This is a question about a coding scheme of numbers that might yield insights in cellular automata and the rules that govern them.
Background
Since Wolfram the rules defining elementary cellular automata are given as decimal numbers, e.g. $254$, $250$, $90$, and $30$ (in the order of appearance in A New Kind of Science). In G. Martinez' paper A Note on Elementary Cellular Automata Classification we find the following lists of equivalent and of classified rules , uniform (class I), periodic (class II), chaotic (class III), complex (class IV).
I am interested in (arithmetic) properties of rule numbers that allow to determine their complexity and equivalence classes. So I played around with coding schemes and arrangements of rule numbers that allow to see at least something.
A natural coding scheme
The specific coding scheme I came up with goes like this: Write rule number $r$ given as a binary sequence $\langle r\rangle = \langle r_7,r_6,r_5,r_4,r_3,r_2,r_1,r_0 \rangle$ as a pair of binary sequences $r^0$, $r^1$ of length $4$ by the following mapping:
$r^{0} = \langle r_0,r_1,r_4,r_5\rangle$
$r^{1} = \langle r_7,r_6,r_3,r_2\rangle$
Writing $r^0$ below $r^1$ reveals that the scheme is quite "natural": $r_1$ describes the behaviour of black cells, $r_0$ describes the behaviour of white cells, and the two sequences are inverse to each other.
The normal coding scheme
Compare this with the "normal" scheme which maps $r$ to $r^0 = \langle r\text{ mod }16\rangle$ and $r^1 = \langle \lfloor r / 16\rfloor \rangle$:
Plotting rules
Rule number $r$ is plotted at position $(r^0,r^1)$ in a $16\times 16$ grid. Coloring the numbers by their rule complexity class (see above) gives an erratic pattern for the "normal" scheme:
Yellow = class I, green = class II, red = class III, blue = class IV.
Accordingly, plotting equivalent rules gives erratic patterns for the normal scheme:
For the "natural" scheme, the class matrix is symmetric and shows some interesting regularities:
And so do the equivalence classes of rules, here for the strong class III rules:
The symmetries and equivalences can be easily understood. The rules obtained from a given rule by
swapping and inverting $r^0$ and $r^1$, i.e. reflecting $(r^0,r^1)$ at the second diagonal
swapping the entries $r^i_1$ and $r^i_2$ for $i = 0,1$
yield respecitively
inverted patterns (when evolving the rules from inverted start conditions)
reflected patterns (when evolving from the same start condition)
and thus equivalent rules. Examples of equivalence classes of size $1$ can be found among the weak class III rules:
My question is: Can the proposed coding scheme be used to define transformations (beyond reflecting and inverting) and (invariance) properties that allow to derive the complexity class of a given rule? If not so: (How) can it be proved, that no coding scheme can do this job?