Self-Dual Codes and Invariant Theory

Transcription

1 Gabriele Nebe Eric M. Rains Neil J.A. Sloane Self-Dual Codes and Invariant Theory With 10 Figures and 34 Tables 4y Springer

2 Preface List of Symbols List of Tables List of Figures v xiv xxv xxvii 1 The Type of a Self-Dual Code Quadratic maps Self-dual and isotropic codes Twisted modules and their representations Twisted rings and their representations Triangular twisted rings Quadratic pairs and their representations Form rings and their representations The Type of a code Triangular form rings Matrix rings of form rings and their representations Automorphism groups of codes Shadows 24 2 Weight Enumerators and Important Types Weight enumerators of codes Mac Williams identity and generalizations The weight enumerator of the shadow Catalogue of important types Binary codes ! 41 2n 41 2s Euclidean codes 42 4 E 42

3 xviii Contents q E (even) 43 <$ q E (odd) 46 qf (odd) Hermitian codes 47 4 H 47 q fl Additive codes 48 4 H+ 48 q H + (even) 49 g?+ (even) 49 q% + (even) 50 & (even) 50 q B+ (odd) 50 qf+ (odd) Codes over Galois rings Z/mZ 51 4 Z 52 m z 53 m\ 54 m\ x 54 mfj x 55 m Codes over more general Galois rings 55 GRQ/,/) E 55 GR(p p,f)f 56 GR(p e,/) E 56 GR(2 e,/) 57 GR(2 e,/)p! 57 GR(2 e,/) E 2 58 GR(p e,/) H 58 GR(p e,/)«, 58 /) H + 59 f)f Linear codes over p-adic integers 60 Z p 60 More general p-adic integers Examples of self-dual codes : Binary codes 60 2j: Singly-even binary self-dual codes 61 2JI: Doubly-even binary self-dual codes E : Euclidean self-dual codes over F q E (even or odd): Euclidean self-dual codes over q

4 2.4.4 (/{p Generalized doubly-even self-dual codes : Euclidean self-dual codes over F H : Hermitian self-dual codes over F q H : Hermitian self-dual linear codes over q H+ : Trace-Hermitian additive codes over F Z : Self-dual codes over Z/4Z Codes over other Galois rings Z p : Codes over the p-adic numbers The Gleason-Pierce Theorem 80 Closed Codes Bilinear forms and closed codes Families of closed codes Codes over commutative rings Codes over quasi-frobenius rings Algebras over a commutative ring Direct summands Representations of twisted rings and closed codes Morita theory New representations from old Subquotients and quotients Direct sums and products Tensor products 100 The Category Quad The category of quadratic groups The internal hom-functor IHom Properties of quadratic rings Morita theory for quadratic rings Morita theory for form rings Witt rings, groups and modules 121 The Main Theorems Parabolic groups Hyperbolic co-unitary groups Generators for the hyperbolic co-unitary group Clifford-Weil groups Scalar elements in C(p) Clifford-Weil groups and full weight enumerators Results from invariant theory Molien series Relative invariants Construction of invariants using differential operators Invariants and designs Symmetrizations 162 xix

5 5.8 Example: Hermitian codes over Fg 167 Real and Complex Clifford Groups Background Runge's theorems The real Clifford group C m The complex Clifford group X m Barnes-Wall lattices Maximal finiteness in real case Maximal finiteness in complex case Automorphism groups of weight enumerators 190 Classical Self-Dual Codes Quasisimple form rings Split type q iin : Linear codes over F, 196 Clifford-Weil groups 198 F 2, Genus F 2, Genus Hermitian type q H : Hermitian self-dual codes over F g 202 Clifford-Weil groups 202 The case q = The case q = Orthogonal (or Euclidean) type, p odd q E (odd): Euclidean self-dual codes over q 207 Clifford-Weil groups (q odd) 207 The case q = The case q = 3, genus The case q = The case q = Symplectic type, p odd g H+ (odd): Hermitian F r -linear codes over q, q = r Clifford-Weil groups (genus g) 214 The case q = 9, genus Characteristic 2, orthogonal and symplectic types q u+ (even): Hermitian F r -linear codes over q, q = r Clifford-Weil groups (genus g) 217 The case q = 4, genus The case q = A, genus The case q = q E (even): Euclidean self-dual F 9 -linear codes 220 Clifford-Weil groups (genus g) 220 The case q = The case q = A 221

6 xxi q^+ (even): Even Trace-Hermitian F r -linear codes 222 Clifford-Weil groups (genus g) 222 The case q = 4, genus q^ (even): Generalized Doubly-even codes over q 224 Clifford-Weil groups (genus g) 224 The case k = 2, arbitrary genus 225 The case k = F 4, genus The case k = F Further Examples of Self-Dual Codes m z : Codes over Z/rnZ Z : Self-dual codes over Z/4Z Z : Type I self-dual codes over Z/4Z Z : Type I self-dual codes over Z/4Z containing Same, with 1 in the shadow f T : Type II self-dual codes over Z/4Z f u : Type II self-dual codes over Z/4Z containing Z : Self-dual codes over Z/8Z Codes over more general Galois rings GR(p e, /) E : Euclidean self-dual GR(p e, /)-linear codes GR(p e, /) H : Hermitian self-dual GR(p e, /)-linear codes GR(p e, 2Z) H+ : Trace-Hermitian GR(p e, i)-linear codes Clifford-Weil groups for GR(4,2) Self-dual codes over F, 2 + q 2 u 243 Lattices Lattices and theta series Preliminary definitions Modular lattices and Atkin-Lehner involutions Shadows Jacobi forms Siegel theta series 261 Jacobi-Siegel theta series and Riemann theta functions 265 Riemann theta functions with Harmonic coefficients Hilbert theta series Positive definite form R-algebras Half-spaces Form orders and lattices Even and odd unimodular lattices Gluing theory for codes Gluing theory for lattices 282

7 xxii Contents 10 Maximal Isotropic Codes and Lattices Maximal isotropic codes Maximal isotropic doubly-even binary codes Maximal isotropic even binary codes Maximal isotropic ternary codes Maximal isotropic additive codes over F Maximal isotropic codes over Z/4Z Maximal even lattices Maximal even lattices of determinant 3 fc Maximal even and integral lattices of determinant 2 k Extremal and Optimal Codes Upper bounds Extremal weight enumerators and the LP bound Self-dual binary codes, 2 n and 2 r Some other types A new definition of extremality Asymptotic upper bounds Lower bounds Tables of extremal self-dual codes Binary codes Type 3: Ternary codes Types 4 E and 4f t : Euclidean self-dual codes over F Type 4 H : Hermitian linear self-dual codes over F Types 4 H+ and 4" + : Trace-Hermitian codes over F Type 4 Z : Self-dual codes over Z/4Z Other types Enumeration of Self-Dual Codes The mass formulae Enumeration of binary self-dual codes 350 Interrelations between types 2i and 2n Type 3: Ternary self-dual codes Types 4 E and 4f x : Euclidean self-dual codes over F Type 4 H : Hermitian self-dual codes over F Type 4 H+ : Trace-Hermitian additive codes over F Type 4 Z : Self-dual codes over Z/4Z Other enumerations Quantum Codes Definitions Additive and symplectic quantum codes Hamming weight enumerators 376

8 xxiii 13.4 Linear programming bounds Other alphabets A table of quantum codes 385 References 391 Index 417