  We propose a new type of short to moderate block-length, linear
error-correcting codes, called moderate-density parity-check (MDPC) codes. The
number of ones of the parity-check matrix of the codes presented is typically
higher than the number of ones of the parity-check matrix of low-density
parity-check (LDPC) codes. But, still lower than those of the parity-check
matrix of classical block codes. The proposed MDPC codes are cyclic and are
designed by constructing idempotents using cyclotomic cosets. The construction
is simple and allows finding short block-length, high-rate codes with good
minimum distance. Inspired by some recent iterative soft-input soft-output
(SISO) decoders used in a context of classical block codes, we propose a low
complexity, efficient, iterative decoder called Auto-Diversity (AD) decoder. AD
decoder is based on belief propagation (BP) decoder and takes advantage of the
fundamental property of automorphism group of the constructed cyclic code.
