GALLAGER S binary low-density parity-check (LDPC)

Transcription

1 1560 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 6, JUNE 2009 Group-Theoretic Analysis of Cayley-Graph-Based Cycle GF(2 p )Codes Jie Huang, Shengli Zhou, Member, IEEE, Jinkang Zhu, Senior Member, IEEE, and Peter Willett, Fellow, IEEE Abstract Using group theory, we analyze cycle GF(2 p )codes that use Cayley graphs as their associated graphs. First, we show that through row and column permutations the parity check matrix H can be put in a concatenation form of row-permuted block-diagonal matrices. Encoding utilizing this form can be performed in linear time and in parallel. Second, we derive a rule to determine the nonzero entries of H and present determinate and semi-determinate codes. Our simulations show that the determinate and semi-determinate codes have better performance than codes with randomly generated nonzero entries for GF(16) and GF(64), and have similar performance for GF(256). The constructed determinate and semi-determinate codes over GF(64) and GF(256) can outperform the binary irregular counterparts of the same block lengths. One distinct advantage for determinate and semi-determinate codes is that they greatly reduce the storage cost of H for decoding. The results in this correspondence are appealing for the implementation of efficient encoders and decoders for this class of promising LDPC codes, especially when the block length is large. Index Terms Cayley graph, cycle code, Galois field, group theory, LDPC. I. INTRODUCTION GALLAGER S binary low-density parity-check (LDPC) codes [1] are excellent error-correcting codes with performance close to the Shannon capacity [2]. LDPC codes over GF(2 p ) have been investigated empirically by Davey and Mackay over the binary-input AWGN channel [3]. LDPC codes of column weight j = 2 are known as cycle codes [4]. Although the distance properties of binary cycle codes are not as good as LDPC codes of column weight j 3 [1], it has been shown in [5] that cycle codes over GF(2 p ) can achieve near-shannon-limit performance with maximum likelihood (ML) decoding as p increases. Further, the performance loss of iterative decoding relative to ML decoding can be minimized due to the sparsity of the parity check matrices of cycle codes [5]. Numerical results demonstrate that cycle GF(2 p ) codes can outperform other LDPC codes, including degree-distribution-optimized binary irregular LDPC codes [5]. For high order fields, the best GF(2 p )-LDPC codes Paper approved by O. Milenkovic, the Editor for Coding Theory and Applications of the IEEE Communications Society. Manuscript received February 2, 2008; revised July 24, This work was supported by the Office of Naval Research grants N and N J. Huang, S. Zhou, and P. Willett are with the Department of Electrical and Computer Engineering, University of Connecticut, 371 Fairfield Road U-2157, Storrs, Connecticut 06269, USA ( jhuang@engr.uconn.edu; shengli@engr.uconn.edu; willett@engr.uconn.edu). J. Zhu is the director of the Personal Communication Networks and Spread Spectrum (PCN&SS) research group, Department of Electronic Engineering and Information Science, University of Science and Technology of China, Hefei, Anhui, China ( jkzhu@ustc.edu.cn). Digital Object Identifier /TCOMM /09$25.00 c 2009 IEEE decoded by belief propagation (BP) tend to be ultra sparse [3], [6], with a good example being the cycle codes that have j = 2. A Fast-Fourier-Transform based q-ary sumproduct algorithm (FFT-QSPA) for decoding LDPC codes