Simple elementary cellular automata with high period? I'm looking for a simple rewrite system which displays high period. 

In order to do that, I've ran a brute-force search on every elementary cellular automata, for a few fixed memory lengths L. The result is that, when L=7, there are rules with the max possible period (128). Yet, for L=8, no matter which rule is used, I couldn't get a period > 180. For L=9, the maximum period of 133.

I've, then, tried a few variations of the core idea. For example, I tried using 3 symbols instead of just 2 and do a similar brute-force search, but the results are similar.

Thus, I ask: is there any similar system with a rewrite rule which displays high period?
Used symbols will be $0, 1$, table $T$ with $18$ elements ($L=18$), and one element for register $B$.
Initialize table with any configuration with at least one element set to $1$.

At generation step do:
B = T[0] Xor T[7]
T[i] = T[i - 1]
T[0] = B  

The array bounds act like wall, if you read from outside you get $0$. This is simple rewrite system, with minimal number of operations without any fancy operations, only rewrite by one calculate xor of two cells  (hey, Rule 90 takes three), and has period $262143$ giving all $18 bit$ numbers excluding $0$.

How to get such system? This is Linear feedback shift register which fits your description of PRNG, has the highest possible period ($2^L-1$) for given number of bits (cells?) minus 1, the best one would have $2^L$.