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Decoding and Finding the Minimum Distance with Gröbner Bases: History and New Insights

Bulygin, Stanislav and Pellikaan, Ruud Woungang, Isaac and Misra, Sudip and Misra, Subhas Chandra (eds.) (2010):
Decoding and Finding the Minimum Distance with Gröbner Bases: History and New Insights.
In: Selected Topics in Information and Coding Theory, World Scientific, pp. 585-622, [Book Section]

Abstract

In this chapter we discuss decoding techniques and finding the minimum distance of linear codes with the use of Gröbner bases. First we give a historical overview of decoding cyclic codes via solving systems of polynomial equations over finite fields. In particular we mention papers of Cooper, Reed, Chen, Helleseth, Truong, Augot, Mora, Sala and others. Some structural theorems that use Gröbner bases in this context are presented. After that we shift to the general situation of arbitrary linear codes. We give an overview of approaches of Fitzgerald and Lax. Then we introduce our method of decoding linear codes that reduces this problem to solving a system of quadratic equations. We discuss open problems and future research possibilities.

Item Type: Book Section
Erschienen: 2010
Editors: Woungang, Isaac and Misra, Sudip and Misra, Subhas Chandra
Creators: Bulygin, Stanislav and Pellikaan, Ruud
Title: Decoding and Finding the Minimum Distance with Gröbner Bases: History and New Insights
Language: ["languages_typename_1" not defined]
Abstract:

In this chapter we discuss decoding techniques and finding the minimum distance of linear codes with the use of Gröbner bases. First we give a historical overview of decoding cyclic codes via solving systems of polynomial equations over finite fields. In particular we mention papers of Cooper, Reed, Chen, Helleseth, Truong, Augot, Mora, Sala and others. Some structural theorems that use Gröbner bases in this context are presented. After that we shift to the general situation of arbitrary linear codes. We give an overview of approaches of Fitzgerald and Lax. Then we introduce our method of decoding linear codes that reduces this problem to solving a system of quadratic equations. We discuss open problems and future research possibilities.

Title of Book: Selected Topics in Information and Coding Theory
Publisher: World Scientific
ISBN: 978-981-283-716-5
Uncontrolled Keywords: Secure Data
Divisions: LOEWE > LOEWE-Zentren > CASED – Center for Advanced Security Research Darmstadt
LOEWE > LOEWE-Zentren
LOEWE
Date Deposited: 30 Dec 2016 20:23
Identification Number: TUD-CS-2009-0136
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