over GF(2 p ) has been proposed in [7], [8]. The complexity of FFT- QSPA is on the order of O(p2 p ), whereas the straightforward implementation of the BP algorithm [3] has complexity on the order of O(2 2p ). Recently, reference [9] presented a low-complexity, low-memory decoding algorithm for LDPC codes over GF(2 p ), which has complexity on the order of O(n m log n m ) where n m 2 p. The complexity of the decoder in [9] is comparable to that of a binary decoder. An universal linear-complexity encoding algorithm for any cycle GF(2 p ) code is available in [10]. With the performance and implementation advantages, cycle GF(2 p ) codes are very promising for practical applications. Different from the Tanner-graph representation of LDPC codes, a cycle GF(2 p ) code with parity check matrix H of size m by n can be represented by its associated graph [10], which has m vertices and n edges. Each vertex represents a constraint node and corresponds to a row of H. Each edge represents a variable node and corresponds to a column of H. As indicated in [11], to achieve good decoding performance the associated graph of a cycle GF(2 p ) code should be a good expander [12] and the girth of the graph should be as large as possible. One kind of good expander graph is the Ramanujan graph [13]. A d-regular Ramanujan graph is definedbythe property that the second largest eigenvalue of its adjacency matrix is no greater than 2 d 1 and thus is known to have good expansion properties, large girth and small diameter. In particular the girth of Ramanujan graphs is asymptotically a factor of 4/3 better than the Erdös-Sachs bound, which in terms of girth appears to be the best d-regular graphs known [13]. Some explicitly constructed Ramanujan graphs [13] are particular instances of a more general kind of graph called Cayley graph [14]. This paper analyzes the structure of Cayley-graphbased cycle GF(2 p ) codes from the group theory point of view and serves as an effort towards the implementation of efficient encoders and decoders for this kind of promising LDPC code. In particular, we present an equivalent concatenation form for the parity check matrix of any Cayley-graph-based cycle GF(2 p ) code. Encoding utilizing this form can be performed in linear time and in parallel. Further, we propose determinate and semi-determinate Cayley-graph-based cycle GF(2 p ) codes, which have very good performance and can greatly reduce the storage cost for decoding. The rest of the paper is organized as follows. Related definitions and notations are collected in Section II. Section III

2 HUANG et al.: GROUP-THEORETIC ANALYSIS OF CAYLEY-GRAPH-BASED CYCLE GF(2 P ) CODES 1561 presents the equivalent concatenation form for the parity check matrix, while Section IV presents the determinate and semideterminate Cayley-graph-based cycle GF(2 p ) codes. Finally, Section V concludes the paper. II. DEFINITIONS AND NOTATIONS Let G be a finite group of size m and S G be a subset of G satisfying S = S 1. The Cayley graph Γ(G, S) is the graph having vertices indexed by G such that (g, g ) is an edge iff g = gs for some s S. Denote the identity element of G by 1 G. Throughout this paper we assume 1 G / S, that is, Γ(G, S) has no loop. Γ(G, S) is an undirected d-regular graph with d = S. WeuseC E (Γ(G, S), H) to denote a cycle GF(2 p ) code with parity check matrix H and associated graph Γ(G, S). IfH is of size m by n, then2n = md. Assume G G and S S, letγ(g,s ) represent the sub-graph of Γ(G, S) having vertices indexed by G such that (g, g ), g G, g G is an edge iff g = gs for some s S. For a sub-graph Γ,letH Γ be the sub-matrix of H restricted to the rows and columns indexed by the vertices and edges of Γ respectively. For two matrices H 1 and H 2,ifH 1 can be transformed into H 2 simply through row and column permutations, we deem H 1 equivalent to H 2 and denote this relationship as H 1 = H2. For any cycle code with associated graph Γ(G, S), the nonzero entries of H can be described by a mapping f : G S GF(2 p )\{0} (1) where f(g, s),g G, s S represents the nonzero entry of H in the row indexed by g and column indexed by (g, gs). For an edge (g, gs) illustrated in Fig. 1(a), assume α = f(g, s) and β = f(gs, s 1 ).Define the gain of the edge as h(g, s) =α 1 β =[f(g, s)] 1 f(gs, s 1 ). (2) We infer from (2) that h(gs, s 1 )=[h(g, s)] 1 (3) f(gs, s 1 )=f(g, s)h(g, s). (4) The sub-matrix of H associated with a single edge can be put in an equivalent vector form h E as [ α h E =. (5) β] Fig. 1(b) illustrates a length-k cycle C = g gs 1 gs 1 s 2...s k 1 g, whereg G and s 1 s 2...s k is a minimal word over S satisfying s 1 s 2...s k =1 G, i.e., any portion of it does not satisfy s i s i+1...s j =1 G where 1 i j k and j i<k 1; andα i, β i, i =1, 2,...,k,representthe corresponding nonzero entries. Then the sub-matrix associated with the cycle C can be put in an equivalent form H C as shownin(6). α β k β 1 α H C = 0 β 2 α (6) β k 1 α k Fig. 1. Illustration of an edge and a cycle. (a) An edge (g, gs). (b) A lengthk cycle g gs 1 gs 1 s 2...s k 1 g, wheres 1 s 2...s k is a minimal word satisfying s 1 s 2...s k =1 G. III. STRUCTURE OF CAYLEY-GRAPH-BASED CYCLE GF(2 p )CODES We use group theory to analyze the cycle GF(2 p ) codes based on Cayley graphs. We have the following results. Lemma 1: Given any cycle code with associated graph Γ(G, S), leto(s) be the order of s in G and G s be the subgroup generated by s for any s S. Define Q = G /O(s). 1) If O(s) > 2, the sub-matrix associated with Γ(G, {s, s 1 }) is equivalent to a matrix having the form H c = diag(h 1 C, H2 C,...,HQ C ), (7) where H i C is a matrix having the form in (6) with k = O(s). 2) If O(s) =2, the sub-matrix associated with Γ(G, s) is equivalent to a matrix having the form H e = diag(h 1 E, h 2 E,...,h Q E ), (8) where h i E is a vector having the form in (5). Proof: The size of G s is O(s) and the group G can be partitioned into the union of disjoint left cosets g 1 G s, g 2 G s,..., g Q G s,whereg i is a representative of the coset g i G s, 1 i Q. 1) If O(s) > 2, i.e., s s 1, the sub-graph Γ(g i G s, {s, s 1 }) forms a length-k cycle determined by the relation s k =1 G with k = O(s). Thus the submatrix associated with Γ(g i G s, {s, s 1 }) is equivalent to H C as shown in (6). Since Γ(G, {s, s 1 }) is the union of Q disjoint components Γ(g i G s, {s, s 1 }), so the submatrix associated with Γ(G, {s, s 1 }) is equivalent to H c as shown in (7). 2) If O(s) =2, i.e., s = s 1, the sub-graph Γ(g i G s,s) forms an edge E i =(g i,g i s). Thus the sub-matrix associated with Γ(g i G s,s) is equivalent to h E as shown in (5). Since Γ(G, s) is the union of Q disjoint components Γ(g i G s,s), so the sub-matrix associated with Γ(G, s) is equivalent to H e as shown in (8). This completes the proof. Lemma 2: Given any cycle code with associated graph Γ(G, S), if O(s 1 ) = 2, O(s 2 ) = 2 and s 1 s 2, let k =2O(s 1 s 2 ) and Q = G /k where O(s1 s 2 ) is the order of s 1 s 2, then the sub-matrix associated with Γ(G, {s 1,s 2 }) is equivalent to a matrix having the form H c in (7).

