Design of nonbinary quasi-cyclic low-density parity-check codes by maximising the minimum distance

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1 TRANSACTIONS ON EMERGING TELECOMMUNICATIONS TECHNOLOGIES Trans. Emerging Tel. Tech. (01) Published online in Wiley Online Library (wileyonlinelibrary.com). DOI /ett.6 SHORT RESEARCH LETTER Design of nonbinary quasi-cyclic low-density parity-check codes by maximising the minimum distance Lei Liu 1 *, Wuyang Zhou 1 and Shengli Zhou 1 Wireless Information Network Lab, University of Science and Technology of China, Hefei, Anhui 00, China Department of Electrical and Computer Engineering, University of Connecticut, Storrs, CT 0669, USA ABSTRACT In this paper, we propose a construction method of nonbinary quasi-cyclic low-density parity-check (QC-LDPC) codes. The shift offset values of the circulant permutation submatrices are selected not only by maximising the girth but also by maximising the minimum distance upper bound. The proposed method provides a more rational optimisation way than only maximising the girth. Copyright 01 John Wiley & Sons, Ltd. *Correspondence L. Liu, Wireless Information Network Lab, University of Science and Technology of China, Hefei, Anhui 00, China. liul@mail.ustc.edu.cn Received 1 April 01; Revised 0 May 01; Accepted July INTRODUCTION Low-density parity-check (LDPC) codes can achieve performance close to the benchmark predicted by the Shannon theory, which can be applied in the optical code division multiple access system [1], the design of Raptor codes [] and joint network and channel coding schemes []. Quasicyclic LDPC (QC-LDPC) codes are particularly appealing to practical systems as the quasi-cyclic structure of the parity-check matrix (H matrix) allows for linear time encoding using only shift registers, renders efficient routing for decoding implementation and enables the storage of the coding matrix with only a few memory units. The shift offset values for circulant permutation submatrices can be obtained via the help of computer search for random-like methods [, ], whereas special algebraic or geometrical structures are utilised for code design in structured methods [6]. From the references mentioned earlier, we found that LDPC codes performance does not only depend on the girth, which is the length of the shortest cycle in the code s Tanner graph representation. There are already many good nonbinary LDPC codes with girth 6 or even. Thus, the girth is not the best parameter to be optimised for LDPC codes. The minimum distance of the code can characterise the code s error floor performance. But the problem of finding the minimum distance of a linear code (including LDPC codes) was proved to be an NP-hard problem by Vardy []. An approximate randomised algorithm, called the nearest nonzero codeword search (NNCS) was proposed by Hu et al. [8] to tackle this problem for binary LDPC codes. And an effective modification of NNCS technique was proposed for computation of the minimum distance of the LDPC codes [9]. The binary NNCS algorithm was extended to the nonbinary field in [10]. In this paper, we introduce the estimated minimum distance upper bound as the design constraint to construct the H matrix of nonbinary QC-LDPC codes.. PRELIMINARIES ON NONBINARY QUASI-CYCLIC LOW-DENSITY PARITY-CHECK CODES In this paper, we focus on LDPC codes with a underlying quasi-cyclic structure. The parity-check matrix H of symbol length nl shall be written as P a 00 P a 01 P a 0.n 1/ 6 H D P a.m 1/0 P a.m 1/1 P a.m 1/.n 1/ (1) where P a ij represents the L L nonbinary circulant permutation matrix with a nonzero element over GF. p / at position r;.r C a ij / mod L, 0 6 r 6 L 1, andzero elsewhere. The offset value a ij satisfies 0 6 a ij 6 L 1, Copyright 01 John Wiley & Sons, Ltd.

