Multiple-Bases Belief-Propagation for Decoding of Short Block Codes

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1 Multiple-Bases Belief-Propagation for Decoding of Short Block Codes Thorsten Hehn, Johannes B. Huber, Stefan Laendner, Olgica Milenkovic Institute for Information Transmission, University of Erlangen-Nuremberg, Germany Department of Electrical and Computer Engineering, University of Colorado, Boulder {hehn, {laendner, Abstract A novel soft-decoding method for algebraic block codes is presented. The algorithm is designed for soft-decision decoding and is based on Belief-Propagation (BP) decoding using multiple bases of the dual code. Compared to other approaches for high-performance BP decoding, this method is conceptually simple and does not change at each stage of the decoding process. With its multiple BP decoders the proposed scheme achieves the performance of a standard BP algorithm with a significantly lower number of iterations per decoder realization. By this means the data delay introduced by decoding is reduced. Moreover, a significant improvement in decoding performance is achieved while keeping the data delay small. It is shown that for selected codes the proposed scheme approaches near maximum likelihood (ML) performance for very small data processing delays. I. INTRODUCTION Classical algebraic codes such as BCH Codes [1] are used in current transmission systems because of their good minimum distance properties, often as an outer code in concatenated coding schemes. Many of these systems use a powerful lowdensity parity-check (LDPC) code and a BP decoder [2] or convolutional codes and a turbo decoder [3] for the inner coding scheme but require an outer algebraic code to overcome the error-floor problem [4]. Algebraic codes are also used as component codes in product code schemes [] where a Chase algorithm is applied to decode them. The approach introduced in this paper uses the BP algorithm to decode algebraic codes. The main advantage of this type of decoding is the fact that only one decoder is required to decode a concatenated scheme consisting of an LDPC code and an outer algebraic code. Furthermore, soft-output is readily available at its output what enables straightforward application in iterative schemes like the one presented in []. However, algebraic codes are not suitable for BP decoding. Even parity-check matrices composed of minimum-weight codewords of the dual code still tend to have densities of one-entries that exceed 20% and the corresponding Tanner graphs have girth 4, cf. [6]. In [] an efficient approach to decode algebraic codes based on ordered statistics decoding is presented, but it is not designed for BP decoding. The random redundant algorithm in [8] uses multiple parity-check matrix representations but imposes a high decoding delay and is scaled by an adaptive damping factor. Several authors [9], [10] proposed other adaptive BP algorithms, which require an additional reduction of the the parity-check matrix after each decoding iteration and hence increase the delay of the data stream caused by channel decoding. Contrary, our approach is inspired by redundant BP decoding when transmitting over the binary erasure channel (BEC) [11]. The ideas valid for the BEC can not be directly applied to the additive white Gaussian noise (AWGN) channel as the number of cycles in the Tanner graph increases when additional parity checks are appended to an existing decoding matrix. This issue is overcome by using multiple parallel BP decoders. These decoders use paritycheck matrices of cyclic form which contain the lowest number of check-equations possible such that all bits are still protected equally. The main contribution of this work is to show that this new parallel decoding approach compares favorably in terms of data delay imposed by the decoder and decoding performance when compared to standard BP decoding and other decoding schemes. Furthermore the approach yields near maximum-likelihood (ML) performance for high-rate codes of length n 12. This paper is organized as follows. Section II briefly introduces the notation used throughout the paper and presents the proposed decoding algorithm as well as the corresponding parity-check matrices. In Section III we investigate the possible performance improvement when the overall complexity is arbitrary