Research Letters in Communications

Volumeย 2008, Article IDย 324503, 4 pages

http://dx.doi.org/10.1155/2008/324503

## Design of Short, High-Rate DVB-S2-Like Semi-Regular LDPC Codes

Dipartimento di Elettronica e Informazione (DEI), Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy

Received 11 October 2008; Accepted 5 December 2008

Academic Editor: Rajeshย Khanna

Copyright ยฉ 2008 Luca Barletta and Arnaldo Spalvieri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This work focuses on high-rate () moderate-length () low-density parity-check codes. High-rate codes allow to maintain good quality of the preliminary decisions that are used in carrier recovery, while a moderate code length allows to keep the latency low. The interleaver of the LDPC matrix that we consider is inspired to the DVB-S2 standard one. A novel approach for avoiding short cycles is analyzed. A modified BP decoding algorithm is applied in order to deal with longer cycles. Simulations and results for the AWGN channel are presented, both for BPSK signalling and for coded modulation based on the partition .

#### 1. Introduction

An LDPC code is a binary block code defined as
the null space of the sparse โparity checkโ matrix ,
where can be viewed as the adjacency matrix of the
bipartite graph ,
the Tanner graph, with meaning that the th *c*-node is connected to the th *v*-node. Richardson et al. [1] studied a method named *Density
Evolution* (DE) for finding the pair of distributions ,
as ,
that allows to reach the fastest BER convergence at some probability error
target ,
under BP algorithm. For short and moderate-length LDPC codes, that is, for ,
the design of via DE shows relatively high error floors (), therefore a different approach is required.
The performance can be improved by conditioning the construction of in such a way that short cycles are avoided. A
technique for the elimination of short cycles in regular LDPC codes has been
recently presented in [2]. However, regular codes are outperformed by irregular
or semiregular codes.

Eroz et al. [3] proposed a particular structure of that leads to irregular codes. This structure has been adopted for standardization of DVB-S2 applications (the DVB-S2 standard includes 2 versions with and ) and belongs to the family of IRA codes [4].

This paper addresses the problem of designing short DVB-S2-like codes, introducing a new technique for the elimination of cycles of length 4 and 6.

The remainder of the paper is outlined as follows. Section 2 introduces the system models taken under consideration and the matrix structure of the DVB-S2 standard. Section 3 explains the features adopted in designing the short-length DVB-S2-like LDPC codes. Section 4 shows simulation results and some comparisons with short codes designed via DE. Finally, some conclusions are drawn in Section 5.

#### 2. DVB-S2-like Parity-Check Matrix

Here follow some recalls of the parity-check matrix structure described in [3].

The LDPC matrix is composed of two
submatrices and , ;
the rightmost one is a lower bidiagonal matrix which offers the advantage of
simple systematic encoding. Specifically, taking modulo-2 additions of
consecutive rows of ,
such that for and ,
one gets the identity matrix .
Another advantage is to avoid short cycles in Tanner graph involving low-degree
*v*-nodes, leading to improved performance as stated in [1].

Submatrix is built as follows: chosen an opportune value that divides both and , randomly generate the first column of , , and get the next columns by cyclic shifting of positions, respectively, for . The second group of columns is created in the same way, just after the first columns of the new group have been generated. This procedure allows to fill the entire submatrix .

#### 3. Code Design

##### 3.1. Short Cycle Elimination

A cycle of
length in the Tanner graph is a set of *v*-nodes and *c*-nodes such that a path of consecutive edges starts from a *v*-node (*c*-node) and ends
at the same *v*-node (*c*-node).

Given two columns and of , a length- cycle is formed when the inner product between the 2 columns is greater than 1:for some . This condition suggests a practical way to check the presence of length- cycle in a parity-check matrix.

Focusing on a column of , , let us define the set of the โcyclic-distancesโ between the nonzero entries of the th columnwhere andFor example, if the nonzero entries of are located at the positions with , then . Also we have for all the columns of , briefly denoted with .