2 and I presents the identity matrix when a ij D 0. And P a ij represents an L L zero matrix when a ij D 1. Consider the m n matrix H m obtained from H by replacing zero matrices and circulant permutation matrices with 0 and 1, respectively. H m is called the mother matrix (or base matrix). The nonzero elements of each submatrix are defined as follows. In each nonbinary circulant permutation matrix, let ij denote the nonzero element in the first row of P a ij,which can be drawn randomly from GF( p )n0, and the nonzero elements for the remaining rows of P a ij are obtained by multiplying the one in the row above it by,where is a primitive element of GF( p )and is an integer. However, generally speaking, the nonzero element in the first row is not equal to the nonzero element in the last row multiplied by ; hence, the codes are not qualified as quasi-cyclic codes. Two constructions have been provided to resolve this issue. Reference [11] introduces a so-called -multiplied circulant permutation matrix, which translates to L D p 1 and D 1. Reference [1] generalised the approach in [11] to a ˇ-multiplied circulant permutation matrix, which translates to Lj. p 1/ and D. p 1/=L. More generally, following [], we select once L is given. Specifically, we choose the smallest for any given L such that L D. p 1/,where is an integer.. CONSTRUCTION OF NONBINARY QUASI-CYCLIC LOW-DENSITY PARITY-CHECK CODES AIMING TO OPTIMISE THE UPPER BOUND OF THE MINIMUM DISTANCE In this section, we first introduce the estimated upper bound of minimum distance for nonbinary QC-LDPC codes [10] and then present a design algorithm for nonbinary QC-LDPC codes..1. Upper bound of the minimum distance for nonbinary quasi-cyclic low-density parity-check codes Choose any m C 1 column blocks of Equation (1) to construct the following matrix 6 NH D () P a i m 1 j 0 P a i m 1 j 1 P a i m 1 jm For each column block index h, h D j 0 ;j 1 ;;j m, define an operator matrix D h to be the determinant (over GF( p )) of the matrix obtained from NH by deleting column block h D h D X.j 0 j 1 j m =h/ P a i 0 j 0 P a i 1 j 1 P a i m 1 jm () where.j 0 j 1 j m =h/ denotes the permutation of all.m C 1/ elements except the hth one. We use x as a weight-1 vector of length L over GF( p ), where different values and positions of the nonzero element in x are used. Define Nc WD.D 0 x/ T ;.D 1 x/ T ;;.D m x/ T T It can be easily verified that NHNc D 0. For example, the product of the first row of NH and Nc is P a i 0 j 0 D 0 x C P a i 0 j 1 D 1 x CCP a i 0 jm D m x D x D 0 ˇ P a i m 1 j 0 P a i m 1 j 1 P a i m 1 jm ˇ because the top two rows are identical. The same argument can be applied to other rows of NH. Therefore, Nc can be easily extended to be a valid codeword by inserting zeros between subblocks. Because we aim to find an upper bound of minimum distance as low as possible, the following steps have been taken. () () We combine any mc1 column blocks from n column blocks to form NH. We use x as a weight-1 vector of length L over GF( p ), where different values and positions of the nonzero element in x are used. For each combination of NH and x, we can find codewords through Equation (), whose weights are found as w.nc/ D w.d 0 x/ CCw.D m x/ (6) We choose the lowest weight given by Equation (6) as the upper bound of the code s minimum distance... Construction method for nonbinary quasi-cyclic low-density parity-check codes Given the parameters in Equation (1), that is, the number of symbol nodes n, the number of parity-check number m, the symbol node degree sequence D s, which should be determined before the construction, and the size of the Trans. Emerging Tel. Tech. (01) 01 John Wiley & Sons, Ltd. DOI /ett

3 submatrix L, in this paper, we focus on nonbinary irregular LDPC codes with mixing column weights of and in their H matrices to achieve linear time encoding [1], and the construction algorithm is summarised as follows. Algorithm We construct the first m columns of H m with weight of by the progressive edge growth (PEG) algorithm. Because the minimum distance of a cycle code (weight of ) is g= [1], where g is the girth of the code, we can determine the shift offset values of the weight- submatrix by the method given in [] in order to maximise the local girth. LOOP FROM j D m TO n 1 LOOP FROM k D 0 TO d sj 1, whered sj is the degree of symbol node s j. IF k D 0 Choose the check node