Signal Space Coding over Rings

Chapter 1: Introduction 6
given signal set S related to an isometric labeling that maps a label code output into a
given signal of the set [1].
Any signal space code is based on the construction of a signal constellation. An effort
has been made on the design of multidimensional constellations, mainly represented
by lattices. Other more elaborate signal sets are based on the design of N-dimensional
constellations. Based on the characterisation of lattice codes, that is, codes defined
over a lattice, Forney and Wei [9] define parameters like shaping gain and coding
gain, useful for characterising a given multidimensional constellation, together with
the Constellation Figure of Merit (CFM).
In general terms, high dimension constellations reduce difficulties in obtaining
rotationally invariant (RI) codes, an important property of a combined coding and
modulation scheme in several applications.
On the other hand, the number of neighbours at the same distance, also called the
kissing number, is another characteristic to be considered for a given constellation. It
can be expected that this number increases while the dimensionality does, especially
when the constellation is GU. GU signal sets are characterised by the fact of having
Voronoi regions of the same shape [1, 10].
Chapter 4 deals with the design of ring-TCM schemes, that is, with a Trellis-Coded
Modulation scheme for which the coding machine is a ring-FSSM, particularly, a
Multilevel Convolutional Encoder that operates over the ring of integers modulo-Q.
This is a multilevel signal space code over rings, using convolutional coding.
Baldini [72, 76, 77, 94], Farrell [72, 78, 82, 90, 91, 92, 94], Acha [38, 39], Carrasco
[38, 39, 78, 82, 91, 92, 94], Lopez [80, 82, 91, 94], Honary [40, 89], and Ahmadian-
Attari [79, 90] among others, have made a great contribution on this area, developing
this coding technique in combined coding and modulation schemes over different
constellations, mainly MPSK, MQAM and Q 2 PSK signal sets, using both block and
convolutional coding. Coding over rings appears also to be a very suitable coding
technique for combined coding and modulation schemes based on GU N-dimensional
signal sets. This is developed in Chapters 4, 5 and 6. Some ring-MCE structures are
studied and modified in order to provide an improvement of the squared Euclidean
free distance of the corresponding ring-TCM scheme. Topology proposed by Baldini
and Farrell [76, 77] will be taken as basic topology to perform these modifications.