Study advances error correction bounds for codes from special curves
A study of the $m$th relative Feng-Rao distances with a special emphasis on $C_{ab}$ curves
Information Theory
Summary
This paper focuses on improving ways to estimate how well certain error-correcting codes perform. These codes are built from mathematical objects called numerical semigroups and special curves known as C_ab curves. The authors expand earlier work to provide better lower bounds on the ability to detect and correct multiple errors simultaneously in these codes. They develop formulas that work for a broad range of these structures and methods to handle tricky cases where generic approaches are less effective.
What this means in practice
- •For communications engineers: Design error-correcting codes with improved bounds for nested code pairs used in data transmission systems relying on algebraic geometry codes.
- •For storage system developers: Create more efficient error detection schemes for data storage by utilizing refined weight bounds from one-point codes on special curves.
Authors
Eduardo Camps-Moreno, Olav Geil, Hiram H. López, Gretchen L. Matthews, Rodrigo San-José
Abstract
In the seminal paper [13], Farrán and Munuera introduced a method to establish generic information on the $m$th Feng-Rao distance via a function on arbitrary numerical semigroups that serves as a lower bound on the generalized Hamming weights of dual one-point algebraic geometry codes whenever the semigroup is Weierstrass. In the present paper, we study what we shall call the primary and dual, respectively, $m$th relative Feng-Rao distances, which serve as lower bounds on relative generalized Hamming weights of nested primary and dual, respectively, one-point codes under similar condition. A unified method for establishing generic formulas for $m$th relative Feng-Rao distances of general numerical semigroups is developed and for a large variety of numerical semigroups generated by two elements, closed formula expressions are established complementing Delgado et al.s work [8] on the $m$th Feng-Rao distances of such semigroups. Finally, for non-singular $C_{ab}$ curves, we develop a method to estimate the $m$th relative generalized Hamming weight of dual and primary one-point codes in those regions where the generic methods may fail to provide best possible results.