i having the lowest check degree under the current Tanner graph of the mother matrix and initialise the shift offset value of P a ij to 0. Choose ij randomly over GF( p )n0 for the circulant matrix P a ij. Thus, all nonzero elements of P a ij are determined. ELSE In the local Tanner graph, there are m k check nodes unconnected to the symbol node s j. For each check node i, choose ij randomly over GF( p )n0 for the circulant matrix P a ij, and all nonzero elements of P a ij are determinate. Then traverse the shift offset value a ij from 0 to L 1 and select the check node and the corresponding shift offset value with the largest weight given by Equation (6) under the local submatrix. END IF END OF LOOP END OF LOOP We call the preceding algorithm as distance-maximised (D-max) algorithm.. EXAMPLE DESIGNS FOR THE DISTANCE-MAXIMISED CODES In this section, we present some example designs by using the D-max algorithm. All H matrices are constructed over GF(8), where n is set to be 1, m is set to be 6, and the symbol node degree sequence D s is Œ;;;;;;;;;;;. We design four H matrices with L D 19; 8; 1 and 6, respectively. Thus, all codes are of rate 1/ and of mean column weight. with short block length of 68, 1008, 16 and 68. That is, the symbol lengths are 8, 6, 9 and 6 over GF.8/, respectively. For the four sizes of submatrix, the H matrices are as follows. (1) Example design for L D 19 When L D 19, theh matrix of the D-max code is as follows I 0 0 I 0 0 I 0 0 P I 0 0 I 0 0 I 0 P P I I I P 1 0 P P I 0 I 0 0 P 11 P 0 P I 0 0 P 1 0 I 0 I 0 P 0 I 0 I I P 6 P 0 () () Example design for L D 8 When L D 8, theh matrix of the D-max code is as follows I 0 0 I 0 0 I 0 0 P 9 I 0 0 I 0 0 I 0 P 0 I 0 P 1 P 9 I I P 1 0 P 0 0 I 60 0 I 0 I 0 0 P P 0 P I 0 0 P 1 0 I 0 I I 0 I P 0 P 0 P 0 (8) () Example design for L D 1 When L D 1, theh matrix of the D-max code is as follows I 0 0 I 0 0 I P 0 0 I 0 0 I 0 0 I 0 P 6 0 P 0 0 I I I 0 P 1 P P I 0 I I P 9 P P 0 0 I 0 0 P 1 0 I 0 I P 0 0 I 0 I 0 0 P 0 0 P (9) () Example design for L D 6 When L D 6, theh matrix of the D-max code is as follows I 0 0 I 0 0 I 0 P 1 0 I 0 0 I 0 0 I 0 P 0 I 0 0 I I I 0 P 0 I 0 P I 0 I 0 P P 0 P I 0 0 P 1 0 I P 0 P 0 0 I 0 I P 0 0 P 8 0 P 1 (10). SIMULATION RESULTS Monte Carlo simulations are used to evaluate the performance of the proposed nonbinary QC-LDPC codes and the compared codes, using binary phase-shift keying transmission over the additive white Gaussian noise (AWGN) channel. The fast Fourier transform q-ary sum-product algorithm [1] is used for decoding with the maximum number of iterations set to be 80. For all simulations, we Trans. Emerging Tel. Tech. (01) 01 John Wiley & Sons, Ltd. DOI /ett

4 Table I. Estimated minimum distance of nonbinary low-density parity-check codes by the error impulse method [10]. Code name VL PEG D-max PEG D-max VL PEG D-max VL PEG D-max Block length Girth Minimum symbol weight Minimum bit weight Upper bound of minimum symbol weight Upper bound of minimum bit weight VL, various length; PEG, progressive edge growth; D-max, distance maximised. run until 0 block errors are found or 00; 000; 000 blocks are transmitted. Table I shows the estimated minimum distance of all simulated codes by the error impulse method [10] and the respective upper bound. It can be observed that the D-max codes have better minimum distance performance than the other two codes with the same block length, and the minimum bit weight grows with the increase of the block length, although the girth of proposed codes is not larger or even smaller than the girth-optimised codes. It conforms that the minimum distance performance does not depend on the girth. Moreover, as the block length increases, the upper bound becomes closer to the estimated minimum distance. Thus, the upper bound is more accurate to be applied as the optimisation goal when the block length of D-max codes becomes longer. Figures 1 and show the performance comparison among the D-max codes, PEG codes [16] and various length (VL) codes [] over AWGN channel and Rayleigh fading channel, respectively. All codes are of rate 1/ and of mean column weight. with the same short block length. Particularly, the nonzero elements of all PEG codes are selected randomly. We observe that the proposed D-max codes have similar waterfall performance as with Block Error Rate VL PEG D max E b /N 0 (db) Figure 1. Performance of