Method and system for routing in low density parity check (LDPC) decoders

[0001] This application is related to, and claims the benefit of the earlier filing date under 35 U.S.C. § 119(e) of, U.S. Provisional Patent Application (Ser. No. 60/393,457) filed Jul. 3, 2002, entitled “Code Design and Implementation Improvements for Low Density Parity Check Codes,” U.S. Provisional Patent Application (Ser. No. 60/398,760) filed Jul. 26, 2002, entitled “Code Design and Implementation Improvements for Low Density Parity Check Codes,” U.S. Provisional Patent Application (Ser. No. 60/403,812) filed Aug. 15, 2002, entitled “Power and Bandwidth Efficient Modulation and Coding Scheme for Direct Broadcast Satellite and Broadcast Satellite Communications,” U.S. Provisional Patent Application (Ser. No. 60/421,505), filed Oct. 25, 2002, entitled “Method and System for Generating Low Density Parity Check Codes,” U.S. Provisional Patent Application (Ser. No. 60/421,999), filed Oct. 29, 2002, entitled “Satellite Communication System Utilizing Low Density Parity Check Codes,” U.S. Provisional Patent Application (Ser. No. 60/423,710), filed Nov. 4, 2002, entitled “Code Design and Implementation Improvements for Low Density Parity Check Codes,” U.S. Provisional Patent Application (Ser. No. 60/440,199) filed Jan. 15, 2003, entitled “Novel Solution to Routing Problem in Low Density Parity Check Decoders,” U.S. Provisional Patent Application (Ser. No. 60/447,641) filed February 14, 2003, entitled “Low Density Parity Check Code Encoder Design,” U.S. Provisional Patent Application (Ser. No. 60/456,220) filed Mar. 20, 2003, entitled “Description LDPC and BCH Encoders,” U.S. Provisional Patent Application Ser. No. 60/469,356 filed May 9, 2003, entitled “Description LDPC and BCH Encoders,” U.S. Provisional Patent Application Ser. No. 60/482,112 filed Jun. 24, 2003 entitled “Description LDPC and BCH Encoders,” and U.S. Provisional Patent Application Ser. No. 60/482,107 filed Jun. 24, 2003, entitled “Description LDPC and BCH Encoders”; the entireties of which are incorporated herein by reference.

[0002] The present invention relates to communication systems, and more particularly to coded systems.

[0003] Communication systems employ coding to ensure reliable communication across noisy communication channels. These communication channels exhibit a fixed capacity that can be expressed in terms of bits per symbol at certain signal to noise ratio (SNR), defining a theoretical upper limit (known as the Shannon limit). As a result, coding design has aimed to achieve rates approaching this Shannon limit. One such class of codes that approach the Shannon limit is Low Density Parity Check (LDPC) codes.

[0004] Traditionally, LDPC codes have not been widely deployed because of a number of drawbacks. One drawback is that the LDPC encoding technique is highly complex. Encoding an LDPC code using its generator matrix would require storing a very large, non-sparse matrix. Additionally, LDPC codes require large blocks to be effective; consequently, even though parity check matrices of LDPC codes are sparse, storing these matrices is problematic.

[0005] From an implementation perspective, a number of challenges are confronted. For example, storage is an important reason why LDPC codes have not become widespread in practice. Also, a key challenge in LDPC code implementation has been how to achieve the connection network between several processing engines (nodes) in the decoder. Further, the computational load in the decoding process, specifically the check node operations, poses a problem.

[0006] Therefore, there is a need for a LDPC communication system that employs simple encoding and decoding processes. There is also a need for using LDPC codes efficiently to support high data rates, without introducing greater complexity. There is also a need to improve performance of LDPC encoders and decoders. There is also a need to minimize storage requirements for implementing LDPC coding. There is a further need for a scheme that simplifies the communication between processing nodes in the LDPC decoder.

[0007] These and other needs are addressed by the present invention, wherein an approach for decoding a structured Low Density Parity Check (LDPC) codes is provided. Structure of the LDPC codes is provided by restricting portion part of the parity check matrix to be lower triangular and/or satisfying other requirements such that the communication between bit nodes and check nodes of the decoder is simplified. Edge values associated with the structured parity check matrix used to generate the LDPC coded signal are retrieved from memory. The edge values specify the relationship of bit nodes and check nodes, and according to one embodiment of the present invention, are stored within the memory according to a predetermined scheme (e.g., contiguous physical memory locations) that permits concurrent retrieval of a set of the edge values. According to another embodiment of the present invention, the edge values having bit nodes of n degrees are stored in a first portion of the memory, and edge values having bit nodes of greater than n degrees are stored in a second portion of the memory. The storage arrangement of the edge values advantageously allows fast retrieval of the edge values during the decoding process.

[0008] Also, the approach can advantageously exploit the unequal error protecting capability of LDPC codes on transmitted bits to provide extra error protection to more vulnerable bits of high order modulation constellations (such as 8-PSK (Phase Shift Keying)). The decoding process involves iteratively regenerating signal constellation bit metrics into an LDPC decoder after each decoder iteration or several decoder iterations. The above arrangement provides a computational efficient approach to decoding LDPC codes.

[0009] According to one aspect of an embodiment of the present invention, a method for decoding a low density parity check (LDPC) coded signal is disclosed. The method includes retrieving edge values associated with a structured parity check matrix used to generate the LDPC coded signal, wherein the edge values specify relationship of bit nodes and check nodes, and are stored according to a predetermined scheme that permits concurrent retrieval of a set of the edge values. The method also includes outputting a decoded signal corresponding to the LDPC coded signal based on the retrieved edge values.

[0010] According to another aspect of an embodiment of the present invention, a decoder for decoding a low density parity check (LDPC) coded signal is disclosed. The decoder includes means for retrieving edge values associated with a structured parity check matrix used to generate the LDPC coded signal. The decoder also includes memory for storing the edge values according to a predetermined scheme that permits concurrent retrieval of a set of the edge values, wherein the edge values specify relationship of bit nodes and check nodes. Further, the decoder includes means for outputting a decoded signal corresponding to the LDPC coded signal based on the retrieved edge values.