3 1562 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 6, JUNE 2009 Proof: Let G s1,s 2 denote the subgroup generated by {s 1,s 2 }, then the size of G s1,s 2 is k and the group G can be partitioned into the union of disjoint left cosets g 1 G s1,s 2, g 2 G s1,s 2,..., g Q G s1,s 2,whereg i is a representative of the coset g i G s1,s 2, 1 i Q. The sub-graph Γ(g i G s1,s 2, {s 1,s 2 }) forms a length-k cycle determined by the relation (s 1 s 2 ) k/2 =1 G. Thus the sub-matrix associated with Γ(g i G s1,s 2, {s 1,s 2 }) is equivalent to H C as shown in (6). Since Γ(G, {s 1,s 2 }) is the union of Q disjoint components Γ(g i G s1,s 2, {s 1,s 2 }), the sub-matrix associated with Γ(G, {s 1,s 2 }) is equivalent to H c as shown in (7). This completes the proof. Denote S = {s S s = s 1 } and let S + be a maximal subset of S such that if s s 1, either s S + or s 1 S +. Hence, S S +.LetS = {s S + s/ S }.Then S = 2 S + S. Theorem 1: Given any cycle code with associated graph Γ(G, S), lett = S and r = S =2r 1 + r 2,wherer 2 = 0 or 1, thend = S =2(t + r 1 )+r 2 and the parity check matrix H is equivalent to a matrix having the form { [H H= c 1, Pc 2 Hc 2,...,Pc t+r 1 ] for d even [H c 1, Pc 2 Hc 2,...,Pc t+r 1, P e H e ] for d odd (9) where P c i, 2 i t + r 1,andP e are permutation matrices, H c i, 1 i t + r 1,hastheformasH c in (7), and H e has the form in (8). Proof: We just give a proof for the case where r 2 = 0, i.e., d is even. The proof for the case where r 2 = 1 is similar. If r 2 =0, suppose S = {s 1,s 2,...,s t } and S = {s t+1,1,s t+1,2,...,s t+r1,1,s t+r1,2}. Arrange the columns of H in such a way that the columns indexed by the edges of Γ(G, {s 1,s 1 1 }) are placed in the first G columns, followed by the G columns indexed by the edges of Γ(G, {s 2,s 1 2 }) until the G columns indexed by the edges of Γ(G, {s t,s 1 t }), then followed by the G columns indexed by the edges of Γ(G, {s t+1,1,s t+1,2 }) until the G columns indexed by the edges of Γ(G, {s t+r1,1,s t+r1,2}). Following Lemma 1 and Lemma 2, the resultant matrix would have a form like [P c 1 Hc 1 Rc 1, Pc 2 Hc 2 Rc 2,...,Pc t+r 1 R c t+r 1 ],wherep c i and R c i, 1 i t + r 1, are permutation matrices. Furthermore, we can permute the rows of H to let P c 1 be the identity matrix and permute the columns of H to let each R c i be the identity matrix. This completes the proof. One direct benefit of the representation in Theorem 1 is to enable efficient encoding as follows. Assume that H has the form in (9) and the size of H is m by n and n m, sot + r 1 1. If the matrix H c 1 is full rank, then the encoding can be accomplished in linear time. Assume a codeword x = [x T c,1, xt c,2,...,xt c,t+r 1 ] T for r 2 = 0 or x =[x T c,1, x T c,2,...,x T c,t+r 1, x T e ] T for r 2 =1. Let the symbols of x c,1 be the parity symbols and others be information symbols. For an even d, a valid codeword x means that H c 1 x c,1 = P c 2 Hc 2 x c,2 + + P c t+r 1 x c,t+r1 (10) while for an odd d it means that H c 1 x c,1 = P c 2H c 2x c,2 + + P c t+r 1 x c,t+r1 + P e H e x e (11) Suppose that H c 1 is as shown in (7) and the right-hand side of (10) or (11) is (b T 1,...,bT Q )T,whereb i is a length-k vector with elements from GF(2 p ). Computation of x c,1 requires solving the Q equations H i C x = b i, 1 i Q, whereh i C has the same equivalent form as in (6). A linear time algorithm for solving these equations has been proposed in the lemma 4 of [10]. Note that solving these Q equations can be performed in parallel, thus encoding can be performed in parallel in linear time. This provides a lot of flexibility in the implementation of efficient encoders, and is quite desirable especially when the block length is large. Note that the universal linear-time encoding algorithm presented in [10] can work only in a serial manner. IV. DETERMINATE