At 2nd-order the construction of a Markov class becomes slightly involved.
It may be difficult to directly infer a rule table from a specification of
theoretical coefficients values because each neighborhood block of
length d may be part of several d+1 blocks each controlled by
a different
coefficient. This
means that the values of the
coefficients may
interact in a complicated way to determined which transitions in
the rule table are consistent with a specification of coefficient values.
Below a two step process which handles these complications is outlined.
In appendix II, direct construction of all reflection-symmetric rules
in a given class of r=2 rules is described.
The first step of the
construction of a 2nd-order class relies on the observation that
both the
coefficients of the mean field theory
and the
coefficients of the 2nd-order
Markov approximation
for d-diameter rules control blocks of the neighborhood size d.
By employing exactly the
method described above for the construction of a mean field class,
a set of rules
with potential membership in a 2nd-order class may be found. Such
rules have the desired
coefficient values, but their
coefficients
values have yet to be determined.
The second step of the construction determines the
coefficient values.
The forward
map from a rule table to a set of Markov coefficients is easily computed.
In the second step of construction,
the forward map is used to determine the
coefficient values
of all cellular automata isolated in the first step. These values are then
checked
against the
coefficient values which define the class in question.
The number of 2nd-order types of n-blocks
is greater than the number of 1st-order types of n-blocks, 
.
Hence, for fixed radius, the number
of
coefficients in the 2nd-order approximation
is greater than the number of
coefficients in the 1st-order approximation.
However, as noted above,
Each of these sets of coefficients
collectively control the same set of d-blocks.
This implies that for fixed radius the
typical values of the
coefficients will be smaller than
the typical values of the
coefficients, and, a fortiori, that the
number of rules isolated in the first step of the construction of a 2nd-order
class will typically be much smaller than the number of rules in a mean field class.