  In an earlier work, the statistical physics associated with
finite--temperature decoding of code ensembles, along with the relation to
their random coding error exponents, were explored in a framework that is
analogous to Derrida's random energy model (REM) of spin glasses, according to
which the energy levels of the various spin configurations are independent
random variables. The generalized REM (GREM) extends the REM in that it
introduces correlations between energy levels in an hierarchical structure. In
this paper, we explore some analogies between the behavior of the GREM and that
of code ensembles which have parallel hierarchical structures. In particular,
in analogy to the fact that the GREM may have different types of phase
transition effects, depending on the parameters of the model, then the
above--mentioned hierarchical code ensembles behave substantially differently
in the various domains of the design parameters of these codes. We make an
attempt to explore the insights that can be imported from the statistical
mechanics of the GREM and be harnessed to serve for code design considerations
and guidelines.