AND SEMI-DETERMINATE CODES In Section III, we presented an equivalent structure of the parity check matrix H for any Cayley-graph-based cycle GF(2 p ) code. We now look into how to decide those nonzero entries of H. We focus our attention to h(g, s) =h(s), thatis, function h is independent of g. In this situation, (3) simplifies to h(s 1 )=[h(s)] 1. (12) On the other hand, if function f is independent of g, (4) simplifies to f(s 1 )=f(s)h(s). (13) We define two kinds of codes as follows. Definition 1: A cycle code with associated graph Γ(G, S) is called semi-determinate if function h is independent of g, i.e., (12) holds true. Definition 2: A cycle code with associated graph Γ(G, S) is called determinate if both functions h and f are independent of g, i.e., both (12) and (13) hold true. Let ξ be a primitive element of GF(2 p ) satisfying q(ξ) = 0 where q(x) = x p + q p 1 x p q 0 is a primitive polynomial of degree p over GF(2). Let Z 2p 1 be the additive group modulo 2 p 1. The mapping M : GF(2 p )\{0} Z 2p 1, ξ i i, i =0, 1,...,2 p 2, (14) is an isomorphism from the multiplicative group of GF(2 p )to Z 2p 1. For a length-k cycle C as illustrated in Fig. 1(b), where s 1 s 2...s k =1 G,leth(s i )=α 1 i β i, i =1, 2,...,k, denote the corresponding gain of the edges. The sub-matrix associated with this cycle is equivalent to H C as shown in (6). As indicated in [10], this cycle is irresolvable iff H C is fullrank, i.e., k i=1 α 1 i β i 1.Sinceh(s i )=α 1 i β i, it becomes k i=1 h(s i) 1, i.e., k M(h(s i )) 0 (mod 2 p 1). (15) i=1 If the cycle is resolvable, i.e., H C is rank-deficient, equation H C x = 0 would have nonzero solutions. With code symbols other than those corresponding to the edges of the cycle been set to zeroes, any nonzero solution of equation H C x = 0 corresponds to a nonzero codeword of symbol weight k. Thus

4 HUANG et al.: GROUP-THEORETIC ANALYSIS OF CAYLEY-GRAPH-BASED CYCLE GF(2 P ) CODES 1563 resolvable cycles with short length correspond to low-weight codewords, which may induce undetected errors during the decoding process. To achieve good decoding performance, we should try to make as many cycles as possible irresolvable, especially those having short length. To achieve this, we can enumerate all the patterns s 1 s 2...s k = 1 G that generate length-k cycles for those k we are concerned about and choose appropriate h(s) through solving a set of inequalities like (15). A. Performance Results for Example Determinate and Semideterminate Codes Example 1: Let the Cayley graph be the Ramanujan graph X 3,7 [13]. The group G is the projective general linear group PGL 2 (7) and the set S is { S := s 1 = s 1 1 = [ ] ,s 2 = [ ] 1 5,s = [ ] , [ ] } With X 3,7 specified in Example 1, we have m = 336, d =4,andn = 672. Each element in S has an order of 8 and the girth of X 3,7 is exactly 8. Using the graph X 3,7 as the associated graph we have constructed some random, semideterminate and determinate codes over GF(16), GF(64) and GF(256). The nonzero entries of H are generated randomly for random codes. The parameters (h(s), f(s)) for semideterminate and determinate codes are listed in Table I, where SemiDET stands for semi-determinate code and DET stands for determinate code. The function h(s) for semideterminate and determinate codes is selected to make as many as possible cycles irresolvable. For example, we obtain h(s 1 ) = ξ 3,h(s 2 ) = ξ 4 to make all cycles of length 8 and 10 irresolvable for GF(16). All the codes listed in Table I can make all cycles of length 8 and 10 irresolvable. We also find that no function h(s) can make all cycles of length 8, 10 and 12 irresolvable simultaneously for GF(16), GF(64) and GF(256). For semi-determinate codes, given the function h(s) the value α = f(g, s) for each edge (g, gs) is generated randomly and