Given three columns , , and , a length- cycle is formed when the sum of two cyclic-distances of two distinct columns equals a cyclic-distance of the third column . An example is shown below (4) with , , and a length-6 cycle is formed: .

A complete search for the sets that avoid the presence of length- cycles is computationally expensive as and the average weight of the columns grow.

The problem of finding the longest LDPC code that is length- and length- cycle free is described as follows: let be the initial set of admissible cyclic-distances (i.e., length- and length- cycle free); when a new column is added in the design of a new set of cyclic-distances is added to the current set according to the following update rule:as runs in ; the objective is to find the maximum s.t. the set is cycle free. The binary operation between two sets and is defined asA new column is an eligible set ifA smart strategy is to choose the cyclic distances from a class of rest modulo a small integer : in this way all cyclic distances will be eligible. For example, choosing the class of rest modulo , namely , the sum of two cyclic-distances drawn from two distinct columns is in the class , so a length- cycle involving columns cannot be created. When the sum of two distances is greater than , the sum is complemented to as illustrated in (3); usually so . This strategy automatically avoid the creation of length- cycles.

For some particular choices of , , and the average column weight, it is possible
that the total number of eligible cyclic distances is not sufficient to build
all the columns. Actually the total number of original
columns can be reduced to ,
where is the cyclic offset between two consecutive
columns in the same *c*-node -subset. Without loss of generality, consider
the first *c*-node -subset and its shifted replica by position denoted with .
Obviously, no cycles are created between the two subsets and ;
not only, but also the matrix for does not contain length- and length- cycles, so an original column can generate columns without any cycle.

##### 3.2. *v*-Node Degree Distribution

In the semiregular construction the *v*-node degree
distribution has the formthat is, the submatrix has a constant column degree .
The parameter can assume only small values to keep the
low-density nature of and hence few cycles in the factor graph.

It is known that there exists a tradeoff between the performance in the cliff region and the performance in the floor region. In [5] it is shown that the floor improves when increases. However the weight cannot be increased above a maximum value that depends on the code length and the rate, in order to maintain the low-density nature of . This makes it difficult to obtain good performance in the floor region with codes of moderate length and high rate.

##### 3.3. Modified BP Algorithm

The BP decoding
algorithm is optimal with tree-like factor graph, a condition practically
reached when is very large. When the block length
is finite, the BP becomes a suboptimal algorithm in
the sense that it no longer performs MAP decoding [6, 7].
The main idea in order to outperform the BP decoder is to include a procedure
of hard decisions on the *v*-nodes' posteriors, called *decimation*. The
modified algorithm resembles that in [8]: in the generic iteration the mutual information from
the *v*-nodes to *c*-nodes is estimated and some hard decisions are taken
on the most reliable *v*-nodes to reach the desired value of for the current iteration. The increment between two consecutive iterations is assumed
to be ,
such that the expected maximum number of iteration is .

#### 4. Results

##### 4.1. BPSK Signalling

In the BPSK system, the receiver computes the *a priori* metrics for
the LDPC decoderwhere is the received vector and is the variance per dimension of the AWG
noise. Several LDPC codes are compared, that are different for code rate ( and ), code length ( and ). Also, we considered LDPC codes designed via
DE. The results obtained with DE are not reported here due to the poor
performance in the floor region.

In Figure 1, there is a comparison between the original BP decoding algorithm and the modified version, working on the same semiregular code with , and . The modified version of the BP algorithm exploits the original version with a constant gain in the cliff region.

In Figure 2 are plotted BER curves versus in dB. The solid curves represent the performance of the codes designed with the short cycle elimination procedure, while the dashed curves represent the performance of the codes designed as described in Section 2. The code with and is 1โdB from capacity and the code with is about โdB from capacity (); the floor does not appear up to .

##### 4.2. Coded Modulation

Let us consider the coded modulation based on the 16-way partition where coded bits address one coset leader in and uncoded bits address one out of subconstellations. The coded vector is drawn from an LDPC encoder of rate ; the normalized redundancy is and the volume to noise ratio (VNR) is

As suggested in [9] it has been adopted a partial Gray mapping rule.