and limited, respectively. Simulation results are presented in Section IV. II. MULTIPLE-BASES BELIEF-PROPAGATION DECODING The presented approach was inspired by current research on BP decoding after transmission over the BEC where redundant parity-check equations are used to improve the decoding performance [11], [12], [13], [14]. In this case, significant performance improvements are possible as the decoder is not affected by the introduction of new cycles. Consequently, one can improve the decoding performance in a straightforward way by presenting a redundant parity-check matrix to the decoder instead of matrix of size (n k) n. We are interested in adapting this idea to the AWGN channel scenario but this can not be done in a straightforward manner. This will be confirmed by simulations in Section IV. We therefore present an idea which relies on using a large number of redundant parity-check equations which are partitioned among a set of BP decoders processing the received data simultaneously. The approach is referred to as Multiple-Bases Belief-Propagation (MBBP) decoding and is introduced next along with the paritycheck matrices employed by the decoders. For a rigorous description of the decoding algorithm, let us first introduce a consistent notation. Let C be a linear [n, k, d] block code of length n, dimension k and minimum distance d, and let u denote the vector of information symbols u ν {0, 1}, ν= 1,...,k of length k. Weusec, c ν { 1, 1}, ν =1,...,n,to denote a codeword consisting of antipodal symbols, which is a realization of the random vector C. Furthermore, the realvalued vector y =(y 1,...,y n ) is used to denote the noisy

2 received word, which is a realization of the random vector Y. The channel decoder determines an estimate û of the information vector. A. Decoding algorithm MBBP decoding runs multiple BP decoders in parallel. Each of the decoders uses a different parity-check matrix representation of the code. Let the parity-check matrix representation used by the l-th decoder be denoted by H l, l =1,...,l and the decoded vector after at most i iterations by ĉ l, l =1,...,l. We define a set S {1,...,l} which contains the indices l of the decoders that converged to a valid codeword. If no decoder finds a valid codeword, we let S = {1,...,l}. The decoder estimates ĉ s, s Sare passed on to a least metric selector (LMS) which determines the best representative using the decision rule ĉ = argmax Pr {Y = y C = ĉ s }.Itis s S well known that this equation can be rewritten [1] as ĉ = n argmin y ν ĉ s,ν 2, if the channel is assumed to be s S ν=1 the AWGN channel. The estimated information vector û can be obtained from ĉ in a straightforward manner. Algorithm 1 lists the pseudocode of the described algorithm, while Figure 1 visualizes MBBP decoding. Algorithm 1 MBBP decoding S := for l := 1,...,l do ĉ l := BPdec(y, H l ) =0then if ĉ l H T l S := S l end if end for if S = then S := {1,...,l} end if ĉ := argmin s S y n y ν ĉ s,ν 2 ν=1 BP-Dec. BP-Dec. BP-Dec. on on... on H 1 H 2 H l 1 S? 2 S? l S? LMS Fig. 1. Parallelized decoding scheme using l different parity-check matrix representations H l, l =1,...,l B. Parity-check matrices For a linear [n, k, d] code C, each set of n k linearly independent codewords of the dual code C constitutes a base for C, i.e. can be used to generate all codewords of the dual û... code. Hence, each of these sets can be used to form a valid parity-check matrix of C, although most of these matrices are not appropriate for BP decoding. In general it is advisable to use (whenever possible) only minimum-weight codewords of the dual code to construct parity-check matrices of full rank for MBBP decoding. By this the low-density property is approximated which helps to keep the number of short cycles in the Tanner graph small. We henceforth concentrate on cyclic and extended cyclic codes and introduce the idea of cyclic group generators [13] which will be used to construct H l, l =1,...,l. Definition 2.1: Let C be an [n, k, d] binary linear cyclic code. Partition the set of codewords of its dual code C into sets consisting only of cyclic shifts of one codeword. Let one of these codewords be the representative of that group, and refer to it as the cyclic group generator (cgg). Let G be the set of cyclic group generators with a Hamming weight equal to the minimum