Designs From Linear Codes

Since the publication of the first edition of this monograph, a generalisation of the Assmus-Mattson theorem for linear codes over finite fields has been developed, two 70-year breakthroughs and a considerable amount of other progress on t-designs from linear codes have been made. This second edition is a substantial revision and expansion of the first edition. Two new chapters and two new appendices have been added, and most chapters of the first edition have been revised.It provides a well-rounded and detailed account of t-designs from linear codes. Most chapters of this book cover the support designs of linear codes. A few chapters deal with designs obtained from linear codes in other ways. Connections among ovals, hyperovals, maximal arcs, ovoids, special functions, linear codes and designs are also investigated. This book consists of both classical and recent results on designs from linear codes.It is intended to be a reference for postgraduates and researchers who work on combinatorics, or coding theory, or digital communications, or finite geometry. It can also be used as a textbook for postgraduates in these subject areas.

Table of Contents

1.Mathematical Foundations

2.Linear Codes over Finite Fields

3.Cyclic Codes over Finite Fields

4.Designs and Codes

5.Designs of Binary Reed-Muller Codes

6.Affine Invariant Codes and Their Designs

7.Weights in Some BCH Codes over GF(q)

8.Designs from Four Types of Linear Codes

9.Designs from BCH Codes

10.Designs from Codes with Regularity

11.Designs from QR and Self-Dual Codes

12.Designs from Arc and MDS Codes

13.Designs from Oviod Codes

14.Quasi-Symmetric Designs from Bent Codes

15.Almost M DS Codes and Their Designs

16.Beyond the Assmus-Mattson Theorem

Appendix A Sporadic Designs from Linear Codes 

Appendix B Designs from Binary Codes with Regularities 

Appendix C Exercises on Mathematical Foundations