Fundamentals of Error-Correcting Codes

May 19, 2026

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Fundamentals of Error-Correcting Codes provides a rigorous introduction to coding theory from both engineering and mathematical perspectives. It covers classical foundations such as linear codes, generator and parity-check matrices, dual codes, weights and distances, and standard constructions for creating new codes. The text extends to modern techniques from specialized literature, including bounds on code size, finite fields, cyclic codes, and major decoding methods for BCH and Reed–Solomon codes, supported by numerous examples and exercises.

LanguageEnglish

CategoryTechnology

FormatPDF

Pages665

Reading Time27 h 56 min

Preface

Basic concepts of linear codes

Three fields

Linear codes, generator and parity check matrices

Dual codes

Weights and distances

New codes from old

Permutation equivalent codes

More general equivalence of codes

Hamming codes

The Golay codes

Reed–Muller codes

Encoding, decoding, and Shannon’s Theorem

Sphere Packing Bound, covering radius, and perfect codes

Bounds on the size of codes

Aq(n,d) and Bq(n,d)

The Plotkin Upper Bound

The Johnson Upper Bounds

The Singleton Upper Bound and MDS codes

The Elias Upper Bound

The Linear Programming Upper Bound

The Griesmer Upper Bound

The Gilbert Lower Bound

The Varshamov Lower Bound

Asymptotic bounds

Finite fields

Polynomials and the Euclidean Algorithm

Primitive elements

Constructing finite fields

Cyclic codes

Factoring xn − 1

Basic theory of cyclic codes

Zeros of a cyclic code

Meggitt decoding of cyclic codes

BCH and Reed–Solomon codes

BCH codes

Reed–Solomon codes

Decoding BCH codes

Duadic codes

FAQs

What perspectives does the book use to teach coding theory?

The book introduces coding theory from both an engineering viewpoint and a mathematical viewpoint, aiming to serve readers with different backgrounds.

Which foundational topics are covered in the early chapters?

It covers basic linear-code structures such as generator and parity-check matrices, dual codes, weights and distances, and several methods for constructing new codes from existing ones.

What later topics does the book cover beyond linear codes?

Later sections include bounds on code size, finite fields, cyclic codes and their decoding, and BCH and Reed–Solomon codes with multiple decoding algorithms.

May 19, 2026

Upload by Ophelia