distance of the dual code d. The cyclic group generator cgg l G, l =1,...,l is used to construct the parity-check matrix H l, l = 1,...,l of size n n by defining the i-th row of H l as the (i 1)- th cyclic shift of cgg l. As matrices of cyclic form are used and equal error protection of symbols is reasonable, n parity checks are necessary. Simulation results have shown that for all cyclic codes considered in this paper the use of redundant rows is advantageous until equal error protection for all symbols is assured. Exemplary results for the [31, 16, ]-BCH are presented in Section IV. For extended cyclic codes like the [24, 12, 8]-Golay code little technical modifications are required to construct the parity-check matrices. III. DECODING PERFORMANCE AND COMPLEXITY -ACASE STUDY We investigate the benefits of using MBBP decoders, which are twofold. On the one hand, a significant performance gain can be obtained when one allows the overall decoding complexity of the set of MBBP decoders to exceed the complexity of one standard BP decoder. On the other hand, it will also be shown that MBBP exploits a limited decoding complexity more efficiently than standard BP. For this purpose, let i be the number of iterations performed by each of the l decoders. We measure the overall complexity in terms of total number of iterations performed, given by I = i l. The performance obtained with the [31, 16, ]-BCH code and the [24, 12, 8]-Golay code will be considered as illustrative examples. In order to obtain the results required for the study on the [31, 16, ]-BCH code, we simulate MBBP decoding with 1 l 1 decoders and I {100, 200,...,100} overall iterations. The upper restriction on l is due to the maximum number of cggs of minimum weight for the chosen codes, cf. Section IV. Figure 2(a) shows the signal-to-noise ratio (SNR, measured as 10 log 10 (E b /N 0 )) which is required to obtain a frame error rate (FER) of when the number of decoders l and the number of iterations per decoder i are varied. It is to be observed that classical BP decoding (l = 1)

3 6.2 of l ([31, 16, ]-BCH) and l 6 ([24, 12, 8]-Golay) lead to the most efficient use of the given complexity i l l l (a) [31, 16, ]-BCH (a) [31, 16, ]-BCH (b) [24, 12, 8]-Golay Fig. 3. Required SNR (10 log 10 (E b /N 0 )) to obtain FER = for varying l when I = i l = 100 is fixed i (b) [24, 12, 8]-Golay 11 l Fig. 2. Required SNR 10 log 10 (E b /N 0 ) to obtain FER =. performs noticeably worse than MBBP decoding with l>1. Note that a very significant reduction of the required SNR can already be obtained when allowing for l =2decoders. Figure 2(a) also shows that the performance gain increases more by increasing l, rather than by increasing i. A low required SNR is achievable when one allows for l decoders and i 0 iterations per decoder. As the required SNR is almost constant within this operational region, one has the freedom to trade-off between the number of parallel decoders and the data delay. For example, the performance obtained by l =,i = 3 (4.3 db) could also be obtained by l =6,i=60(4.0 db) orl =9,i=40(4.1 db). Let us now consider MBBP decoding of the [31, 16, ]-BCH code for limited overall complexity. We choose I = 100 as this is a reasonable setting for standard BP decoding. The markers in Figure 2(a) and Figure 3(a) show the required SNR to obtain FER = for 1 l 1 when I = i l = 100 is fixed. It can be observed that the required SNR reaches a minimum for a certain choice of l and i, which is l and i 20 for the considered case of I = 100. Similar results are obtained for the [24, 12, 8]-Golay code, cf. Figure 2(b) and Figure 3(b). It is to be mentioned that a higher overall complexity is required to obtain the desired FER at a low SNR. For the case of limiting the overall complexity described by I = 100, a choice IV. RESULTS We present results for block codes of high rate and rates 1/2 with n 12. In order to evaluate the proposed approach we compare the MBBP performance simulation results to the results of an ML decoder performing a full search on all codewords, where we consider the bit error rate (BER) as well as the FER as performance measure. However, since it is not feasible to simulate the performance of an ML decoder for all considered codes, we use the union bound which requires the knowledge of the weight distribution of the code. Based