The Asymptotic Bit Error Probability of LDPC Codes for the Binary Erasure Channel with Finite Iteration Number
We consider communication over the binary erasure channel (BEC) using low-density parity-check (LDPC) code and belief propagation (BP) decoding. The bit error probability for infinite block length is known by density evolution  and it is well known that a difference between the bit error probability at finite iteration number for finite block length and for infinite block length is asymptotically , where is a specific constant depending on the degree distribution, the iteration number and the erasure probability. Our main result is to derive an efficient algorithm for calculating for regular ensembles. The approximation using is accurate for -regular ensembles even in small block length.
In this paper, we consider irregular low-density parity-check (LDPC) codes  with a degree distribution pair . The bit error probability of LDPC codes over the binary erasure channel (BEC) under belief propagation (BP) decoding is determined by three quantities; the block length , the erasure probability and the iteration number . Let denote the bit error probability of LDPC codes with block length over the BEC with erasure probability at iteration number . For infinite block length, can be calculated easily by density evolution  and there exists threshold parameter such that for and for . Despite the ease of analysis for infinite block length, finite-length analysis is more complex. For finite block length and infinite iteration number, can be calculated exactly by stopping sets analysis . For finite block length and finite iteration number, can also be calculated exactly in a combinatorial way . The exact finite-length analysis becomes computationally challenging as block length increasing. An alternative approach which approximates the bit error probability is therefore employed. For asymptotic analysis of the bit error probability, two regions of can be distinguished in the error probability; the high error probability region called waterfall and the low error probability region called error floor. In terms of block length, they correspond to the small block length region and the large block length region. This paper deals with the bit error probability for large block length both below and above threshold with finite iteration number.
For infinite iteration number, the asymptotic analysis for error floor was shown by Amraoui as following :
as . This equation means that for ensembles with , is a good approximation of where is sufficiently large.
Our main result is following.
For regular LDPC codes with finite iteration number
This analysis is the first asymptotic analysis for finite iteration number.
Ii Main result
The error probability of a bit in fixed tanner graph at the -th iteration is determined by neighborhood graph of depth of the bit [5, 9]. Since the probability of neighborhood graphs which have cycles is we focus on the neighborhood graphs with no cycle and single cycle for calculating the coefficient of in the bit error probability. Let denote the coefficient of in the bit error probability due to cycle-free neighborhood graphs and denote the coefficient of in the bit error probability due to single-cycle neighborhood graphs. Then the coefficient of in the bit error probability can be expressed as following: . can be calculated efficiently for irregular ensembles and can be expressed simply for regular ensembles.
The expected probability of erasure message for infinite block length can be calculated by density evolution.
Proposition 1 (Density evolution ).
Let denote erasure probability of messages into check node at the -th iteration and denote erasure probability of messages into variable node at the -th iteration for infinite block length. Then
The coefficient of in the bit error probability due to single-cycle neighborhood graphs can be calculated using density evolution.
Theorem 1 (The coefficient of in the bit error probability due to single-cycle neighborhood graphs).
The complexity of the computation of is in time and in space.
can be expressed simply for regular ensembles since of uniqueness of the cycle-free neighborhood graph.
Theorem 2 (The coefficient of in the bit error probability due to cycle-free neighborhood graphs for regular ensembles).
for the -regular LDPC ensemble is expressed as following
The probability of the unique cycle-free neighborhood graph of depth is
where . The coefficient of in the probability is
and the error probability of the root node is . Then we obtain the statement of the theorem. ∎
Due to the above theorems, for regular ensembles can be calculated efficiently.
Proposition 2 (The bit error probability decays exponentially ).
Assume . Then for any , there exists some iteration number such that for any
Although if then converges to and converges to as , if then and grow exponentially as due to the above proposition. Thus convergence of is non-trivial. In practice it is necessary to use high precision floating point tools for calculating .
Iii Outline of the proof of theorem 1
The bit error probability of an ensemble with iteration number is defined as following:
where denotes a set of all neighborhood graphs of depth , denotes the probability of the neighborhood graph and denotes the error probability of the root node in the neighborhood graph . The coefficient of in the bit error probability with iteration number due to single-cycle neighborhood graphs is defined as following:
where denotes a set of all single-cycle neighborhood graphs of depth .
