  We derive bounds on the asymptotic density of parity-check matrices and the
achievable rates of binary linear block codes transmitted over memoryless
binary-input output-symmetric (MBIOS) channels. The lower bounds on the density
of arbitrary parity-check matrices are expressed in terms of the gap between
the rate of these codes for which reliable communication is achievable and the
channel capacity, and the bounds are valid for every sequence of binary linear
block codes. These bounds address the question, previously considered by Sason
and Urbanke, of how sparse can parity-check matrices of binary linear block
codes be as a function of the gap to capacity. Similarly to a previously
reported bound by Sason and Urbanke, the new lower bounds on the parity-check
density scale like the log of the inverse of the gap to capacity, but their
tightness is improved (except for a binary symmetric/erasure channel, where
they coincide with the previous bound). The new upper bounds on the achievable
rates of binary linear block codes tighten previously reported bounds by
Burshtein et al., and therefore enable to obtain tighter upper bounds on the
thresholds of sequences of binary linear block codes under ML decoding. The
bounds are applied to low-density parity-check (LDPC) codes, and the
improvement in their tightness is exemplified numerically. The upper bounds on
the achievable rates enable to assess the inherent loss in performance of
various iterative decoding algorithms as compared to optimal ML decoding. The
lower bounds on the asymptotic parity-check density are helpful in assessing
the inherent tradeoff between the asymptotic performance of LDPC codes and
their decoding complexity (per iteration) under message-passing decoding.