[0011] According to another aspect of an embodiment of the present invention, a memory accessible by a low density parity check (LDPC) decoder for decoding a LDPC coded signal is disclosed. The memory includes a first portion storing a first group of edge values associated with a structured parity check matrix used to generate the LDPC coded signal, the first group of edges being connected to bit nodes of n degrees. Additionally, the memory includes a second portion storing a second group of edge values associated with the structured parity check matrix used to generate the LDPC coded signal, the second group of edges being connected to bit nodes of greater than n degrees, wherein a set of edge values from the first group or the second group is retrieved to output a decoded signal.

[0012] Still other aspects, features, and advantages of the present invention are readily apparent from the following detailed description, simply by illustrating a number of particular embodiments and implementations, including the best mode contemplated for carrying out the present invention. The present invention is also capable of other and different embodiments, and its several details can be modified in various obvious respects, all without departing from the spirit and scope of the present invention. Accordingly, the drawing and description are to be regarded as illustrative in nature, and not as restrictive.

[0030] A system, method, and software for efficiently decoding structured Low Density Parity Check (LDPC) codes are described. In the following description, for the purposes of explanation, numerous specific details are set forth in order to provide a thorough understanding of the present invention. It is apparent, however, to one skilled in the art that the present invention may be practiced without these specific details or with an equivalent arrangement. In other instances, well-known structures and devices are shown in block diagram form in order to avoid unnecessarily obscuring the present invention.

[0031] FIG. 1 is a diagram of a communications system configured to utilize Low Density Parity Check (LDPC) codes, according to an embodiment of the present invention. A digital communications system 100 includes a transmitter 101 that generates signal waveforms across a communication channel 103 to a receiver 105. In this discrete communications system 100, the transmitter 101 has a message source that produces a discrete set of possible messages; each of the possible messages has a corresponding signal waveform. These signal waveforms are attenuated, or otherwise altered, by communications channel 103. To combat the noise channel 103, LDPC codes are utilized.

[0032] The LDPC codes that are generated by the transmitter 101 enables high speed implementation without incurring any performance loss. These structured LDPC codes output from the transmitter 101 avoid assignment of a small number of check nodes to the bit nodes already vulnerable to channel errors by virtue of the modulation scheme (e.g., 8-PSK).

[0033] Such LDPC codes have a parallelizable decoding algorithm (unlike turbo codes), which advantageously involves simple operations such as addition, comparison and table look-up. Moreover, carefully designed LDPC codes do not exhibit any sign of error floor.

[0034] According to one embodiment of the present invention, the transmitter 101 generates, using a relatively simple encoding technique, LDPC codes based on parity check matrices (which facilitate efficient memory access during decoding) to communicate with the receiver 105. The transmitter 101 employs LDPC codes that can outperform concatenated turbo+RS (Reed-Solomon) codes, provided the block length is sufficiently large.

[0035] FIG. 2 is a diagram of an exemplary transmitter in the system of FIG. 1. A transmitter 200 is equipped with an LDPC encoder 203 that accepts input from an information source 201 and outputs coded stream of higher redundancy suitable for error correction processing at the receiver 105. The information source 201 generates k signals from a discrete alphabet, X. LDPC codes are specified with parity check matrices. On the other hand, encoding LDPC codes require, in general, specifying the generator matrices. Even though it is possible to obtain generator matrices from parity check matrices using Gaussian elimination, the resulting matrix is no longer sparse and storing a large generator matrix can be complex.

[0036] Encoder 203 generates signals from alphabet Y to a modulator 205 using a simple encoding technique that makes use of only the parity check matrix by imposing structure onto the parity check matrix. Specifically, a restriction is placed on the parity check matrix by constraining certain portion of the matrix to be triangular. The construction of such a parity check matrix is described more fully below in FIG. 6. Such a restriction results in negligible performance loss, and therefore, constitutes an attractive trade-off.

[0037] Modulator 205 maps the encoded messages from encoder 203 to signal waveforms that are transmitted to a transmit antenna 207, which emits these waveforms over the communication channel 103. Accordingly, the encoded messages are modulated and distributed to a transmit antenna 207. The transmissions from the transmit antenna 207 propagate to a receiver, as discussed below.

[0038] FIG. 3 is a diagram of an exemplary receiver in the system of FIG. 1. At the receiving side, a receiver 300 includes a demodulator 301 that performs demodulation of received signals from transmitter 200. These signals are received at a receive antenna 303 for demodulation. After demodulation, the received signals are forwarded to a decoder 305, which attempts to reconstruct the original source messages by generating messages, X′, in conjunction with a bit metric generator 307. With non-Gray mapping, the bit metric generator 307 exchanges probability information with the decoder 305 back and forth (iteratively) during the decoding process, which is detailed in FIG. 10. Alternatively, if Gray mapping is used (according to one embodiment of the present invention), one pass of the bit metric generator is sufficient, in which further attempts of bit metric generation after each LDPC decoder iteration are likely to yield limited performance improvement; this approach is more fully described with respect to FIG. 11. To appreciate the advantages offered by the present invention, it is instructive to examine how LDPC codes are generated, as discussed in FIG. 4.

[0039] FIG. 4 is a diagram of a sparse parity check matrix, in accordance with an embodiment of the present invention. LDPC codes are long, linear block codes with sparse parity check matrix H_(n−k)xn. Typically the block length, n, ranges from thousands to tens of thousands of bits. For example, a parity check matrix for an LDPC code of length n=8 and rate ½ is shown in FIG. 4. The same code can be equivalently represented by the bipartite graph, per FIG. 5.

[0040] FIG. 5 is a diagram of a bipartite graph of an LDPC code of the matrix of FIG. 4. Parity check equations imply that for each check node, the sum (over GF (Galois Field)(2)) of all adjacent bit nodes is equal to zero. As seen in the figure, bit nodes occupy the left side of the graph and are associated with one or more check nodes, according to a predetermined relationship. For example, corresponding to check node m₁, the following expression exists n₁+n₄+n₅+n₈=0 with respect to the bit nodes.

