Signal Space Coding over Rings

3.4 Parameters for a constellation 59
3.4.1 Signal-to-Noise ratio efficiency 59
3.4.2 Coding and shaping 60
3.4.3 Coding and shaping gain 60
3.5 Coset codes 62
3.5.1 Introduction 62
3.5.2 Lattice partitioning and cosets 64
3.6 Coding gain of the encoding procedure: The normalised redundancy 66
3.7 Generalised Group Alphabets 68
3.7.1 Introduction 68
3.7.2 Definition of a Group Alphabet 68
3.7.3 Distance properties in a GGA 69
3.7.4 Set partitioning of a GGA 70
3.7.5 Chain partitions 72
3.8 Geometrically Uniform (GU) Codes 72
3.8.1 Introduction 72
3.8.2 Isometries 73
3.8.3 Symmetry groups 74
3.9 Geometrically uniform signal sets 75
3.9.1 Introduction 75
3.9.2 Generating groups 75
3.9.3 Examples of GU signal sets 75
3.9.4 Properties of GU signal sets 76
3.9.5 Geometrically uniform partitions 78
3.9.6 Isometric labelings 78
3.10 Signal Space codes 80
3.10.1 Introduction 80
3.10.2 Definition of a Signal Space code 81
3.10.3 Definition of a Multilevel Signal Space code 82
3.11 Signal Space 83
3.12 Sets of orthogonal functions 84
3.12.1 Introduction 84
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