on comprehensive studies, this bound is known to be tight for the error rates of interest. The union bound is of the form BER 1 ( ) n n δ=d A δ δ Q 2 k n δ E b N 0 and FER ( n δ=d A δ Q 2 k n δ E b N 0 ), where A δ denotes the number of codewords of weight δ. If the considered code C is a high-rate code, we determine the weight distribution of C and use the MacWilliams identities [1] to yield the weight distribution of C. The evaluated codes include the [31, 16, ]-BCH, [63, 4, ]- BCH, and [12, 113, ]-BCH, as well as the [24, 12, 8] extended Golay code. A. [31, 16, ]-BCH code There exist 1 cggs of minimum weight 8 in the dual of the [31, 16, ] BCH code which can be used to create the paritycheck matrices H l, l 1. Based on the results from Section III, we choose l =6decoders, each of them performing at most i =60iterations. Figure 4 and Figure show the decoding performance in terms of BER and FER, respectively. For comparison with standard BP decoding, the performance of standard BP decoders using the matrices H l, l =1,...,l is depicted. These decoders perform not more than i =60and i = 100 iterations, respectively. It is to be observed that all parity-check matrix representations yield equal performance results for fixed i and are significantly outperformed by MBBP decoding. To exemplify the decoding performance of matrices of size (n k) n, we show results obtained by decoding with a full-rank matrix of minimum size, H l, l =1,...,lwith

4 Hl, l =1,...,l, standard BP, 66 iter MBBP decoding, l =3, 66 iter Hs1, standard BP, 100 iter Hs2, standard BP, 100 iter H l, l =1,...,l, standard BP, 100 iter Hl, l =1,...,l, standard BP, 60 iter MBBP decoding, l =6, 60 iter Fig. 4. BER performance obtained with the [31, 16, ] BCH code. Fig. 6. BER performance obtained with the [63, 4, ] BCH code. Hl, l =1,...,l, standard BP, 66 iter MBBP decoding, l =3, 66 iter Hs1, standard BP, 100 iter Hs2, standard BP, 100 iter H l, l =1,...,l, standard BP, 100 iter Hl, l =1,...,l, standard BP, 60 iter MBBP decoding, l =6, 60 iter Fig.. FER performance obtained with the [31, 16, ] BCH code. Fig.. FER performance obtained with the [63, 4, ] BCH code. H l(j, :) = H l (j, :), j =1,...,n k, l =1,...,l. Results for the stacked matrices H s1 =[H 1;...; H l] and H s2 = [H 1 ;...; H l ] demonstrate that a straightforward adaptation of the approach applicable in the BEC case does not lead to good decoding performance. For a better classification of the results, performance values for the Chase decoding algorithm- 2 [1] are presented as well. This algorithm is frequently used for soft-decision decoding of algebraic codes and performs about 0. dbworse compared to MBBP decoding, measured at BER = 10 and FER =, respectively. To assess the performance of MBBP decoding more generally, the union bound on ML decoding as well as simulated results for full search ML decoding are shown. It is worth mentioning that MBBP decoding almost meets the ML decoding performance. For a BER of 10 or a FER of, MBBP performs only 0.06 db worse than ML decoding. B. [63, 4, ]-BCH code The dual code of the [63, 4, ]-BCH contains only 3 cggs of minimum weight 16. In this case, l =3and i =66is a good choice for MBBP decoding. Figure 6 and Figure show simulation results in terms of BER and FER. Again, for comparison, the performance results obtained by standard BP decoding with 100 and 66 iterations, as well as the union bound for ML decoding are given. It is to be observed that with a total number of only I = 198 iterations, MBBP decoding performs very close to the bound for ML decoding. To be precise, MBBP decoding allows to decode this short code of rate 0.1 at db with a residual BER of 10. This is of interest for practical implementations as it is not feasible to decode the [63, 4, ]-BCH code by full search with a reasonable complexity. Due to the presence of 18 parity symbols it is also not straightforward to apply ML decoding using the trellis of the dual code [16]. For comparison we also include simulation results for. C. [12, 113, ]-BCH code The minimum-weight codewords of the dual of the [12, 113, ]-BCH code have weight 6 and can be partitioned into 36 cyclic groups. The resulting large check node degrees result in a poor decoding performance when using standard BP decoding which allows us to demonstrate the decoding performance advantage of MBBP decoding. Figure 8 shows that MBBP decoding leads to an 1dBperformance improvement compared to standard BP decoding. Note that at the same time, MBBP decoding reduces the decoder delay from 100 to 30 iterations. Moreover, MBBP decoding almost meets the bound on ML decoding. D. [24, 12, 8]-Golay code The [24, 12, 8]-Golay code is an extended cyclic code. In this case, the codewords of the dual code with minimum

