Codes and Rings: Theory and Practice

Codes and Rings: Theory and Practice is a systematic review of literature that focuses on codes over rings and rings acting on codes. Since the breakthrough works on quaternary codes in the 1990s, two decades of research have moved the field far beyond its original periphery. This book fills this gap by consolidating results scattered in the literature, addressing classical as well as applied aspects of rings and coding theory.

New research covered by the book encompasses skew cyclic codes, decomposition theory of quasi-cyclic codes and related codes and duality over Frobenius rings. Primarily suitable for ring theorists at PhD level engaged in application research and coding theorists interested in algebraic foundations, the work is also valuable to computational scientists and working cryptologists in the area.

Consolidates 20+ years of research in one volume, helping researchers save time in the evaluation of disparate literatureDiscusses duality formulas in the context of Frobenius ringsReviews decomposition of quasi-cyclic codes under ring actionEvaluates the ideal and modular structure of skew-cyclic codesSupports applications in data compression, distributed storage, network coding, cryptography and across error-correction

1. Introduction 2. Motivation 3. Rings 4. Distances 5. Few Weights Codes 6. Linear Codes 7. Self-dual codes 8. Cyclic codes 9. Quasi-cyclic codes 10. Quasi-Twisted Codes 11. Generalized Quasi-Cyclic Codes 12. Skew Cyclic Codes 13. MDE Codes 14. Convoluted Codes 15. Character Sums