  We describe and analyze the joint source/channel coding properties of a class
of sparse graphical codes based on compounding a low-density generator matrix
(LDGM) code with a low-density parity check (LDPC) code. Our first pair of
theorems establish that there exist codes from this ensemble, with all degrees
remaining bounded independently of block length, that are simultaneously
optimal as both source and channel codes when encoding and decoding are
performed optimally. More precisely, in the context of lossy compression, we
prove that finite degree constructions can achieve any pair $(R, D)$ on the
rate-distortion curve of the binary symmetric source. In the context of channel
coding, we prove that finite degree codes can achieve any pair $(C, p)$ on the
capacity-noise curve of the binary symmetric channel. Next, we show that our
compound construction has a nested structure that can be exploited to achieve
the Wyner-Ziv bound for source coding with side information (SCSI), as well as
the Gelfand-Pinsker bound for channel coding with side information (CCSI).
Although the current results are based on optimal encoding and decoding, the
proposed graphical codes have sparse structure and high girth that renders them
well-suited to message-passing and other efficient decoding procedures.
