  In this paper, an expression for the asymptotic growth rate of the number of
small linear-weight codewords of irregular doubly-generalized LDPC (D-GLDPC)
codes is derived. The expression is compact and generalizes existing results
for LDPC and generalized LDPC (GLDPC) codes. Ensembles with check or variable
node minimum distance greater than 2 are shown to be have good growth rate
behavior, while for other ensembles a fundamental parameter is identified which
discriminates between an asymptotically small and an asymptotically large
expected number of small linear-weight codewords. Also, in the latter case it
is shown that the growth rate depends only on the check and variable nodes with
minimum distance 2. An important connection between this new result and the
stability condition of D-GLDPC codes over the BEC is highlighted. Such a
connection, previously observed for LDPC and GLDPC codes, is now extended to
the case of D-GLDPC codes. Finally, it is shown that the analysis may be
extended to include the growth rate of the stopping set size distribution of
irregular D-GLDPC codes.
