We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Published online by Cambridge University Press: 05 July 2011
Abstract
In this note, we present a sketch of several interplays between Gröbner bases theory and coding theory. For readers who are not so familiar to coding theory, some introductory explanations on error-correcting codes and coding theory are included. The main topics are some problems of encoding and decoding of algebraic codes which are related to Gröbner bases. Some simple examples of codes are referred to. In particular, recent developments in coding theory which have been done around multidimensional or multivariate codes have initiated and strengthened the connections, and several new problems and relevant algorithms have been explored in coding theory.
Introduction
In this note, we give a sketch of several interactions between coding theory (or rather algebraic coding theory) and Gröbner bases theory. Precisely, coding theory is the theory of error-correcting codes, which are used widely in digital communication and storage systems to transmit or to store digital information error-free or as correct as possible. Although error-correcting codes are probabilistic by nature in the sense that they are used to battle against random noise in transmission or storage channels, practical codes are constructed in deterministic fashions with algebraic methods, without which error control cannot be achieved with low complexity and cost. As a result, almost all of them are algebraic. Therefore we discuss nothing but the algebraic error-correcting codes.
To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.
