On algebras admitting a complete set of near weights, evaluation codes, and Goppa codes

نویسندگان

  • Cícero Carvalho
  • Ercilio da Silva
چکیده

In 1998 Høholdt, van Lint and Pellikaan introduced the concept of a " weight function " defined on a F q-algebra and used it to construct linear codes, obtaining among them the algebraic-geometric (AG) codes supported on one point. Later, in 1999, it was proved by Matsumoto that all codes produced using a weight function are actually AG codes supported on one point. Recently, " near weight functions " (a generalization of weight functions), also defined on a F q-algebra, were introduced to study codes supported on two points. In this paper we show that an algebra admits a set of m near weight functions having a compatibility property, namely, the set is a " complete set " , if and only if it is the ring of regular functions of an affine geometrically irreducible algebraic curve defined over F q whose points at infinity have a total of m rational branches. Then the codes produced using the near weight functions are exactly the AG codes supported on m points. A formula for the minimum distance of these codes is presented with examples which show that in some situations it compares better than the usual Goppa bound.

برای دانلود رایگان متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

One-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 کامل

Parameters 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 کامل

On Codes from Norm-Trace Curves

The main results of this paper are derived by using only simple Gröbner basis techniques. We present a new construction of evaluation codes from MiuraKamiya curves Cab. We estimate the minimum distance of the codes and estimate the minimum distance of a class of related one-point geometric Goppa codes. With respect to these estimates the new codes perform at least as well as the related geometr...

متن کامل

Weierstrass Pairs and Minimum Distance of Goppa Codes

We prove that elements of the Weierstrass gap set of a pair of points may be used to define a geometric Goppa code which has minimum distance greater than the usual lower bound. We determine the Weierstrass gap set of a pair of any two Weierstrass points on a Hermitian curve and use this to increase the lower bound on the minimum distance of particular codes defined using a linear combination o...

متن کامل

Generalized Hamming Weights of q-ary Reed-Muller Codes

The order bound on generalized Hamming weights is introduced in a general setting of codes on varieties which comprises both the one point geometric Goppa codes as the q-ary Reed-Muller codes. For the latter codes it is shown that this bound is sharp and that they satisfy the double chain condition.

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

عنوان ژورنال:
  • Des. Codes Cryptography

دوره 53  شماره 

صفحات  -

تاریخ انتشار 2009