Effects of Single-Cycle Structure on Iterative Decoding of Low-Density Parity-Check Codes

We consider communication over the binary erasure channel (BEC) using low-density parity-check (LDPC) codes and belief propagation (BP) decoding. For fixed numbers of BP iterations, the bit error probability approaches a limit as the blocklength tends to infinity, and the limit is obtained via density evolution. The finite-blocklength correction behaves like α(ε,
t
)/
n
+Θ(
n
^-2
) as the blocklength
n
tends to infinity where α(ε,
t
) denotes a specific constant determined by the code ensemble considered, the number
t
of iterations, and the erasure probability ε of the BEC. In this paper, we derive a set of recursive formulas which allows the evaluation of the constant α(ε,
t
) for standard irregular ensembles. The dominant difference α(ε,
t
)/
n
can be considered as effects of cycle-free and single-cycle structures of local graphs. Furthermore, it is confirmed via numerical simulations that estimation of the bit error probability using α(ε,
t
) is accurate even for small blocklengths.