the value β = f(gs, s 1 ) is determined according to (4). For any determinate code, note that each row of H contains the same set of nonzero entries {f(s 1 ),f(s 1 1 ),f(s 2),f(s 1 2 )}. Given the function h(s), the function f(s) can be determined based on the search algorithm proposed by Mackay which maximizes the marginal entropy of one element of the syndrome vector [15]. Usually the obtained set of nonzero entries is not globally optimal. However, all the determinate codes listed in Table I contain exactly the globally optimal set of nonzero entries in each row. By Theorem 1 the parity check matrix H has an equivalent form as [H c 1, P c 2H c 2] where H c 1 and H c 2 both have an equivalent form as diag(h 1 C, H2 C,...,HQ C ) with Q =42and Hi C having the equivalent form as (6) with k =8. H c 1 has been rendered full rank in all the constructed codes. Thus encoding utilizing the equivalent form can be done in parallel with 42 subblocks of the codeword computed simultaneously. The code rate of all the constructed codes is 0.5 and the block length is 672 symbols, or equivalently 672p bits TABLE I PARAMETERS OF THE CODES INVESTIGATED IN EXAMPLE 1, THE PRIMITIVE POLYNOMIALS q(x) OF GF(16), GF(64), AND GF(256) ARE x 4 + x +1, x 6 + x +1, AND x 8 + x 4 + x 3 + x 2 +1, RESPECTIVELY. h(s), f(s) SemiDET, GF(16) h(s 1 )=ξ 3,h(s 2 )=ξ 4 DET, GF(16) h(s 1 )=ξ 3,h(s 2 )=ξ 4, f(s 1 )=ξ 0,f(s 2 )=ξ 7 SemiDET, GF(64) h(s 1 )=ξ 15,h(s 2 )=ξ 8 DET, GF(64) h(s 1 )=ξ 15,h(s 2 )=ξ 8, f(s 1 )=ξ 0,f(s 2 )=ξ 42 SemiDET, GF(256) h(s 1 )=ξ 8,h(s 2 )=ξ 11 DET, GF(256) h(s 1 )=ξ 8,h(s 2 )=ξ 11, f(s 1 )=ξ 0,f(s 2 )=ξ 172 for GF(2 p ). Fig. 2 shows the simulation results of these codes. The curves for the random codes are obtained by averaging over multiple code realizations. Also plotted are the performance of binary irregular rate-1/2 LDPC codes of the same block lengths. These binary LDPC codes are constructed as follows. We first obtain the the bit-node edge distribution x x x x x x x 14 and the check-node edge distribution x x 8 from Table II in [17], which are derived using a density evolution based optimization algorithm. This distribution pair achieves an impressive iterative decoding threshold of db. We then construct the parity check matrices from the edge distributions using the progressive edge growth (PEG) algorithm [16]. The girth of the three binary irregular LDPC codes constructed by the PEG algorithm is 6, hence, no length-four cycle exists. For all the simulations, we assume BPSK transmission over the AWGN channel. For decoding, we use the FFT-QSPA algorithm given in [7], [8] for non-binary LDPC codes and the standard sum-product algorithm given in [2] for binary LDPC codes. The maximum number of iterations is set to 80. For all the simulations we run until 20 block errors are found or 2,000,000 blocks are transmitted. For GF(16), no determinate and semi-determinate code shows undetected errors of weight 8 and 10 whereas we have observed undetected errors of weight 8 and 10 for random codes. This verifies our design rule. For random codes undetected errors of weight 8 and 10 are significant for all range of E b /N 0, even dominant at high E b /N 0 as observed. We observe from Fig. 2 that at block-error-rate (BLER) of 10 4 determinate and semi-determinate codes achieve several tenths of db (range from 0.3 to 0.6) gain against the random code. The performance gain of determinate and semi-determinate codes over GF(16) against the corresponding random code results from the fact that it has low probability to render most of cycles of length 8 and 10 irresolvable with randomly generated nonzero entries from GF(16). As p increases it becomes highly probable to render most of short length cycles irresolvable with randomly generated nonzero entries from GF(2 p ), and consequently we would expect that the performance gain decreases. This can be verified from Fig. 2 where the performance gain of determinate codes over GF(64)