codes over additive white Gaussian noise channel. VL, various length; PEG, progressive edge growth; D-max, distance maximised. Block Error Rate VL PEG D max E b /N 0 (db) Figure. Performance of codes over Rayleigh fading channel. VL, various length; PEG, progressive edge growth; D-max, distance maximised. codes constructed by the VL and PEG algorithm. However, the larger minimum distance from Table I guarantees the proposed code achieves lower error floor than the other two codes, which can be observed from the codes of block lengths 68 and 1008 at block error rate of 10 6 over both AWGN channel and Rayleigh fading channel. Note that the code of block length 68 is only 1.9 db away from the Shannon limit at block length error rate of 10 6 over AWGN channel. 6. CONCLUSIONS In this paper, we propose a construction method to design the H matrices of nonbinary QC-LDPC codes. Comparing with existing algorithms, in the proposed method, a novel evaluation criterion based on maximising both the girth and the upper bound of the minimum distance is introduced to provide us a more rational optimisation way than only maximising the girth. Simulation results show that the proposed codes perform very well even for the short block length codes, and the larger minimum distances lower down the error floors of the proposed codes. Thus, the proposed method provides a novel design of nonbinary QC-LDPC codes to achieve better error floor performance. Trans. Emerging Tel. Tech. (01) 01 John Wiley & Sons, Ltd. DOI /ett

5 ACKNOWLEDGEMENTS The work of L. Liu and W. Zhou is supported by the National Programs for High Technology Research and Development (The Key Technologies Research for Satellite Mobile Communication System) and the National Major Special Projects in Science and Technology of China under grant 011ZX The work of S. Zhou is supported by ONR grant N (PECASE) and the NSF grant ECCS REFERENCES 1. Sahuguede S, Julien-Vergonjanne A, Cances J. Performance of OCDMA system with FEC based on interference statistical distribution analysis. European Transactions on Telecommunications 010; 1() Yuan L, An J. Design of UEP-Raptor codes over BEC. European Transactions on Telecommunications 010; 1(1) 0.. Johnson S, Ong L, Kellett C. Joint channel-network coding strategies for networks with low-complexity relays. European Transactions on Telecommunications 011; () Fossorier M. Quasi-cyclic low-density parity-check codes from circulant permutation matrices. IEEE Transactions on Information Theory 00; 0(8) Huang J, Liu L, Zhou W, Zhou S. Large-Girth nonbinary QC-LDPC codes of various lengths. IEEE Transactions on Communications 010; 8(1) Tanner R, Sridhara D, Fuja T. A class of groupstructured LDPC codes, In Proceedings of the Sixth International Symposium on Communications Theory and Applications, Ambleside, England, July 1 16, 001; Vardy A. The intractability of computing the minimum distance of a code. IEEE Transactions on Information Theory 199; (6) Hu X-Y, Fossorier M, Eleftheriou E. On the computation of the minimum distance of low-density paritycheck codes, In Proceedings of IEEE International Conference on Communications, Paris, France, vol., June 00; Daneshgaran F, Laddomada M, Mondin M. An algorithm for the computation of the minimum distance of LDPC codes. European Transactions on Telecommunications 006; 1(1) Liu L, Huang J, Zhou W, Zhou S. Computing the minimum distance of nonbinary LDPC codes. IEEE Transactions on Communications 01; 60() Lin S, Song S, Lan L, Zeng L, Tai Y-Y. Constructions of nonbinary quasi-cyclic LDPC codes a finite field approach. IEEE Transactions on Communications 008; 6(). 1. Peng R-H, Chen R-R. Design of nonbinary quasi-cyclic LDPC cycle codes, In Proceedings of ITW 0, Tahoe City, CA, September 6, 00; Huang J, Zhou S, Willett P. Near-Shannon-limit lineartime-encodable nonbinary irregular LDPC codes, In Proceedings of IEEE Global Telecommunications Conference, Honolulu, Hawaii, November 0 December, 009; Peterson WW, Weldon EJ, Jr. Error-Correcting Codes, (nd edn). MIT Press Cambridge, MA, Song H, Cruz JR. Reduced-complexity decoding of q-ary LDPC codes for magnetic recording. IEEE Transactions on Magnetics 00; Hu X-Y, Eleftheriou E, Arnold D-M. Regular and irregular progressive edge-growth Tanner graphs. IEEE Transactions on Information Theory 00; 1(1) Trans. Emerging Tel. Tech. (01) 01 John Wiley & Sons, Ltd. DOI /ett