Completely Regular Codes in Distance Regular Graphs

by Minjia Shi (Editor), Patrick Solé (Editor)

Overview

The concept of completely regular codes was introduced by Delsarte in his celebrated 1973 thesis, which created the field of Algebraic Combinatorics. This notion was extended by several authors from classical codes over finite fields to codes in distance-regular graphs. Half a century later, there was no book dedicated uniquely to this notion. Most of Delsarte examples were in the Hamming and Johnson graphs. In recent years, many examples were constructed in other distance regular graphs including q-analogues of the previous, and the Doob graph.

Completely Regular Codes in Distance Regular Graphs provides, for the first time, a definitive source for the main theoretical notions underpinning this fascinating area of study. It also supplies several useful surveys of constructions using coding theory, design theory and finite geometry in the various families of distance regular graphs of large diameters.

Features

• Written by pioneering experts in the domain
• Suitable as a research reference at the master’s level
• Includes extensive tables of completely regular codes in the Hamming graph
• Features a collection of up-to-date surveys

Product Details

ISBN-13:	9781032494449
Publisher:	CRC Press
Publication date:	03/13/2025
Series:	Chapman & Hall/CRC Monographs and Research Notes in Mathematics
Pages:	520
Product dimensions:	7.00(w) x 10.00(h) x (d)

About the Author

Minjia Shi earned his Ph.D. from the Institute of Computer Network Systems, Hefei University of Technology, China, in 2010. From August 2012 to August 2013, he was a visiting researcher with the School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore. From July 2016 to August 2016, he was a visiting researcher with Telecom Paris Tech, Paris, France. Later, he visited the Sobolev Institute of Mathematics in 2020. He has been a Professor at the School of Mathematical Sciences, Anhui University, since 2017. He is the author of more than 100 journal articles and two books. His current research interests include algebraic coding theory and cryptography.

Patrick Solé received the Ingénieur and Docteur-Ingénieur degrees both from École Nationale Supérieure des Télécommunications, Paris, France, in 1984 and 1987, respectively, and the habilitation à diriger des recherches from Université de Nice-Sophia Antipolis, Sophia Antipolis, France, in 1993.

He has held visiting positions at Syracuse University, Syracuse, NY, from 1987 to 1989, Macquarie University, Sydney, Australia, from 1994 to 1996, and Lille University, Lille, France, from 1999 to 2000.

Since 1989, he has been a permanent member of the CNRS and became Directeur de Recherche in 1996. He is currently a member of the CNRS lab I2M, Marseilles, France.

His research interests include coding theory (codes over rings, quasi-cyclic codes), interconnection networks (graph spectra, expanders), vector quantization (lattices), and cryptography (boolean functions, secret sharing schemes). He is the author of over 300 journal articles and 3 books.

Table of Contents

1. Completely regular codes and equitable partitions. 2. Completely regular codes over finite fields. 3. Completely regular codes in the Johnson graph. 4. Codes over rings and modules. 5. Group actions on codes in graphs. 6. Some completely regular codes in Doob graphs. 7. Completely regular codes: tables of small parameters for binary and ternary Hamming graphs.