In this talk, we investigate a particular family of graphs with large
girth and look at their applications to low-density parity-check (LDPC)
codes. Specifically, we study a family of Cayley graphs with large girth
over finite fields with prime order, constructed by Margulis. We
analyse Margulis' lower bound for the girth of these graphs and compute
the exact girth for several prime numbers. Our results show that the
actual girth exceeds the bound by an average of about 87% in all cases
studied. We show an improvement to Margulis' bound which leads to an
average error rate of about 56%. Additionally, we present further
observations which suggest that an even better bound may still exist.
Furthermore, we show how LDPC codes can be constructed based on these
graphs and simulate their error correction performance over an additive
white Gaussian noise channel.