[0041] Returning the receiver 303, the LDPC decoder 305 is considered a message passing decoder, whereby the decoder 305 aims to find the values of bit nodes. To accomplish this task, bit nodes and check nodes iteratively communicate with each other. The nature of this communication is described below.

[0042] From check nodes to bit nodes, each check node provides to an adjacent bit node an estimate (“opinion”) regarding the value of that bit node based on the information coming from other adjacent bit nodes. For instance, in the above example if the sum of n₄, n₅ and n₈ “looks like” 0 to m₁, then m₁ would indicate to n₁ that the value of n₁ is believed to be 0 (since n₁+n₄+n₅+n₈=0); otherwise m₁ indicate to n₁ that the value of n₁ is believed to be 1. Additionally, for soft decision decoding, a reliability measure is added.

[0043] From bit nodes to check nodes, each bit node relays to an adjacent check node an estimate about its own value based on the feedback coming from its other adjacent check nodes. In the above example n₁ has only two adjacent check nodes m₁ and m₃. If the feedback coming from m₃ to n₁ indicates that the value of n₁ is probably 0, then n₁ would notify m₁ that an estimate of n₁'s own value is 0. For the case in which the bit node has more than two adjacent check nodes, the bit node performs a majority vote (soft decision) on the feedback coming from its other adjacent check nodes before reporting that decision to the check node it communicates. The above process is repeated until all bit nodes are considered to be correct (i.e., all parity check equations are satisfied) or until a predetermined maximum number of iterations is reached, whereby a decoding failure is declared.

[0044] FIG. 6 is a diagram of a sub-matrix of a sparse parity check matrix, wherein the sub-matrix contains parity check values restricted to the lower triangular region, according to an embodiment of the present invention. As described previously, the encoder 203 (of FIG. 2) can employ a simple encoding technique by restricting the values of the lower triangular area of the parity check matrix. According to an embodiment of the present invention, the restriction imposed on the parity check matrix is of the form:
H_(n−k)xn=[A_(n−k)xkB_{(n−k)x(n−k)}]
,where B is lower triangular.

[0045] Any information block i=(i₀, i₁, . . . , i_k−1) is encoded to a codeword c=(i₀, i₁, . . . , i_k−1, p₀, p₁, . . . , p_n−k−1) using Hc^T=0, and recursively solving for parity bits; for example,
a₀₀i₀+a₀₁i₁+ . . . +a_0,k−1i_k−1+P₀=0=>Solve P₀,
a₁₀i₀+a₁₁i₁+ . . . +a_1,k−1i_k−1+b₁₀p₀+p₁=0=>Solve p₁
and similarly for p₂, p₃, . . . ,p_n−k−1.

[0046] FIG. 7 is a graph showing performance between codes utilizing unrestricted parity check matrix (H matrix) versus restricted H matrix of FIG. 6. The graph shows the performance comparison between two LDPC codes: one with a general parity check matrix and the other with a parity check matrix restricted to be lower triangular to simplify encoding. The modulation scheme, for this simulation, is 8-PSK. The performance loss is within 0.1 dB. Therefore, the performance loss is negligible based on the restriction of the lower triangular H matrices, while the gain in simplicity of the encoding technique is significant. Accordingly, any parity check matrix that is equivalent to a lower triangular or upper triangular under row and/or column permutation can be utilized for the same purpose.

[0047] FIGS. 8A and 8B are, respectively, a diagram of a non-Gray 8-PSK modulation scheme, and a Gray 8-PSK modulation, each of which can be used in the system of FIG. 1. The non-Gray 8-PSK scheme of FIG. 8A can be utilized in the receiver of FIG. 3 to provide a system that requires very low Frame Erasure Rate (FER). This requirement can also be satisfied by using a Gray 8-PSK scheme, as shown in FIG. 8B, in conjunction with an outer code, such as Bose, Chaudhuri, and Hocquenghem (BCH), Hamming, or Reed-Solomon (RS) code.

[0048] Under this scheme, there is no need to iterate between the LDPC decoder 305 (FIG. 3) and the bit metric generator 307, which may employ 8-PSK modulation. In the absence of an outer code, the LDPC decoder 305 using Gray labeling exhibit an earlier error floor, as shown in FIG. 9 below.

[0049] FIG. 9 is a graph showing performance between codes utilizing Gray labeling versus non-Gray labeling of FIGS. 8A and 8B. The error floor stems from the fact that assuming correct feedback from LDPC decoder 305, regeneration of 8-PSK bit metrics is more accurate with non-Gray labeling since the two 8-PSK symbols with known two bits are further apart with non-Gray labeling. This can be equivalently seen as operating at higher Signal-to-Noise Ratio (SNR). Therefore, even though error asymptotes of the same LDPC code using Gray or non-Gray labeling have the same slope (i.e., parallel to each other), the one with non-Gray labeling passes through lower FER at any SNR.

[0050] On the other hand, for systems that do not require very low FER, Gray labeling without any iteration between LDPC decoder 305 and 8-PSK bit metric generator 307 may be more suitable because re-generating 8-PSK bit metrics before every LDPC decoder iteration causes additional complexity. Moreover, when Gray labeling is used, re-generating 8-PSK bit metrics before every LDPC decoder iteration yields only very slight performance improvement. As mentioned previously, Gray labeling without iteration may be used for systems that require very low FER, provided an outer code is implemented.

[0051] The choice between Gray labeling and non-Gray labeling depends also on the characteristics of the LDPC code. Typically, the higher bit or check node degrees, the better it is for Gray labeling, because for higher node degrees, the initial feedback from LDPC decoder 305 to 8-PSK (or similar higher order modulation) bit metric generator 307 deteriorates more with non-Gray labeling.

