Signal Space Coding over Rings

Chapter 1: Introduction 2
signal of the signal space. Hence, optimisation of each one of these entities can lead
to a closer approximation of the error-free transmission predicted by Shannon. The
so-called Shannon limit, which is a bound for the parameter average bit energy-to-
noise power spectral density, Eb / N 0 , is found to be equal to − 1.
59 dB , and
determines the gap between the performance of any system and that of the ideal
system.
If coding is considered as performed using a coding machine, the mapping procedure
states rules for the assignment of labels or symbols (outputs of the coding machine) to
signals of the signal space. Consequently, the message space is transformed into a
sequence space, constituted from finite or infinite label sequences that are generated
by the coding machine. As defined by Forney [1], a label code is a subset of the
sequence space. Considering a coding system as composed of three entities, the label
code, or coding machine, the signal space, and the mapping procedure for assigning
labels to signals, Forney [1] defined the so-called signal space codes.
Since Shannon's paper has been published, a lot of effort has been made to find an
efficient coding technique to reduce the gap to the Shannon limit. At first, most of this
effort has been made on the optimisation of the coding machine itself.
The idea of combining coding and modulation suggested by Massey [43] appeared as
a new method for optimising the communication system's performance over the
AWGN channel and other channels. Coding and modulation are combined in one
entity, putting attention on the correspondence between the coding machine's output,
and the signal of the signal space. One of the most relevant contributions to the design
of combined coding and modulation schemes has been the work of Ungerboeck [44,
45], who proposed a novel scheme combining convolutional coding with MPSK, to
provide an improvement over the corresponding uncoded system without sacrificing
the bandwidth of the transmission. His rules for assigning signal labels to transitions
of a trellis constitute a first step in the optimisation of the mapping procedure.
Thus, the three main entities of a communication system mentioned above are subject
to an optimisation. Forney's definition of a signal space code is the more generalised
framework of a combined coding and modulation scheme. In his paper [1], attention
is put onto the mapping procedure and its relationship to the signal space, laying on
the analysis of the group theory. Conditions for the definition of geometrical