  We construct and analyze a family of low-density parity check (LDPC) quantum
codes with a linear encoding rate, polynomial scaling distance and efficient
decoding schemes. The code family is based on tessellations of closed,
four-dimensional, hyperbolic manifolds, as first suggested by Guth and
Lubotzky. The main contribution of this work is the construction of suitable
manifolds via finite presentations of Coxeter groups, their linear
representations over Galois fields and topological coverings. We establish a
lower bound on the encoding rate~k/n of~13/72 = 0.180... and we show that the
bound is tight for the examples that we construct. Numerical simulations give
evidence that parallelizable decoding schemes of low computational complexity
suffice to obtain high performance. These decoding schemes can deal with
syndrome noise, so that parity check measurements do not have to be repeated to
decode. Our data is consistent with a threshold of around 4% in the
phenomenological noise model with syndrome noise in the single-shot regime.