[0052] When 8-PSK (or similar higher order) modulation is utilized with a binary decoder, it is recognized that the three (or more) bits of a symbol are not received “equally noisy”. For example with Gray 8-PSK labeling, the third bit of a symbol is considered more noisy to the decoder than the other two bits. Therefore, the LDPC code design does not assign a small number of edges to those bit nodes represented by “more noisy” third bits of 8-PSK symbol so that those bits are not penalized twice.

[0053] FIG. 10 is a flow chart of the operation of the LDPC decoder using non-Gray mapping, according to an embodiment of the present invention. Under this approach, the LDPC decoder and bit metric generator iterate one after the other. In this example, 8-PSK modulation is utilized; however, the same principles apply to other higher modulation schemes as well. Under this scenario, it is assumed that the demodulator 301 outputs a distance vector, d, denoting the distances between received noisy symbol points and 8-PSK -symbol points to the bit metric generator 307, whereby the vector components are as follows:

[0054] $\begin{array}{cc}{d}_i=-\frac{E_s}{N_0}\left\{{\left(r_x-s_{i,x}\right)}^2+{\left(r_y-s_{i,y}\right)}^2\right\}& i=0,1,\dots 7.\end{array}$

[0055] The 8-PSK bit metric generator 307 communicates with the LDPC decoder 305 to exchange a priori probability information and a posteriori probability information, which respectively are represented as u, and a. That is, the vectors u and a respectively represent a priori and a posteriori probabilities of log likelihood ratios of coded bits.

[0056] The 8-PSK bit metric generator 307 generates the a priori likelihood ratios for each group of three bits as follows. First, extrinsic information on coded bits is obtained:
e_j=a_j−u_j j=0,1,2.
Next, 8-PSK symbol probabilities, p_i i=0,1, . . . ,7, are determined.
*y_j=−f(0,e_j) j=0,1,2 where f(a,b)=max(a,b)+LUT_f(a,b) with LUT_f(a,b)=ln(1+e^−|a−b|)
*x_j=y_j+e_j j=0,1,2
*p₀=x₀+x₁+x₂ p₄=y₀+x₁+x₂
p₁=x₀+x₁+y₂ p₅=y₀+x₁+y₂
p₂=x₀+y₁+x₂ p₆=y₀+y₁+x₂
p₃=x₀+y₁+y₂ p₇=y₀+y₁+y₂

[0057] Next, the bit metric generator 307 determines a priori log likelihood ratios of the coded bits as input to LDPC decoder 305, as follows:
u₀=f(d₀+p₀,d₁+p₁,d₂+p₂,d₃+p₃)−f(d₄+p₄,d₅+p₅,d₆+p₆,d₇+p₇)−e₀
u₁=f(d₀+p₀,d₁+p₁,d₄+p₄,d₅+p₅)−f(d₂+p₂,d₃+p₃,d₆+p₆,d₇+p₇)−e₁
u₂=f(d₀+p₀,d₂+p₂,d₄+p₄,d₆+p₆)−f(d₁+p₁,d₃+p₃,d₅+p₅,d₇+p₇)−e₂

[0058] It is noted that the function f(.) with more than two variables can be evaluated recursively; e.g. f(a,b,c) =f(f(a,b),c).

[0059] The operation of the LDPC decoder 305 utilizing non-Gray mapping is now described. In step 1001, the LDPC decoder 305 initializes log likelihood ratios of coded bits, v, before the first iteration according to the following (and as shown in FIG. 12A):
=v_n→kiu_n, n=0,1, . . . ,N−1, i=1,2, . . . ,deg(bit node n)
Here, v_n→ki denotes the message that goes from bit node n to its adjacent check node k_i, u_n denotes the demodulator output for the bit n and N is the codeword size.

[0060] In step 1003, a check node, k, is updated, whereby the input v yields the output w. As seen in FIG. 12B, the incoming messages to the check node k from its d_c adjacent bit nodes are denoted by v_n1→k, v_n2→k, . . . , v_ndc→k. The goal is to compute the outgoing messages from the check node k back to d_c adjacent bit nodes. These messages are denoted by w_k→n1, w_k→n2, w_k→ndc, where
w_k→ni=g(v_n1→k, v_n2→k, . . . , v_ni−1→k, v_ni+1→k, . . . , v_ndc→k).
The function g( ) is defined as follows:
g(a,b)=sign(a)×sign(b)×{min(|a|,|b|)}+LUT_g(a,b),
where LUT_g(a,b)=ln(1+e^−|a+b|)−ln(1+e^−|a−b|) .Similar to function f, function g with more than two variables can be evaluated recursively.

[0061] Next, the decoder 305, per step 1205, outputs a posteriori probability information (FIG. 12C), such that:

[0062] $a_n=u_n+\sum _j{w}_{k_j\to n}.$

[0063] Per step 1007, it is determined whether all the parity check equations are satisfied. If these parity check equations are not satisfied, then the decoder 305, as in step 1009, re-derives 8-PSK bit metrics and channel input u_n. Next, the bit node is updated, as in step 1011. As shown in FIG. 14C, the incoming messages to the bit node n from its d_v adjacent check nodes are denoted by w_k1→n, w_k2→n, . . . , w_kdv→n The outgoing messages from the bit node n are computed back to d_v adjacent check nodes; such messages are denoted by v_n→k1, v_n→k2, . . . , v_n→kdv, and computed as follows:

[0064] $v_{n\to k_i}=u_n+\sum _{j\ne i}{w}_{k_j\to n}$
In step 1013, the decoder 305 outputs the hard decision (in the case that all parity check equations are satisfied):

[0065] ${\hat{c}}_n=\{\begin{array}{cc}0,& a_n\ge 0\\ 1,& a_n<0\end{array}$
Stop if Hĉ^T=0

[0066] The above approach is appropriate when non-Gray labeling is utilized. However, when Gray labeling is implemented, the process of FIG. 11 is executed.

