Generalized weights and other coding theoretic invariants
Publisher
Neuchâtel : Université de Neuchâtel
Date issued
2024
Number of pages
154 p.
Subjects
Coding theory sum-rank metric codes convolutional codes generalized weights support MacWilliams’ Extension Theorem extremal codes
Abstract
Invariants play a crucial role in coding theory. They are in fact a valuable tool for classifying codes. Specifically, invariants help distinguish between non-equivalent codes. This distinction is relevant because equivalent codes exhibit the same decoding properties. However, determining whether two codes are equivalent is a hard problem.
In this thesis, we focus on a family of invariants called generalized weights. They were introduced in 1977 for linear block codes, but only became widely studied in 1991, when Wei proved that they measure the code robustness against wiretapping. The goal of this work is to define and study similar invariants for other classes of codes, such as codes over finite rings, sum-rank metric codes, and convolutional codes.
The first part of the thesis is dedicated to the notion of support for linear codes over finite commmutative rings. We investigate how to associate a combinatorial object, called latroid, with a code, and we explore which properties and invariants of the code can be recovered from it. We also show that the associated ideal is determined by the associated latroid.
The central chapters of the thesis concern sum-rank metric codes. First, we prove an anticode bound for this class of codes and we classify all the codes attaining this bound, i.e., the optimal anticodes. Then, we define the generalized weights for sum-rank metric codes in terms of optimal anticodes. We study their basic properties and we compute them in the case of maximum sum-rank distance codes.
In the final part of the thesis, we deal with convolutional codes. We give a definition of generalized weights that takes into account the module structure of this family of codes. After studying their basic properties, we prove that they can be computed by an algorithm that terminates in finite time. Then, we give upper and lower bounds for the generalized weights of maximum distance separable and maximum distance profile codes. We also discuss the notion of generalized column distances.
In this thesis, we focus on a family of invariants called generalized weights. They were introduced in 1977 for linear block codes, but only became widely studied in 1991, when Wei proved that they measure the code robustness against wiretapping. The goal of this work is to define and study similar invariants for other classes of codes, such as codes over finite rings, sum-rank metric codes, and convolutional codes.
The first part of the thesis is dedicated to the notion of support for linear codes over finite commmutative rings. We investigate how to associate a combinatorial object, called latroid, with a code, and we explore which properties and invariants of the code can be recovered from it. We also show that the associated ideal is determined by the associated latroid.
The central chapters of the thesis concern sum-rank metric codes. First, we prove an anticode bound for this class of codes and we classify all the codes attaining this bound, i.e., the optimal anticodes. Then, we define the generalized weights for sum-rank metric codes in terms of optimal anticodes. We study their basic properties and we compute them in the case of maximum sum-rank distance codes.
In the final part of the thesis, we deal with convolutional codes. We give a definition of generalized weights that takes into account the module structure of this family of codes. After studying their basic properties, we prove that they can be computed by an algorithm that terminates in finite time. Then, we give upper and lower bounds for the generalized weights of maximum distance separable and maximum distance profile codes. We also discuss the notion of generalized column distances.
Notes
Acceptée sur proposition du jury :
Prof. Elisa Gorla, Université de Neuchâtel, directrice de thèse
Prof. Jérémy Blanc, Université de Neuchâtel, membre interne
Dr. Lisa Seccia, Université de Neuchâtel, membre interne
Prof. Joachim Rosenthal, University of Zurich, expert externe
Prof. Hiram H. López Valdez, Virginia Tech, expert externe
Soutenue le 26 juin 2024
Prof. Elisa Gorla, Université de Neuchâtel, directrice de thèse
Prof. Jérémy Blanc, Université de Neuchâtel, membre interne
Dr. Lisa Seccia, Université de Neuchâtel, membre interne
Prof. Joachim Rosenthal, University of Zurich, expert externe
Prof. Hiram H. López Valdez, Virginia Tech, expert externe
Soutenue le 26 juin 2024
Publication type
doctoral thesis
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