Contents -- Preface -- Sage: A Basic Overview for Coding Theory and Cryptography D. Joyner -- 0.1. Introduction -- 0.2. What is Sage? -- 0.2.1. Functionality of selected components of Sage -- 0.2.2. History -- 0.2.3. Why Python? -- 0.2.4. The CLI -- 0.2.5. The GUI -- 0.2.6. Open source philosophy -- 0.3. Coding theory functionality in Sage -- 0.3.1. General constructions -- 0.3.2. Coding theory functions -- 0.3.3. Weight enumerator polynomial -- 0.3.4. More code constructions -- 0.3.5. Automorphism group of a code -- 0.3.6. Even more code constructions -- 0.3.7. Block designs and codes -- 0.3.8. Special constructions -- 0.3.9. Coding theory bounds -- 0.3.10. Asymptotic bounds -- 0.4. Cryptography in Sage -- 0.4.1. Classical cryptography -- 0.4.2. Algebraic cryptosystems -- 0.4.3. RSA -- 0.4.4. Discrete logs -- 0.4.5. Diffle-Hellman -- 0.4.6. Linear feedback shift registers -- 0.4.7. BBS streamcipher -- 0.4.8. Blum-Goldwasser cryptosystem -- 0.5. Miscellaneous topics -- 0.5.1. Duursma zeta functions -- 0.5.2. Self-dual codes -- 0.5.3. Cool example (on self-dual codes) -- 0.6. Coding theory not implemented in Sage -- References -- Aspects of Random Network Coding O. Geil and C. Thomsen -- 1.1. Introduction -- 1.2. The network coding problem -- 1.2.1. Linear network coding for multicast -- 1.2.2. A polynomial time algorithm for solving the multicast problem -- 1.3. Random network coding -- 1.3.1. The algebraic approach -- 1.3.2. The combinatorial approach -- 1.3.2.1. Flow bounds -- 1.3.2.2. The bounds by Balli, Yan, and Zhang -- 1.4. Bibliographic notes -- References -- Steganography from a Coding Theory Point of View C. Munuera -- 2.1. Introduction -- 2.1.1. What is steganography? -- 2.1.2. Digital steganography -- 2.1.3. Steganography, cryptography and watermarking -- 2.1.4. About this chapter -- 2.2. Steganographic systems -- 2.2.1. The cover.

2.2.2. Steganographic schemes -- 2.2.3. Selection rules -- 2.2.4. Parameters -- 2.2.5. Proper stegoschemes -- 2.3. Error-Correcting codes -- 2.3.1. Correcting errors -- 2.3.2. Linear codes over fields -- 2.3.3. An example: binary Hamming codes -- 2.3.4. Generalized Hamming weights for linear codes -- 2.4. Linking the problems -- 2.4.1. Stegoschemes and error-correcting codes -- 2.4.2. Group codes and stegoschemes -- 2.4.3. Linear stegoschemes over rings Zq -- 2.4.4. Linear stegoschemes over fields -- 2.5. Bounds -- 2.5.1. The domain of stegoschemes -- 2.5.2. Balls and entropy -- 2.5.3. A Hamming-like bound -- 2.5.4. Asymptotic bounds -- 2.5.5. Perfect stegoschemes -- 2.5.6. Another new problem for coding theory -- 2.6. Nonshared selection rules -- 2.6.1. Wet paper codes -- 2.6.2. Solvability and the weight hierarchy of codes -- 2.6.3. The rank of random matrices -- 2.7. The ZZW embedding construction -- 2.7.1. Description of the method -- 2.7.2. Asymptotic behavior -- 2.8. Bibliographical notes and further reading -- Acknowledgments -- References -- An Introduction to LDPC Codes I. Marquez-Corbella and E. Martınez-Moro -- 3.1. Introduction -- 3.2. Representation for LDPC codes -- 3.2.1. Tanner graph -- 3.3. Communication channels -- 3.4. Decoding algorithms -- 3.4.1. Maximum-Likelihood Decoding -- 3.4.2. Iterative decoding -- 3.4.2.1. Min-Sum Decoding -- 3.4.2.2. Sum-Product Decoding -- 3.4.3. Linear Programming Decoding -- 3.5. Connections between LP and ML decoding -- 3.6. Pseudocodewords -- 3.7. Connections between LP and iterative decoding -- References -- Numerical Semigroups and Codes M. Bras-Amoros -- 4.1. Introduction -- 4.2. Numerical semigroups -- 4.2.1. Paradigmatic example: Weierstrass semigroups on algebraic curves -- 4.2.1.1. Algebraic curves -- 4.2.1.2. Weierstrass semigroup -- 4.2.1.3. Examples -- 4.2.2. Basic notions and problems.