[0067] FIG. 11 is a flow chart of the operation of the LDPC decoder of FIG. 3 using Gray mapping, according to an embodiment of the present invention. When Gray labeling is used, bit metrics are advantageously generated only once before the LDPC decoder, as re-generating bit metrics after every LDPC decoder iteration may yield nominal performance improvement. As with steps 1001 and 1003 of FIG. 10, initialization of the log likelihood ratios of coded bits, v, are performed, and the check node is updated, per steps 1101 and 1103. Next, the bit node n is updated, as in step 1105. Thereafter, the decoder outputs the a posteriori probability information (step 1107). In step 1109, a determination is made whether all of the parity check equations are satisfied; if so, the decoder outputs the hard decision (step 1111). Otherwise, steps 1103–1107 are repeated.

[0068] FIG. 13A is a flowchart of process for computing outgoing messages between the check nodes and the bit nodes using a forward-backward approach, according to an embodiment of the present invention. For a check node with d_c adjacent edges, the computation of d_c(d_c−1) and numerous g(.,.) functions are performed. However, the forward-backward approach reduces the complexity of the computation to 3(d_c−2), in which d_c−1 variables are stored.

[0069] Referring to FIG. 12B, the incoming messages to the check node k from d_c adjacent bit nodes are denoted by v_n1→k, v_n2→k, . . . , v_ndc→k. It is desired that the outgoing messages are computed from the check node k back to d_c adjacent bit nodes; these outgoing messages are denoted by w_k→n1, w_k→n2, . . . , w_k→ndc.

[0070] Under the forward-backward approach to computing these outgoing messages, forward variables, f₁, f₂, . . . , f_dc, are defined as follows:
f₁=v_1→k
f₂=g(f₁, v_2→k)
f₃=g(f₂, v_3→k)
: : :
f_dc=g(f_dc−1, v_dc→k)
In step 1301, these forward variables are computed, and stored, per step 1303.

[0071] Similarly, backward variables, b₁,b₂, . . . , b_dc, are defined by the following:
b_dc=v_dc→k
b_dc−1=g(b_dc, v_dc−1→k)
: : :
b₁=g(b₂, v_1→k)
In step 1305, these backward variables are then computed. Thereafter, the outgoing messages are computed, as in step 1307, based on the stored forward variables and the computed backward variables. The outgoing messages are computed as follows:
w_k→1=b₂
w_k→i=g(f_i−1, b_i+1) i=2,3, . . . , d_c−1
w_k→dc=f_dc−1

[0072] Under this approach, only the forward variables, g₂, g₃, g_dc, are required to be stored. As the backward variables b_i are computed, the outgoing messages, w_k→i, are simultaneously computed, thereby negating the need for storage of the backward variables.

[0073] The computation load can be further enhance by a parallel approach, as next discussed.

[0074] FIG. 13B is a flowchart of process for computing outgoing messages between the check nodes and the bit nodes using a parallel approach, according to an embodiment of the present invention. For a check node k with inputs v_n1→k, v_n2→k, . . . , v_ndc→k from d_c adjacent bit nodes, the following parameter is computed, as in step 1311:
γ_k=f(v_n1→k, v_n2→k, . . . , v_ndc→k).

[0075] It is noted that the g(.,.) function can also be expressed as follows:

[0076] $g\left(a,b\right)=\ln \frac{1+e^{a+b}}{e^a+e^b}.$

[0077] Exploiting the recursive nature of the g(.,.) function, the following expression results:

[0078] $\begin{array}{c}{\gamma }_k=\ln \frac{1+e^{g\left(v_{n_1\to k},\dots ,v_{n_{i-1}\to k},v_{n_{i+1}\to k},\dots ,v_{n_{\mathrm{dc}}\to k}\right)+v_{n_i\to k}}}{e^{g\left(v_{n_1\to k},\dots ,v_{n_{i-1}\to k},v_{n_{i+1}\to k},\dots ,v_{n_{\mathrm{dc}}\to k}\right)}+e^{v_{n_i\to k}}}\\ =\ln \frac{1+e^{w_{k\to n_i}+v_{n_i\to k}}}{e^{w_{k\to n_i}}+e^{v_{n_i\to k}}}\end{array}$

[0079] Accordingly, w_k→ni can be solved in the following manner:

[0080] $w_{k\to n_i}=\ln \frac{e^{v_{n_i\to k}+{\gamma }_k}-1}{e^{v_{n_i\to k}-{\gamma }_k}-1}-{\gamma }_k$

[0081] The ln(.) term of the above equation can be obtained using a look-up table LUT_x that represents the function ln |e^x−1| (step 1313). Unlike the other look-up tables LUT_f or LUT_g, the table LUT_x would likely requires as many entries as the number of quantization levels. Once γ_k is obtained, the calculation of w_k→ni for all n_i can occur in parallel using the above equation, per step 1315.

[0082] The computational latency of γ_k is advantageously log₂(d_c).

[0083] FIGS. 14A–14C are graphs showing simulation results of LDPC codes generated in accordance with various embodiments of the present invention. In particular, FIGS. 14A–14C show the performance of LDPC codes with higher order modulation and code rates of ¾ (QPSK, 1.485 bits/symbol), ⅔ (8-PSK, 1.980 bits/symbol), and ⅚ (8-PSK, 2.474 bits/symbol).

[0084] Two general approaches exist to realize the interconnections between check nodes and bit nodes: (1) a fully parallel approach, and (2) a partially parallel approach. In fully parallel architecture, all of the nodes and their interconnections are physically implemented. The advantage of this architecture is speed.

[0085] The fully parallel architecture, however, may involve greater complexity in realizing all of the nodes and their connections. Therefore with fully parallel architecture, a smaller block size may be required to reduce the complexity. In that case, for the same clock frequency, a proportional reduction in throughput and some degradation in FER versus Es/No performance may result.

[0086] The second approach to implementing LDPC codes is to physically realize only a subset of the total number of the nodes and use only these limited number of “physical” nodes to process all of the “functional” nodes of the code. Even though the LDPC decoder operations can be made extremely simple and can be performed in parallel, the further challenge in the design is how the communication is established between “randomly” distributed bit nodes and check nodes. The decoder 305 (of FIG. 3), according to one embodiment of the present invention, addresses this problem by accessing memory in a structured way, as to realize a seemingly random code. This approach is explained with respect to FIGS. 15A and 15B.

