  We analyze the performance of decoders for the 2D and 4D toric code which are
local by construction. The 2D decoder is a cellular automaton decoder
formulated by Harrington which explicitly has a finite speed of communication
and computation. For a model of independent $X$ and $Z$ errors and faulty
syndrome measurements with identical probability we report a threshold of
$0.133\%$ for this Harrington decoder. We implement a decoder for the 4D toric
code which is based on a decoder by Hastings arXiv:1312.2546 . Incorporating a
method for handling faulty syndromes we estimate a threshold of $1.59\%$ for
the same noise model as in the 2D case. We compare the performance of this
decoder with a decoder based on a 4D version of Toom's cellular automaton rule
as well as the decoding method suggested by Dennis et al.
arXiv:quant-ph/0110143 .