At the receiver metrics for the th received symbol and each of the Gray-mapped bits are computed according tofor and , such that metrics are provided to the LDPC decoder.

Four different LDPC codes are compared; they differ for code rate ( and lead to overall redundancies of and , resp.), code length ( and mean and 2D symbols, resp.). The LDPC decoder computes iterations of the modified BP algorithm. In Figure 3 are plotted BER performance curves versus SNR in dB. The longer semiregular code shows good performance and error floor below . Figure 4 shows some WER curves versus the volume to noise ratio , which is recalled in (10), and the sphere lower bound on the WER for the finite dimensionality of the lattice [10].

#### 5. Conclusion

The design of the moderate length instance of the DVB-S2-like parity-check matrix has been considered. Several codes designed with a semiregular approach are worked out and their performance is obtained by running few iterations of the decoding algorithm. The proposed edge assignment and the modified BP algorithm, with a decimation procedure, allow to lower the error floor compared to the same codes designed via density evolution.

#### References

- T. J. Richardson, M. A. Shokrollahi, and R. L. Urbanke, โDesign of capacity-approaching irregular low-density parity-check codes,โ
*IEEE Transactions on Information Theory*, vol. 47, no. 2, pp. 619โ637, 2001. View at Publisher ยท View at Google Scholar ยท View at MathSciNet - J. Fan, Y. Xiao, and K. Kim, โDesign LDPC codes without cycles of length 4 and 6,โ
*Research Letters in Communications*, vol. 2008, Article ID 354137, 5 pages, 2008. View at Publisher ยท View at Google Scholar - M. Eroz, F.-W. Sun, and L.-N. Lee, โAn innovative low-density parity-check code design with near-Shannon-limit performance and
simple implementation,โ
*IEEE Transactions on Communications*, vol. 54, no. 1, pp. 13โ17, 2006. View at Publisher ยท View at Google Scholar - H. Jin, A. Khandekar, and R. McEliece, โIrregular repeat-accumulate codes,โ in
*Proceedings of the 2nd International Symposium on Turbo Codes and Related Topics*, pp. 1โ8, Brest, France, September 2000. - M. Yang, W. E. Ryan, and Y. Li, โDesign of efficiently encodable moderate-length high-rate irregular LDPC codes,โ
*IEEE Transactions on Communications*, vol. 52, no. 4, pp. 564โ571, 2004. View at Publisher ยท View at Google Scholar - H. Pishro-Nik and F. Fekri, โOn decoding of low-density parity-check codes over the binary erasure channel,โ
*IEEE Transactions on Information Theory*, vol. 50, no. 3, pp. 439โ454, 2004. View at Publisher ยท View at Google Scholar ยท View at MathSciNet - C. Measson, A. Montanari, and R. Urbanke, โMaxwell construction: the hidden bridge between iterative and maximum a posteriori decoding,โ
*IEEE Transactions on Information Theory*, vol. 54, no. 12, pp. 5277โ5307, 2008. View at Publisher ยท View at Google Scholar - Q. Wang and C. He, โApproaching 1.53-dB shaping gain with LDGM quantization codes,โ in
*Proceedings of IEEE Global Telecommunications Conference (GLOBECOM '07)*, pp. 1571โ1576, Washington, DC, USA, November 2007. View at Publisher ยท View at Google Scholar - U. Wachsmann, R. F. H. Fischer, and J. B. Huber, โMultilevel codes: theoretical concepts and practical design rules,โ
*IEEE Transactions on Information Theory*, vol. 45, no. 5, pp. 1361โ1391, 1999. View at Publisher ยท View at Google Scholar ยท View at MathSciNet - V. Tarokh, A. Vardy, and K. Zeger, โUniversal bound on the performance of lattice codes,โ
*IEEE Transactions on Information Theory*, vol. 45, no. 2, pp. 670โ681, 1999. View at Publisher ยท View at Google Scholar ยท View at MathSciNet