[0087] FIGS. 15A and 15B are diagrams of the top edge and bottom edge, respectively, of memory organized to support structured access as to realize randomness in LDPC coding, according to an embodiment of the present invention. Structured access can be achieved without compromising the performance of a truly random code by focusing on the generation of the parity check matrix. In general, a parity check matrix can be specified by the connections of the check nodes with the bit nodes. For example, the bit nodes can be divided into groups of a fixed size, which for illustrative purposes is 392. Additionally, assuming the check nodes connected to the first bit node of degree 3, for instance, are numbered as a, b and c, then the check nodes connected to the second bit node are numbered as a+p, b+p and c+p, the check nodes connected to the third bit node are numbered as a+2p, b+2p and c+2p etc.; where p=(number of check nodes)/392. For the next group of 392 bit nodes, the check nodes connected to the first bit node are different from a, b, c so that with a suitable choice of p, all the check nodes have the same degree. A random search is performed over the free constants such that the resulting LDPC code is cycle-4 and cycle-6 free. Because of the structural characteristics of the parity check matrix of the present invention, the edge information can stored to permit concurrent access to a group of relevant edge values during decoding.

[0088] In other words, the approach of the present invention facilitates memory access during check node and bit node processing. The values of the edges in the bipartite graph can be stored in a storage medium, such as random access memory (RAM). It is noted that for a truly random LDPC code during check node and bit node processing, the values of the edges would need to be accessed one by one in a random fashion. However, such a conventional access scheme would be too slow for a high data rate application. The RAM of FIGS. 15A and 15B are organized in a manner, whereby a large group of relevant edges can be fetched in one clock cycle; accordingly, these values are placed “together” in memory, according to a predetermined scheme or arrangement. It is observed that, in actuality, even with a truly random code, for a group of check nodes (and respectively bit nodes), the relevant edges can be placed next to one another in RAM, but then the relevant edges adjacent to a group of bit nodes (respectively check nodes) will be randomly scattered in RAM. Therefore, the “togetherness,” under the present invention, stems from the design of the parity check matrices themselves. That is, the check matrix design ensures that the relevant edges for a group of bit nodes and check nodes are simultaneously placed together in RAM.

[0089] As seen in FIGS. 15A and 15B, each box contains the value of an edge, which is multiple bits (e.g., 6). Edge RAM, according to one embodiment of the present invention, is divided into two parts: top edge RAM 1501 (FIG. 15A) and bottom edge RAM 1503 (FIG. 15B). Bottom edge RAM contains the edges between bit nodes of degree 2, for example, and check nodes. Top edge RAM contains the edges between bit nodes of degree greater than 2 and check nodes. Therefore, for every check node, 2 adjacent edges are stored in the bottom edge RAM 1503, and the rest of the edges are stored in the top edge RAM 1501. For example, the size of the top edge RAM 1501 and bottom edge RAM 1503 for various code rates are given in Table 1

[0090]

	TABLE 1
	½	⅔	¾	⅚
Top Edge	400 × 392	440 × 392	504 × 392	520 × 392
RAM
Bottom	160 × 392	110 × 392	72 × 392	52 × 392
Edge RAM

[0091] Based on Table 1, an edge RAM of size 576×392 is sufficient to store the edge metrics for all the code rates of ½, ⅔, ¾, and ⅚.

[0092] As noted, under this exemplary scenario, a group of 392 bit nodes and 392 check nodes are selected for processing at a time. For 392 check node processing, q=d_c−2 consecutive rows are accessed from the top edge RAM 1501, and 2 consecutive rows from the bottom edge RAM 1503. The value of d_c depends on the specific code, for example d_c=7 for rate ½, d_c=10 for rate ⅔, d_c=16 for rate ¾ and d_c=22 for rate ⅚ for the above codes. Of course other values of d_c for other codes are possible. In this instance, q+2 is the degree of each check node.

[0093] For bit node processing, if the group of 392 bit nodes has degree 2, their edges are located in 2 consecutive rows of the bottom edge RAM 1503. If the bit nodes have degree d>2, their edges are located in some d rows of the top edge RAM 1501. The address of these d rows can be stored in non-volatile memory, such as Read-Only Memory (ROM). The edges in one of the rows correspond to the first edges of 392 bit nodes, the edges in another row correspond to the second edges of 392 bit nodes, etc. Moreover for each row, the column index of the edge that belongs to the first bit node in the group of 392 can also be stored in ROM. The edges that correspond to the second, third, etc. bit nodes follow the starting column index in a “wrapped around” fashion. For example, if the j^th edge in the row belongs to the first bit node, then the (j+1)st edge belongs to the second bit node, (j+2)nd edge belongs to the third bit node, . . . , and (j−1)st edge belongs to the 392^th bit node.

[0094] In Tables 2–5, the row index and the starting column index of top edge RAM 1501 are specified for every group of 392 bit nodes of degree 3 or larger, for the respective code rates of ⅔, ⅚, ½, and ¾. Each row in the Tables 2–5 represents a group of 392 bit nodes. The first number denotes the row index and the second number denotes the starting column index. For example in Table 2, the first row completely determines the addresses of adjacent edges for the first group of 392 bit nodes of degree 13. Specifically, the entry 0/0 indicates that the first adjacent edges for all of the 392 bit nodes are stored in row number 0. Moreover in that row, the column indexed 0 carries the information for the first adjacent edge of the first bit node, column indexed 1 carries the information for the first adjacent edge of the second bit node etc., and finally column indexed 391 carries the information for the first adjacent edge of the 392th bit node.

[0095] Similarly the entry 433/323 specifies that the second adjacent edges for all of the 392 bit nodes are stored in row number 433. Moreover in that row, the column indexed 323 carries the information for the second adjacent edge of the first bit node, column indexed 324 carries the information for the second adjacent edge of the second bit node etc. The column indexed 322 carries the information for the second adjacent edge of the 392th bit node.

