We consider two graph-based techniques for correcting the errors introduced
into a transmission by a noisy channel. In the first technique, which is based
on decoding with a layered graph known as a trellis, we examine generalization
of heuristically good codes called lexicodes. We propose and analyze a
method for designing generalized lexicodes that minimize or
constrain various trellis decoding parameters. The second technique is
based on decoding with factor graphs, which are smaller than trellis representations and
have enjoyed much attention in the recent literature. We look at a specific
type of factor graphs known as Tanner graphs, for which a decoding algorithm
is known. This decoding algorithm is proven to converge always to a
maximum-likelihood decoding if and only if the corresponding Tanner graph is cycle-free.
However, we show that cycle-free Tanner graphs cannot support good codes, because of their
weak structure. This weakness has a direct affect on codes of Tanner graphs
with cycles, which we analyze as well.
To my wife and my whole family
for that which words cannot convey.