Preface page xiii 1 Basic concepts of linear codes 1.1 Three fields 1.2 Linear codes, generator and parity check matrices 1.3 Dual codes 1.4 Weights and distances 1.5 New codes from old 1.5.1 Puncturing codes 1.5.2 Extending codes 1.5.3 Shortening codes 1.5.4 Direct sums 1.5.5 The (u | u + v) construction 1.6 Permutation equivalent codes 1.7 More general equivalence of codes 1.8 Hamming codes 1.9 The Golay codes 1.9.1 The binary Golay codes 1.9.2 The ternary Golay codes 1.10 Reed-Muller codes 1.11 Encoding, decoding, and Shannon's Theorem 1.11.1 Encoding 1.11.2 Decoding and Shannon's Theorem 1.12 Sphere Packing Bound, covering radius, and perfect codes 2 Bounds on the size of codes 2.1 A q (n, d) and B q (n, d) 5 2.2 The Plotkin Upper Bound viii Contents 2.3 The Johnson Upper Bounds 2.3.1 The Restricted Johnson Bound 2.3.2 The Unrestricted Johnson Bound 2.3.3 The Johnson Bound for A q (n, d) 2.3.4 The Nordstrom-Robinson code 2.3.5 Nearly perfect binary codes 2.4 The Singleton Upper Bound and MDS codes 2.5 The Elias Upper Bound 2.6 The Linear Programming Upper Bound 2.7 The Griesmer Upper Bound 2.8 The Gilbert Lower Bound 2.9 The Varshamov Lower Bound 2.10 Asymptotic bounds 2.10.1 Asymptotic Singleton Bound 2.10.2 Asymptotic Plotkin Bound 2.10.3 Asymptotic Hamming Bound 2.10.4 Asymptotic Elias Bound 2.10.5 The MRRW Bounds 2.10.6 Asymptotic Gilbert-Varshamov Bound 2.11 Lexicodes 4 Cyclic codes 4.1 Factoring x n − 1 4.2 Basic theory of cyclic codes 4.3 Idempotents and multipliers 4.4 Zeros of a cyclic code 4.5 Minimum distance of cyclic codes 4.6 Meggitt decoding of cyclic codes 4.7 Affine-invariant codes ix Contents 5 BCH and Reed-Solomon codes 5.1 BCH codes 5.2 Reed-Solomon codes 5.3 Generalized Reed-Solomon codes 5.4 Decoding BCH codes 5.4.1 The Peterson-Gorenstein-Zierler Decoding Algorithm 5.4.2 The Berlekamp-Massey Decoding Algorithm 5.4.3 The Sugiyama Decoding Algorithm 5.4.4 The Sudan-Guruswami Decoding Algorithm 5.5 Burst errors, concatenated codes, and interleaving 5.6 Coding for the compact disc 5.6.1 Encoding 5.6.2 Decoding 6 Duadic codes 6.1 Definition and basic properties 6.2 A bit of number theory 6.3 Existence of duadic codes 6.4 Orthogonality of duadic codes 6.5 Weights in duadic codes 6.6 Quadratic residue codes 6.6.1 QR codes over fields of characteristic 2 6.6.2 QR codes over fields of characteristic 3 6.6.3 Extending QR codes 6.6.4 Automorphisms of extended QR codes 7 Weight distributions 7.1 The MacWilliams equations 7.2 Equivalent formulations 7.3 A uniqueness result 7.4 MDS codes 7.5 Coset weight distributions 7.6 Weight distributions of punctured and shortened codes 7.7 Other weight enumerators 7.8 Constraints on weights 7.9 Weight preserving transformations 7.10 Generalized Hamming weights x Contents 8 Designs 8.1 t-designs 8.2 Intersection numbers 8.3 Complementary, derived, and residual designs 8.4 The Assmus-Mattson Theorem 8.5 Codes from symmetric 2-designs 8.6 Projective planes 8.7 Cyclic projective planes 8.8 The nonexistence of a projective plane of order 10 8.9 Hadamard matrices and...

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This paper presents a revised enumeration of the binary self-dual codes of length up to 32 given by Conway and Pless in 1978-1980. The list of 85 doubly-even selfdual codes of length 32 is essentially correct, but several of their descriptions need amending. The principal change is that there are 731 (not 664) inequivalent selfdual codes of length 30. Furthermore, there are three (not two) [28, 14,6] and 13 (not eight) [30, 15, 61 self-dual codes. Some additional information is provided about the self-dual codes of length less than 32.

A complete classification is given of all [22, 111 and [24, 121 binary self-dual codes. For each code we give the order of its group, the number of codes equivalent to it, and its weight distribution. There is a unique [24, 12, 61 selfdual code. Several theorems on the enumeration of self-dual codes are used, including formulas for the number of such codes with minimum distance > 4, and for the sum of the weight enumerators of all such codes of length n. Selforthogonal codes which are generated by code words of weight 4 are completely characterized.