5 1564 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 6, JUNE 2009 Block Error Rate GF(64) (4032,2016) GF(256) (5376,2688) (2688,1344) (4032,2016) (5376,2688) Random Semi Determinate Determinate Binary Irregular GF(16) (2688,1344) E /N (db) b 0 Block Error Rate GF(64) (6048,4032) Random Semi Determinate Determinate GF(16) (4032,2688) GF(256) (8064,5376) E b /N 0 (db) Fig. 2. Performance of nonbinary cycle codes with Ramanujan graph X 3,7 as their associated graphs. Also included are the curves for the binary irregular LDPC codes. The Shannon capacity is 0.188dB for code rate of 1/2. Fig. 3. Performance of nonbinary cycle codes with Ramanujan graph X 5,7 as their associated graphs. The Shannon capacity is 1.084dB for code rate of 2/3. against the random counterpart is about 0.25dB at BLER of 10 5 and all the codes over GF(256) show similar performance at BLER above We observe from Fig. 2 that cycle codes over GF(64) and GF(256) achieve better performance than the corresponding binary irregular counterparts. Specially, at BLER of 10 5 the cycle codes over GF(256) achieve 0.2dB gain against the corresponding binary irregular LDPC code and are only 1.0dB away from the Shannon capacity. This verifies the good performance of cycle codes over binary extension fields that has been proved in [5]. Compared with existing codes in literature, such as those in [18], [19], the proposed codes achieve performance gains. In reference [18], a rate-1/2 128-ary (2032, 1016) quasicyclic LDPC code has been constructed, which has block length of bits. This code achieves BLER of 10 5 at E b /N 0 of 2.0dB. Our codes over GF(256) (GF(64), resp.) achieve BLER of 10 5 at E b /N 0 of 1.2dB (1.4dB, resp.), which is 0.8dB (0.6dB, resp.) better than the 128-ary (2032, 1016) quasi-cyclic LDPC code reported in [18], even though the block lengths of our codes are much smaller. Compared with Mackay s code which has rate of 1/2 and block length of 8000 bits [19], our codes over GF(256) (GF(64), resp.) gain about 0.5dB (0.4dB, resp.) at BLER of Now consider the Cayley graph to be the Ramanujan graph X 5,7 [13]. The girth of X 5,7 is 6, and the parameters are m = 336, d =6,andn = Using the graph X 5,7 as the associated graph we have constructed some random, semideterminate, and determinate codes over GF(16), GF(64), and GF(256). The code rate of all the constructed codes is 2/3 and the block length is 1008 symbols, or equivalently 1008p bits for GF(2 p ). Fig. 3 shows the simulated performance results of these codes. Similar observations can be made as the rate 1/2 case based on graph X 3,7. At BLER of 10 5,thecycle codes over GF(256) are only 0.9dB away from the Shannon capacity. We have also simulated codes constructed based on the Ramanujan graph X 7,11. All the codes based on X 7,11 have a code rate of 3/4. Simulation results reveal that at BLER of 10 5, the cycle codes over GF(256) are within 1dB away from the corresponding Shannon capacity. Hence, high-rate nonbinary cycle codes also have very good performance. B. Reduction on the Storage Requirement Combined with the structure proposed in Theorem 1, the storage of H for determinate and semi-determinate C E (Γ(G, S), H) codes can be greatly simplified. For illustration, costs in bits for storage of H for decoding are shown in Table II. It needs p and log m bits to store an element of GF(2 p ) and the interleavers respectively, log m and log n bits to store the row and column index for a nonzero entry of H respectively, where log is a base-2 logarithm operation and x is the least integer no less than x. For general cycle GF(2 p ) and general C E (Γ(G, S), H) codes, we need 2np bits to store the