Channel coding lies at the heart of digital communication and data storage. Fully updated to include current innovations in the field, including a new chapter on polar codes, this detailed introduction describes the core theory of channel coding, decoding algorithms, implementation details, and performance analyses. This edition includes over 50 new end-of-chapter problems to challenge students and numerous new figures and examples throughout.

The authors emphasize a practical approach and clearly present information on modern channel codes, including polar, turbo, and low-density parity-check (LDPC) codes, as well as detailed coverage of BCH codes, Reed–Solomon codes, convolutional codes, finite geometry codes, and product codes for error correction, providing a one-stop resource for both classical and modern coding techniques.

Assuming no prior knowledge in the field of channel coding, the opening chapters begin with basic theory to introduce newcomers to the subject. Later chapters then begin with classical codes, continue with modern codes, and extend to advanced topics such as code ensemble performance analyses and algebraic LDPC code design.

• 300 varied and stimulating end-of-chapter problems test and enhance learning, making this an essential resource for students and practitioners alike.

• Provides a one-stop resource for both classical and modern coding techniques.

• Starts with the basic theory before moving on to advanced topics, making it perfect for newcomers to the field of channel coding.

• 180 worked examples guide students through the practical application of the theory.

Channel coding lies at the heart of digital communication and data storage. Fully updated to include current innovations in the field, including a new chapter on polar codes, this detailed introduction describes the core theory of channel coding, decoding algorithms, implementation details, and performance analyses. This edition includes over 50 new end-of-chapter problems to challenge students and numerous new figures and examples throughout.

The authors emphasize a practical approach and clearly present information on modern channel codes, including polar, turbo, and low-density parity-check (LDPC) codes, as well as detailed coverage of BCH codes, Reed–Solomon codes, convolutional codes, finite geometry codes, and product codes for error correction, providing a one-stop resource for both classical and modern coding techniques.

Assuming no prior knowledge in the field of channel coding, the opening chapters begin with basic theory to introduce newcomers to the subject. Later chapters then begin with classical codes, continue with modern codes, and extend to advanced topics such as code ensemble performance analyses and algebraic LDPC code design.

• 300 varied and stimulating end-of-chapter problems test and enhance learning, making this an essential resource for students and practitioners alike.

• Provides a one-stop resource for both classical and modern coding techniques.

• Starts with the basic theory before moving on to advanced topics, making it perfect for newcomers to the field of channel coding.

• 180 worked examples guide students through the practical application of the theory.

Channel coding lies at the heart of digital communication and data storage. Fully updated to include current innovations in the field, including a new chapter on polar codes, this detailed introduction describes the core theory of channel coding, decoding algorithms, implementation details, and performance analyses. This edition includes over 50 new end-of-chapter problems to challenge students and numerous new figures and examples throughout.

The authors emphasize a practical approach and clearly present information on modern channel codes, including polar, turbo, and low-density parity-check (LDPC) codes, as well as detailed coverage of BCH codes, Reed–Solomon codes, convolutional codes, finite geometry codes, and product codes for error correction, providing a one-stop resource for both classical and modern coding techniques.

Assuming no prior knowledge in the field of channel coding, the opening chapters begin with basic theory to introduce newcomers to the subject. Later chapters then begin with classical codes, continue with modern codes, and extend to advanced topics such as code ensemble performance analyses and algebraic LDPC code design.

• 300 varied and stimulating end-of-chapter problems test and enhance learning, making this an essential resource for students and practitioners alike.

• Provides a one-stop resource for both classical and modern coding techniques.

• Starts with the basic theory before moving on to advanced topics, making it perfect for newcomers to the field of channel coding.

• 180 worked examples guide students through the practical application of the theory.

Channel coding lies at the heart of digital communication and data storage. Fully updated to include current innovations in the field, including a new chapter on polar codes, this detailed introduction describes the core theory of channel coding, decoding algorithms, implementation details, and performance analyses. This edition includes over 50 new end-of-chapter problems to challenge students and numerous new figures and examples throughout.

The authors emphasize a practical approach and clearly present information on modern channel codes, including polar, turbo, and low-density parity-check (LDPC) codes, as well as detailed coverage of BCH codes, Reed–Solomon codes, convolutional codes, finite geometry codes, and product codes for error correction, providing a one-stop resource for both classical and modern coding techniques.