[0096] Similarly, other entries in the first row of Table 2 determine the addresses of the remaining adjacent edges for the first group of 392 bit nodes. Likewise, the entries in the second row of Table 2 determine the addresses of the adjacent edges for the second group of 392 bit nodes, etc.

[0097]

TABLE 2
Row Index/Starting Column Index (Rate 2/3)
0/0	433/323	242/150	91/117	323/112	147/93	35/105	227/232	196/311	292/180	52/244	180/250	20/335
8/0	121/326	178/109	299/157	195/338	99/232	251/107	411/263	364/199	28/218	276/370	108/80	84/130
16/0	281/359	18/112	83/180	115/264	163/149	355/321	11/206	268/100	436/79	252/316	420/280	380/335
24/0	345/122	146/365	107/40	283/363	123/368	379/340	3/156	124/15	220/187	356/127	188/71	156/82
32/0	425/177	234/46	267/219	67/224	171/275	219/306	387/87	372/56	140/31	36/339	116/36	316/288
40/0	417/214	122/188	339/58	235/72	187/26	75/302	19/362	164/285	132/109	148/189	60/65	412/303
48/0	89/312	362/214	43/21	419/219	427/378	395/10	347/167	68/221	260/310	396/54	308/268	388/176
56/0	73/69	434/266	155/277	435/360	363/183	51/165	331/181	12/232	404/193	172/175	324/349	348/98
64/0	177/354	34/172	243/141	139/362	259/151	179/166	307/56	76/367	244/121	100/299	428/12	284/133
72/0	145/264	194/335	131/362	403/326	315/180	275/137	203/86	204/303	4/5	228/360	300/76	92/17
80/0	377/382	394/243	27/109	59/237	371/175	211/358	291/353	340/161	212/94	332/333	44/117	236/200
88/0	65/365	378/142
96/0	57/285	226/108
104/0	97/161	250/133
112/0	129/184	114/44
120/0	337/130	50/178
128/0	401/389	170/258
136/0	25/330	82/372
144/0	321/309	162/170
152/0	185/38	386/128
160/0	49/376	90/331
168/0	265/293	314/166
176/0	297/86	282/193
184/0	217/117	42/210
192/0	201/124	306/86
200/0	313/377	138/97
208/0	193/247	202/163
216/0	209/377	186/212
224/0	233/238	26/22
232/0	329/152	410/271
240/0	9/245	106/170
248/0	409/190	58/289
256/0	113/375	154/44
264/0	33/232	274/268
272/0	153/339	218/145
280/0	289/319	98/4
288/0	41/209	130/23
296/0	385/42	210/267
304/0	17/7	258/227
312/0	169/166	290/330
320/0	241/107	66/111
328/0	137/39	418/182
336/0	249/137	354/218
344/0	161/73	2/79
352/0	105/280	266/282
360/0	257/69	298/51
368/0	81/185	338/118
376/0	369/228	370/202
384/0	225/71	74/136
392/0	1/314	346/289
400/0	353/286	322/166
408/0	305/81	330/301
416/0	273/170	402/282
424/0	393/227	10/312
432/0	361/379	426/364
5/0	350/140	263/166
13/0	102/110	87/335
21/0	174/333	215/219
29/0	422/227	31/273
37/0	406/168	175/11
45/0	254/42	279/201
53/0	230/347	47/291
61/0	214/139	55/92
69/0	358/131	199/344
77/0	86/374	183/298
85/0	118/118	407/25
93/0	318/221	39/66
101/0	54/256	79/202
109/0	374/195	119/162
117/0	238/89	207/243
125/0	366/78	95/96
133/0	46/216	351/9
141/0	326/99	127/87
149/0	134/75	319/102
157/0	158/154	15/65
165/0	286/158	143/362
173/0	190/146	191/205
181/0	62/4	343/262
189/0	94/239	271/38
197/0	198/207	231/297
205/0	22/32	167/205
213/0	246/385	303/246
221/0	390/368	439/220
229/0	334/207	247/262
237/0	398/378	63/211
245/0	150/340	359/100
253/0	294/75	415/189
261/0	222/321	391/78
269/0	166/343	159/105
277/0	126/93	239/166
285/0	110/113	151/373
293/0	302/144	71/18
301/0	262/368	111/193
309/0	414/332	375/389
317/0	142/256	103/242
325/0	278/22	7/154
333/0	342/192	423/330
341/0	14/181	431/16
349/0	38/367	383/16
357/0	270/91	223/195
365/0	182/211	287/313
373/0	310/170	135/230
381/0	78/15	295/220
389/0	430/353	335/91
397/0	30/141	367/216
405/0	382/36	311/98
413/0	206/377	255/372
421/0	438/225	399/148
429/0	70/182	327/105
437/0	6/277	23/94

[0098]

