In two dimensions, "Toom's rule" is known to be a cellular automaton which can fault-tolerantly store one bit of information. This means that, if we start with the all-0 configuration on an $N\times N$ grid, and apply the cellular automaton together with some local uniform probabilistic noise for some number of time steps $T$, the probability of ending up in a majority-1 state will go to $0$ with increasing $N$ (typically exponentially), for all $T$. More precisely, for any kind of noise "direction", there's a non-zero threshold such that the above statement is true for noise of strength below that threshold. As far as I know, Toom has proved this (in a technically slightly different formulation) for on-site symmetric and biased noise, but it seems plausible to me that it is true for any other type of uniform local noise, e.g., affecting two neighboring bits (please comment if you know more on this).
I was wondering how the situation is in one spacial dimension. It is pretty clear intuitively, and I believe also rigorously proven to some extent, that local majority voting won't do the job. I came accross the "GKL rule", which seems to be able to perform approximate global majority voting even in the presence of noise, according to https://writings.stephenwolfram.com/2021/05/the-problem-of-distributed-consensus/. On the other hand, I found https://www.cs.purdue.edu/homes/park/interest-ca.html, which seems to suggest that GKL is not robust to biased noise since "islands of 1s" which are created in a "sea of 0s" by noise have the tendency to expand.
I also found https://arxiv.org/abs/math/0003117 by Peter Gacs, which seems to be doing what I'm asking for. However, this paper does way more than that (asynchronous, adverserial noise), and I find it very hard to extract the information I am looking for. Does this paper really suggest a 1-dimensional uniform cellular automaton which can remember a bit subject to noise? If yes, how many local states does it have, what's the spacial range of the update rule, and on which page(s) of the paper is the update rule described? Which noise "directions" is it proven to be robust against, and which noise directions is it believed to be robust against?
Finally, the cellular automaton by Gacs (if it is what it is) is horribly complicated, and I find it hard to believe that if a fault-tolerant cellular automaton in one dimension exists, it would need to be that complicated. It appears to me that the complexity is at least partly for the sake being able to rigorously prove things. Thus, are there any simple cellular automata which can fault-tolerantly remember a bit, with the fault-tolerance demonstrated by numerical simulation instead of rigorous proof? Btw, I'd also allow for the unperturbed cellular automaton itself to be probabilistic.