4.2.2.1. Genus, conductor, gaps, non-gaps, enumeration -- 4.2.2.2. Generators, Apery set -- 4.2.2.3. Frobenius' coin exchange problem -- 4.2.2.4. Hurwitz question -- 4.2.2.5. Wilf conjecture -- 4.2.3. Classification -- 4.2.3.1. Symmetric and pseudo-symmetric numerical semigroups -- 4.2.3.2. Arf numerical semigroups -- 4.2.3.3. Numerical semigroups generated by an interval -- 4.2.3.4. Acute numerical semigroups -- 4.2.4. Characterization -- 4.2.4.1. Homomorphisms of semigroups -- 4.2.4.2. The operation, the sequence, and the sequence -- 4.2.4.3. Characterization of a numerical semigroup by -- 4.2.4.4. Characterization of a numerical semigroup by -- 4.2.4.5. Characterization of a numerical semigroup by -- 4.2.5. Counting -- 4.3. Numerical semigroups and codes -- 4.3.1. One-point codes and their decoding -- 4.3.1.1. One-point codes -- 4.3.1.2. Decoding one-point codes -- 4.3.2. The sequence, classical codes, and Feng-Rao im- proved codes -- 4.3.2.1. The sequence and the minimum distance of classical codes -- 4.3.2.2. On the order bound on the minimum distance -- 4.3.2.3. The sequence and Feng-Rao improved codes -- 4.3.2.4. On the improvement of the Feng-Rao improved codes -- 4.3.3. Generic errors and the sequence -- 4.3.3.1. Generic errors -- 4.3.3.2. Conditions for correcting generic errors -- 4.3.3.3. Comparison of improved codes and classical codes correct- ing generic errors -- 4.3.3.4. Comparison of improved codes correcting generic errors and Feng-Rao improved codes -- Further reading -- Acknowledgments -- References -- Codes, Arrangements and Matroids R. Jurrius and R. Pellikaan -- 5.1. Introduction -- 5.2. Error-correcting codes -- 5.2.1. Codes and Hamming distance -- 5.2.2. Linear codes -- 5.2.3. Generator matrix -- 5.2.4. Parity check matrix -- 5.2.5. Inner product and dual codes -- 5.2.6. The Hamming and simplex codes.

5.2.7. Singleton bound and MDS codes -- 5.3. Weight enumerators and error probability -- 5.3.1. Weight spectrum -- 5.3.2. The decoding problem -- 5.3.3. The q-ary symmetric channel -- 5.3.4. Error probability -- 5.4. Codes, projective systems and arrangements -- 5.5. The extended and generalized weight enumerator -- 5.5.1. Generalized weight enumerators -- 5.5.2. Extended weight enumerator -- 5.5.3. Puncturing and shortening of codes -- 5.5.4. Connections -- 5.5.5. MDS-codes -- 5.6. Matroids and codes -- 5.6.1. Matroids -- 5.6.2. Graphs, codes and matroids -- 5.6.3. The weight enumerator and the Tutte polynomial -- 5.6.4. Deletion and contraction of matroids -- 5.6.5. MacWilliams type property for duality -- 5.7. Posets and lattices -- 5.7.1. Posets, the Mobius function and lattices -- 5.7.2. Geometric lattices -- 5.7.3. Geometric lattices and matroids -- 5.8. The characteristic polynomial -- 5.8.1. The characteristic and coboundary polynomial -- 5.8.2. The M obius polynomial and Whitney numbers -- 5.8.3. Minimal codewords and subcodes -- 5.8.4. The characteristic polynomial of an arrangement -- 5.8.5. The characteristic polynomial of a code -- 5.8.6. Examples and counterexamples -- 5.9. Overview of polynomial relations -- 5.10. Further reading and open problems -- 5.10.1. Multivariate and other polynomials -- 5.10.2. The coset leader weight enumerator -- 5.10.3. Graph codes -- 5.10.4. The reconstruction problem -- 5.10.5. Questions concerning the M obius polynomial -- 5.10.6. Monomial conjectures -- 5.10.7. Complexity issues -- 5.10.8. The zeta function -- References.