First we consider the bit error probability of the root node of the neighborhood graph in Fig. 1. The variable nodes in depth 1 have degree to . Then the coefficient of in is given as
The error probability of the message from the channel to the root node is . The error probabilities of the message from the left check node, the right check node and the middle check node to the root node are , and , respectively. Then the error probability of the root node is given as
The coefficient of term of the bit error probability due to is given as
After summing out the left and right subgraphs,
After summing out degrees , and ,
At last, after summing out the root node and the middle check node,
The coefficient of in the bit error probability for iteration number due to neighborhood graphs with the right graph type in Fig. 1 is given as
in the same way. Notice that is the coefficient of of the probability of neighborhood graphs with the right graph type in Fig. 1.
Single-cycle neighborhood graphs can be classified to six types in Fig. 2. Summing up the bit error probability due to all these types, we obtain . Left two types correspond to , middle two types correspond to and right two types correspond to .
Iv Numerical calculations and simulations
There is a question that how large block length is necessary for using for a good approximation of . It is therefore interesting to compare with numerical simulations. In the proof, we count only the error probability due to cycle-free neighborhood graphs and single-cycle neighborhood graphs. Thus it is expected that the approximation is accurate only at large block length where the probability of the multicycle neighborhood graphs is sufficiently small. Contrary to the expectation, the approximation is accurate already at small block length in Fig. 3. Although there is a large difference in small block length near the threshold, the approximation is accurate at block length 801 which is not large enough.
For the ensembles with , the approximation is not accurate at far below the threshold in Fig. 4. Since decreases to as for the ensembles the higher order terms caused by multicycle stopping sets has a large contribution to the bit error probability. It is expected that the approximation is even accurate for the ensembles from which stopping sets with small number of cycles are expurgated.
The limiting value of , is also interesting. For , calculate where sufficiently large in Fig. 5 and Fig. 6. For the -regular ensemble below the threshold, and take almost the same value. It implies that below threshold takes the same value at two limits; then and then . For the ensembles with , is almost where is smaller than threshold.
At last, notice that takes non-trivial values slightly below threshold. For the -regular ensemble, is negative at , positive at and has absolute value which is too small to be measured at . at .
Although the asymptotic analysis of the bit error probability for finite block length and finite iteration number given in this paper is very accurate at -regular, much work remains to be done. First there remains the problem to computing for irregular ensembles. It would also be interesting to generalize this algorithm to other ensembles and other channels.
In the binary memoryless symmetric channel (BMS) parametrized by , We consider instead of since of lack of monotonicity. The asymptotic analysis of the bit error probability with the best iteration number under BP decoding was shown by Montanari for small as following :
as , where and is a random variable corresponding to the sum of the i.i.d. channel log-likelihood ratio. It implies that if , the asymptotic bit error probability under BP decoding is equal to that of maximum likelihood (ML) decoding. Although the condition of the proof in  implies the convergence of values corresponding to and in this paper, in general if , they do not converge, where is Bhattacharyya constant. Although the condition of is strong, the approximation is very accurate for all smaller than threshold. We have the problem to prove the convergence of for the BEC and the BMS for any .
A iteration number is also important. The approximation is not accurate for too large iteration number. A sufficient (and necessary) iteration number for a given block length and a ensemble is very important to improve the analysis in this paper.
-  R. G. Gallager, Low-Density Parity-check Codes, MIT Press, 1963
-  M. Luby, M. Mitzenmacher, A. Shokrollahi, D. A. Spielman, and V. Stemann, ”Practical loss-resilient codes,” In Proceedings of the 29th annual ACM Symposium on Theory of Computing, pages 150-159, 1997
-  T. Richardson and R. Urbanke, ”The capacity of low-density parity check codes under message-passing decoding,” IEEE Trans. Inform. Theory, vol. 47, no. 2, pp.599-618, Feb. 2001
-  T. Richardson and R. Urbanke, ”Finite-length density evolution and the distribution of the number of iterations for the binary erasure channel”
-  A. Montanari ”The asymptotic error floor of LDPC ensembles under BP decoding,” 44th Allerton Conference on Communications, Control and Computing, Monticello, October 2006
-  C. Di, D. Proietti, T. Richardson, E. Telatar and R. Urbanke, ”Finite length analysis of low-density parity-check codes,” IEEE Trans. Inform, Theory, vol. 48, no. 6, pp.1570-1579, Jun. 2002
-  A. Orlitsky, K. Viswanathan, and J. Zhang, ”Stopping set distribution of LDPC code ensembles,” IEEE Trans. Inform, Theory, vol. 51, no. 3, pp.929-953, Mar. 2005
-  A. Amraoui ”Asymptotic and finite-length optimization of LDPC codes,” Ph.D. Thesis Lausanne 2006
-  T. Richardson and R. Urbanke, Modern Coding Theory, draft available at http://lthcwww.epfl.ch/index.php