values of the 2n nonzero entries of H. To perform sum-product decoding for general cycle GF(2 p ) code [3], we need 2n( log m + log n ) bits to store the row and column indices for the 2n nonzero entries. For general, determinate and semi-determinate C E (Γ(G, S), H) codes, we need no more than 2n log m bits to store the interleavers and their inverses corresponding to matrices {P c i } and Pe, and no more than d/2 log m bits to store the parameter Qs corresponding to matrices {H c i } and He. We also need pd bits to store the f(s) function for determinate C E (Γ(G, S), H) code and p d/2 bits to store the h(s) function for semideterminate C E (Γ(G, S), H) code. Besides this, for semideterminate C E (Γ(G, S), H) code we need np bits to store the n values of α = f(g, s) corresponding to every edge (g, gs), thevaluesβ = f(gs, s 1 ) corresponding to every edge can be determined according to (4). The costs for storage of h(s), f(s) and parameter Qs are negligible. It can be verified from Table II that the

6 HUANG et al.: GROUP-THEORETIC ANALYSIS OF CAYLEY-GRAPH-BASED CYCLE GF(2 P ) CODES 1565 TABLE II COSTS IN BITS FOR STORAGE OF H FOR DECODING General cycle GF(2 p ) code General C E (Γ(G, S), H) Semi-Determinate C E (Γ(G, S), H) Determinate C E (Γ(G, S), H) Row/Column Indices or Interleavers and their inverses Nonzero entries of H h(s) or f(s) Parameter Qs Total cost 2n( log m + log n ) 2np 2n(p + log m + log n ) 2n log m 2np d/2 log m 2n(p + log m )+ d/2 log m 2n logm np p d/2 d/2 logm n(p + 2 logm )+ d/2 ( logm + p) 2n log m pd d/2 log m 2n log m + d/2 log m + pd The size of H is m by n with 2n = dm, n m. p and log m bits are used to store an element of GF(2 p ) and the interleavers respectively. log m and log n bits are used to store the row and column index for a nonzero entry of H respectively. TABLE III EVALUATION OF STORAGE COST REDUCTION FOR THE CODES IN EXAMPLE 1 GF(2 4 ) GF(2 6 ) GF(2 8 ) Semi-Determinate C E (Γ(G, S), H) against a general cycle GF(2 p ) code 52.1% 51.9% 51.8% Semi-Determinate C E (Γ(G, S), H) against a general C E (Γ(G, S), H) 15.3% 19.9% 23.4% Determinate C E (Γ(G, S), H) against a general cycle GF(2 p ) code 60.8% 63.9% 66.4% Determinate C E (Γ(G, S), H) against a general C E (Γ(G, S), H) 30.6% 39.8% 46.9% storage cost reduction rate of both determinate and semideterminate C E (Γ(G, S), H) codes against a general cycle GF(2 p ) code is more than 50%. The storage cost reduction of a determinate (resp. semi-determinate) C E (Γ(G, S), H) code against a general C E (Γ(G, S), H) code is about 2np (resp. np) bits. Example 2: In Table III, we illustrate the storage cost reduction for the codes investigated in Example 1, where m = 336, n = 672, d = 4. We see from Table III that significant savings on the storage are achieved. This is very attractive in the implementation of efficient decoders, especially when the block length is large. V. CONCLUSION Through group-theoretic analysis, an equivalent concatenation form was presented for the parity check matrix of any Cayley-Graph-Based cycle GF(2 p ) code. Encoding utilizing this form can be performed in linear time and in parallel. Determinate and semi-determinate Cayley-Graph-Based cycle GF(2 p ) codes were presented and shown by simulations to have very good performance. In addition, the storage cost of H for decoding can be greatly reduced for these codes. The properties presented in this correspondence are quite desirable for the implementation of efficient encoders and decoders for this class of promising LDPC codes, especially when the block length is large. REFERENCES [1] R. G. Gallager, Low Density Parity Check Codes. Cambridge, MA: MIT Press, [2] D. J. C. Mackay, Good error-correcting codes based on very sparse matrices, IEEE Trans. Inform. 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