Assuming no prior knowledge in the field of channel coding, the opening chapters begin with basic theory to introduce newcomers to the subject. Later chapters then begin with classical codes, continue with modern codes, and extend to advanced topics such as code ensemble performance analyses and algebraic LDPC code design.

• 300 varied and stimulating end-of-chapter problems test and enhance learning, making this an essential resource for students and practitioners alike.

• Provides a one-stop resource for both classical and modern coding techniques.

• Starts with the basic theory before moving on to advanced topics, making it perfect for newcomers to the field of channel coding.

• 180 worked examples guide students through the practical application of the theory.

Channel coding lies at the heart of digital communication and data storage. Fully updated to include current innovations in the field, including a new chapter on polar codes, this detailed introduction describes the core theory of channel coding, decoding algorithms, implementation details, and performance analyses. This edition includes over 50 new end-of-chapter problems to challenge students and numerous new figures and examples throughout.

The authors emphasize a practical approach and clearly present information on modern channel codes, including polar, turbo, and low-density parity-check (LDPC) codes, as well as detailed coverage of BCH codes, Reed–Solomon codes, convolutional codes, finite geometry codes, and product codes for error correction, providing a one-stop resource for both classical and modern coding techniques.

Assuming no prior knowledge in the field of channel coding, the opening chapters begin with basic theory to introduce newcomers to the subject. Later chapters then begin with classical codes, continue with modern codes, and extend to advanced topics such as code ensemble performance analyses and algebraic LDPC code design.

• 300 varied and stimulating end-of-chapter problems test and enhance learning, making this an essential resource for students and practitioners alike.

• Provides a one-stop resource for both classical and modern coding techniques.

• Starts with the basic theory before moving on to advanced topics, making it perfect for newcomers to the field of channel coding.

• 180 worked examples guide students through the practical application of the theory.

Channel coding lies at the heart of digital communication and data storage. Fully updated to include current innovations in the field, including a new chapter on polar codes, this detailed introduction describes the core theory of channel coding, decoding algorithms, implementation details, and performance analyses. This edition includes over 50 new end-of-chapter problems to challenge students and numerous new figures and examples throughout.

The authors emphasize a practical approach and clearly present information on modern channel codes, including polar, turbo, and low-density parity-check (LDPC) codes, as well as detailed coverage of BCH codes, Reed–Solomon codes, convolutional codes, finite geometry codes, and product codes for error correction, providing a one-stop resource for both classical and modern coding techniques.

Assuming no prior knowledge in the field of channel coding, the opening chapters begin with basic theory to introduce newcomers to the subject. Later chapters then begin with classical codes, continue with modern codes, and extend to advanced topics such as code ensemble performance analyses and algebraic LDPC code design.

• 300 varied and stimulating end-of-chapter problems test and enhance learning, making this an essential resource for students and practitioners alike.

• Provides a one-stop resource for both classical and modern coding techniques.

• Starts with the basic theory before moving on to advanced topics, making it perfect for newcomers to the field of channel coding.

• 180 worked examples guide students through the practical application of the theory.

Channel coding lies at the heart of digital communication and data storage. Fully updated to include current innovations in the field, including a new chapter on polar codes, this detailed introduction describes the core theory of channel coding, decoding algorithms, implementation details, and performance analyses. This edition includes over 50 new end-of-chapter problems to challenge students and numerous new figures and examples throughout.

The authors emphasize a practical approach and clearly present information on modern channel codes, including polar, turbo, and low-density parity-check (LDPC) codes, as well as detailed coverage of BCH codes, Reed–Solomon codes, convolutional codes, finite geometry codes, and product codes for error correction, providing a one-stop resource for both classical and modern coding techniques.

Assuming no prior knowledge in the field of channel coding, the opening chapters begin with basic theory to introduce newcomers to the subject. Later chapters then begin with classical codes, continue with modern codes, and extend to advanced topics such as code ensemble performance analyses and algebraic LDPC code design.

• 300 varied and stimulating end-of-chapter problems test and enhance learning, making this an essential resource for students and practitioners alike.

• Provides a one-stop resource for both classical and modern coding techniques.

