums. This new bound is used to estimate the covering radius of long binary Goppa codes. A new lower bound is also derived on the minimum distance of the dual of a binary Goppa code, similar to that for BCH codes. This is an example of the use of a number-theory bound for the problem of the estimation of minimum distance of codes, as posed in research problem 9.9 of Mac Williams and Sloane, The Theory of Error Correcting Codes.
منابع مشابه
Artin-Schreier Curves, Exponential Sums, and Coding Theory
Lachaud, G., Artin-Schreier curves, exponential sums, and coding theory, Theoretical Computer Science 94 (1992) 295-310. This is a survey of some results recently obtained on the distribution of the weights of some classical linear codes on the one hand, such as the dual of the Melas code, and the geometric BCH codes discovered by Goppa (subfield subcodes of Goppa codes) on the other hand. Thes...
متن کاملParameters of Goppa codes revisited
We discuss parameters of Goppa codes, such as minimum distance, covering radius, distance distribution, and generalized Hamming weights. By a variation on the exponential sums method and combinatorial arguments, we sharpen known bounds.
متن کاملHomogeneous Weights and Exponential Sums
In this paper, we give a formula as an exponential sum for a homogeneous weight defined by Constantinescu and Heise [3] in the case of Galois rings (or equivalently, rings of Witt vectors) and use this formula to estimate the weight of codes obtained from algebraic geometric codes over rings.
متن کاملOne-point Goppa Codes on Some Genus 3 Curves with Applications in Quantum Error-Correcting Codes
We investigate one-point algebraic geometric codes CL(D, G) associated to maximal curves recently characterized by Tafazolian and Torres given by the affine equation yl = f(x), where f(x) is a separable polynomial of degree r relatively prime to l. We mainly focus on the curve y4 = x3 +x and Picard curves given by the equations y3 = x4-x and y3 = x4 -1. As a result, we obtain exact value of min...
متن کاملBinary Spherical Geometric Codes
Let q be a power of an odd prime number and Fq be the finite field with q elements. We will construct a binary spherical code from an algebraic curve C defined over Fq and a rational divisor G on C, as the twist by the quadratic character 11 of the Goppa code L(G). The computation of the parameters of this code is based on the study of some character sums. 0. Introduction. In a previous paper (...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2010