5 Hl, l =1,...,l, standard BP, 30 iter MBBP decoding, l =24, 30 iter Fig. 8. BER performance obtained with the [12, 113, ] BCH code. Hl, l =1,...,l, standard BP, 0 iter MBBP decoding, l =1, 0 iter Fig. 10. BER performance obtained with the [24, 12, 8] Golay code. Hl, l =1,...,l, standard BP, 0 iter MBBP decoding, l =1, 0 iter Hl, l =1,...,l, standard BP, 30 iter MBBP decoding, l =24, 30 iter Fig. 9. FER performance obtained with the [12, 113, ] BCH code. Fig. 11. FER performance obtained with the [24, 12, 8] Golay code. weight 8 are partitioned into 33 sets. Each of these sets consists of codewords that are cyclic shifts of each other when restricted to the first 23 positions. Note that we assume that the overall parity-bit is at position 24 in the codewords. In this case, l =1, i =0leads to good decoding results that are about 0.2 db away from the union bound on ML decoding. This result allows us to compare MBBP to the approach introduced in [8], which shows equal decoding performance. Yet MBBP decoding does not utilize a damping factor and needs a little less iterations allowed (100 compared to 1200 [8]). Also, MBBP has a completely parallel structure and keeps the data delay low as i =0is set for each decoder. Results for codes with higher dimensions as well as further interesting decoding approaches using multiple bases will be presented in a full version of the paper. REFERENCES [1] S. Lin and D.J. Costello. Error Control Coding. Pearson Education, Inc., Second edition, [2] J. Pearl. Probabilistic reasoning in intelligent systems: networks of plausible inference. Morgan Kaufmann Publishers, [3] C. Berrou, A. Glavieux, and P. Thitimajshima. Near Shannon limit errorcorrecting coding and decoding. In Proc. Int. Communications Conf., [4] T. Richardson. Error-floors of LDPC codes. In Proceedings of the 41st Annual Allerton Conference on Communication, Control and Computing, pages , Sept [] R. M. Pyndiah. Near-optimum decoding of product codes: Block turbo codes. IEEE Trans. on Comm., 46(8): , August [6] T.R. Halford, K.M. Chugg, and A.J. Grant. Which codes have 4-cyclefree tanner graphs? In Proceedings of Int. Symp. on Inform. Theory (ISIT), pages 81 8, Seattle, WA, July [] M.P.C. Fossorier and S. Lin. Soft-decision decoding of linear block codes based on ordered statistics. IEEE Trans. Inform. Theory, 41: , September 199. [8] T.R. Halford and K.M. Chugg. Random redundant soft-in soft-out decoding of linear block codes. In Proceedings of Int. Symp. on Inform. Theory (ISIT), pages , Seattle, WA, July [9] A. Kothiyal, O. Y. Takeshita, W. Jin, and M.P.C. Fossorier. Iterative reliability-based decoding of linear block codes with adaptive belief propagation. IEEE Communications Letters, 9(12): , December 200. [10] J. Jiang and K.R. Narayanan. Iterative soft decision decoding of reed solomon codes based on adaptive parity check matrices. In Proc. IEEE Int. Symp. on Inform. Theory, page 261, [11] M. Schwartz and A. Vardy. On the stopping distance and stopping redundancy of codes. IEEE Trans. on Inform. Theory, 2(3): , March [12] J. Han and P. Siegel. Improved upper bounds on stopping redundancy. IEEE Trans. on Inform. Theory, 3(1):90 104, January 200. [13] T. Hehn, S. Laendner, O. Milenkovic, and J. B. Huber. The stopping redundancy hierarchy of cyclic codes. In Proceedings of the 44th Annual Allerton Conference on Communication, Control and Computing, [14] T. Hehn, O. Milenkovic, S. Laendner, and J.B. Huber. Permutation decoding and the stopping redundancy hierarchy of linear block codes. Accepted for presentation at IEEE Int. Symp. on Inform. Theory (ISIT 200), Nice, France, June 200. [1] F.J. MacWilliams and N.J. Sloane. The Theory of Error-Correcting Codes. North-Holland Publishing Company, 19. [16] J.L. Wolf. Efficient maximum likelihood decoding of linear block codes using a trellis. IEEE Trans. on Inform. Theory, 24:6 80, 198.

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