• Starts with the basic theory before moving on to advanced topics, making it perfect for newcomers to the field of channel coding.

• 180 worked examples guide students through the practical application of the theory.

We give a brief introduction to sphere-packing in n-dimensions and, in particular, to lattice packings. The notions of density and kissing number are explained. The 24-dimensional Leech lattice Λ is defined following Conway’s approach in his Three lectures on exceptional groups, see Conway (1971), with vectors in the lattice invariably displayed as they appear in the MOG. We explore the factor space Λ/2Λ, a 24-dimensional vector space over ${\mathbb{Z}}_2$, and show that its non-zero elements may be taken to be vectors of type 2 and 3 together with sets of 24 mutually orthogonal vectors of type 4 (and their negatives). These last elements are known as frames of reference or crosses; the six orbits on crosses under permutations of M24 and sign changes on $\mathcal{C}$-sets are described explicitly in the text, as are the orbits on vectors of types 2, 3 and 4. Finally, we explain how Λ can be defined in terms of a single Lorentz-type vector with 25 space-like coordinates and one time-like coordinate.

The binary Golay code $\mathcal{C}$ is defined as the 12-dimensional vector space over ${\mathbb{Z}}_2$ spanned by the 759 octads interpreted as vectors with eight 1s and 16 0s. The MOG is constructed by considering two 3-dimensional spaces over ${\mathbb{Z}}_2$, the Point space and the Line space, whose codewords are of length 8, and gluing three copies together in such a way as to obtain a 12-dimensional subspace of the 24-dimensional space P(Ω), consisting of all subsets of Ω. The minimal weight codewords in this 24-dimensional space are shown to have weight 8 and to total 759. The construction thus proves that a Steiner system S(5, 8, 24) exists, and provides a unique label for each codeword in the binary Golay code. We exhibit a natural isomorphism between the 24-dimensional space P(Ω) factored by $\mathcal{C}$ and the dual space ${\mathcal{C}}^{\star }$, and identify its elements as 24 monads, 276 duads, 2024 triads and $\left(\genfrac{}{}{0pt}{}{24}{4}\right)/6=1771$ sextets; this last division by 6 occurs because two tetrads 4 whose union is an octad are congruent modulo $\mathcal{C}$.

A sextet is a partition of the 24 points of Ω into six tetrads such that the union of any two of them is an octad. We find all ways in which two sextets can intersect one another and use this knowledge to force any Steiner system S(5, 8, 24) to assume the form of the one given by the MOG. In so doing we show that if a Steiner system S(5, 8, 24) exists then the order of its group of automorphisms is 244, 823, 040 and that it acts quintuply transitively on the 24 points. That the MOG does define an S(5, 8, 24) was proved in Chapter 4.

What is the minimal test to decide whether a permutation π ∈ S24 lies in our preferred copy of M24? The space $\mathcal{C}$ is 12 dimensional and so if we choose a basis of 12 codewords of $\mathcal{C}$, apply π to each codeword in the basis and verify that the image is also in $\mathcal{C}$ then π ∈ M24. The 12-dimensional subspace $\mathcal{C}$ is self-orthogonal with respect to the usual inner product, and so $\mathcal{C}={\mathcal{C}}^{\perp }$. Thus a vector is in $\mathcal{C}$ if, and only if, it is orthogonal to every codeword in a basis of $\mathcal{C}$. Now one codeword in our basis may be chosen to be the all 1s vector that is clearly fixed by any permutation; the other 11 can be chosen to be octads. In this chapter we show that we can do much better than this. In fact we show that we can choose 8 octads that are contained in one, and only one, copy of $\mathcal{C}$, but that any set of 7 octads is contained in no copy of $\mathcal{C}$ or in more than one. To this set of 8 octads we add a further 3 to form a basis together with the all 1s codeword. We now have a minimal test for membership of M24: apply π to each of the 8 octads; if the image in each case intersects each of the 11 octads in the basis evenly, then π is in M24, otherwise it is not. When working with M24 we often require an element possessing certain properties. In this chapter we show how to construct elements of shape 18.28, 212 and 16.36. We also reproduce a diagram due to Todd and Conway showing the orbits of M24 on the subsets of Ω.