Proof of Conjectures on the True Dimension of Some Binary Goppa Codes
نویسنده
چکیده
There is a classical lower bound on the dimension of a binary Goppa code. We survey results on some specific codes whose dimension exceeds this bound, and prove two conjectures on the true dimension of two classes of such codes.
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 کاملCodes Derived from Binary Goppa Codes
We present a new family of binary codes derived from the family of classical Goppa codes. We generalize properties of Goppa codes to this family and deduce bounds on the dimension and on the minimum distance, and the existence of a polynomial-time decoding algorithm up to a constructed error-correcting capability. Asymptotically these codes have the same parameters as Goppa codes.
متن کاملOn the gonality of curves, abundant codes and decoding
Let X be a curve defined over the finite field Fq with q elements. The genus of X is denoted by g(X ), or more often by g. Let P1, . . . , Pn be n distinct rational points on the curve X . Let D be the divisor P1 + . . . + Pn. Let G be a divisor on X of degree m. The code CL(D,G) is defined as the image of L(G) in F n q , under the evaluation map f 7−→ (f(P1), . . . , f(Pn)). Goppa [5] showed t...
متن کاملClass of Binary Generalized Goppa Codes Perfect in Weighted Hamming Metric
The class of the binary generalized Goppa codes is offered. It is shown that the codes of this class are on the Hamming bound constructed for a weighted metric.
متن کاملExponential Sums and Goppa Codes: I
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...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Des. Codes Cryptography
دوره 36 شماره
صفحات -
تاریخ انتشار 2005