TABLE 3
Row Index/Starting Column Index (Rate 5/6)
0/0	221/158	442/14	503/323	283/150	104/117	384/112	165/93	45/105	266/232	226/311	347/180	67/244
20/0	101/369	162/326	323/359	23/112	124/180	144/264	205/149	405/321	6/206	306/100	507/79	287/316
40/0	201/285	302/12	63/134	243/68	264/238	344/375	105/259	345/213	246/75	66/148	327/100	167/220
60/0	381/141	422/112	443/125	223/47	204/375	504/214	145/188	385/58	206/72	166/26	87/302	7/362
80/0	461/383	82/80	143/61	463/106	284/196	4/94	85/104	285/235	386/3	426/218	27/38	107/161
100/0	41/310	482/66	343/376	403/166	324/265	404/236	245/230	445/63	186/343	486/88	427/202	267/362
120/0	61/31	502/317	123/25	163/139	424/269	164/309	25/56	505/260	406/279	346/148	367/315	47/382
140/0	421/362	462/206	263/297	83/384	244/287	184/132	225/140	125/14	506/216	106/311	447/87	487/264
160/0	441/191	382/360	423/282	203/2	84/58	64/347	425/249	5/267	466/232	46/275	127/385	187/26
180/0	501/296	222/324	3/73	43/6	364/319	444/204	185/82	65/259	26/90	286/155	307/181	147/366
200/0	301/325	102/119	383/285	103/84	304/121	484/352	365/102	485/107	86/9	366/76	387/229	467/52
220/0	341/331	322/242	483/275	303/293	464/166	44/283	305/232	465/86	126/193	146/184	207/38	407/117
240/0	21/314	362/289	363/211	183/120	24/286	224/166	325/186	265/144	446/81	326/301	227/4	247/199
260/0	161/91	142/78
280/0	241/209	2/119
300/0	141/87	342/147
320/0	281/55	42/46
340/0	261/213	182/145
360/0	181/264	62/88
380/0	1/96	262/184
400/0	361/30	282/126
420/0	81/202	202/206
440/0	481/156	242/263
460/0	401/170	22/126
480/0	321/42	402/21
500/0	121/272	122/337
8/0	289/369	190/223
28/0	369/313	130/127
48/0	189/92	290/241
68/0	509/124	210/56
88/0	489/23	430/101
108/0	309/208	510/162
128/0	349/147	330/242
148/0	29/263	490/54
168/0	69/312	250/377
188/0	249/315	270/116
208/0	49/176	170/58
228/0	109/337	10/55
248/0	469/65	110/187
268/0	209/105	470/362
288/0	229/164	150/80
308/0	9/293	410/374
328/0	329/122	310/152
348/0	149/124	390/382
368/0	389/160	230/92
388/0	169/357	370/368
408/0	449/296	90/377
428/0	269/32	70/212
448/0	409/59	450/257
468/0	89/291	30/234
488/0	429/130	350/95
508/0	129/276	50/38
11/0	292/349	133/372
31/0	492/271	253/248
51/0	192/149	273/378
71/0	352/265	153/37
91/0	332/244	293/199
111/0	152/354	393/243
131/0	312/144	213/184
151/0	92/219	173/11
171/0	392/182	473/325
191/0	232/219	193/30
211/0	372/157	13/63
231/0	12/108	333/359
251/0	112/33	513/88
271/0	72/207	413/9
291/0	272/100	93/357
311/0	432/166	233/272
331/0	412/265	33/210
351/0	132/155	493/50
371/0	512/292	453/214
391/0	172/387	53/114
411/0	32/233	433/177
431/0	252/113	373/52
451/0	212/347	353/90
471/0	52/89	73/198
491/0	452/285	313/233
511/0	472/103	113/84
14/0	55/43	36/361
34/0	355/70	116/287
54/0	115/137	196/57
74/0	95/161	416/206
94/0	295/273	336/209
114/0	255/184	296/287
134/0	435/11	376/38
154/0	155/356	16/379
174/0	135/251	76/10
194/0	235/314	256/293
214/0	75/296	216/326
234/0	275/314	356/116
254/0	455/133	156/165
274/0	375/292	436/283
294/0	515/227	456/337
314/0	315/111	176/155
334/0	175/98	276/334
354/0	335/7	476/87
374/0	415/161	136/15
394/0	395/338	496/98
414/0	495/82	396/269
434/0	195/312	516/187
454/0	475/100	56/356
474/0	15/163	316/195
494/0	35/197	96/145
514/0	215/301	236/381
17/0	18/321	459/4
37/0	398/236	139/8
57/0	338/117	439/84
77/0	438/97	499/93
97/0	298/292	19/215
117/0	218/224	419/275
137/0	98/229	299/27
157/0	378/133	339/232
177/0	118/191	359/271
197/0	478/272	159/386
217/0	498/262	119/219
237/0	418/282	519/297
257/0	238/33	379/339
277/0	258/230	179/350
297/0	158/27	59/188
317/0	518/249	399/229
337/0	278/333	279/330
357/0	138/276	239/49
377/0	178/34	219/304
397/0	458/344	199/181
417/0	58/312	479/158
437/0	358/377	259/364
457/0	78/157	319/380
477/0	318/75	99/57
497/0	38/296	79/26
517/0	198/115	39/342

[0099]

TABLE 4
Row Index/Starting Column Index (Rate 1/2)
240/0	306/249	387/194	98/132	268/80	219/33	64/252	108/54
245/0	146/169	37/233	183/243	233/207	9/336	54/91	363/391
250/0	196/123	242/31	63/103	118/277	344/177	339/46	173/219
255/0	216/36	287/288	318/43	83/327	34/28	354/114	53/84
260/0	316/69	377/8	323/110	308/250	314/209	214/101	298/134
265/0	256/186	257/166	23/196	68/68	234/41	144/249	333/11
270/0	201/192	32/38	213/255	203/124	84/285	264/12	263/134
275/0	36/100	247/220	388/286	273/339	89/334	154/192	223/148
280/0	396/141	132/112	283/125	163/47	79/375	14/214	138/188
285/0	31/189	82/65	18/303	313/92	299/317	129/18	373/356
290/0	381/332	332/312	258/214	43/21	364/219	274/378	198/10
295/0	121/66	217/124	338/346	48/380	189/155	199/79	278/224
300/0	71/260	22/248	193/240	288/248	284/237	224/268	38/263
305/0	46/232	282/193	293/175	378/349	169/98	184/165	168/31
310/0	156/116	87/62	208/390	113/287	69/354	269/172	343/141
315/0	311/12	192/133	143/43	58/75	124/176	324/24	383/346
320/0	16/118	372/259	368/265	133/59	309/321	289/272	104/80
325/0	336/114	7/90	123/190	228/181	114/324	319/240	244/246
330/0	226/71	112/218	358/348	398/83	179/121	119/366	394/197
335/0	21/383	142/80	158/61	218/106	329/196	4/94	49/104
340/0	221/97	2/252	178/174	13/190	164/166	99/130	204/9
345/0	126/76	67/120	78/183	243/53	134/140	149/197	359/239
350/0	96/221	392/290	28/163	353/297	279/147	294/343	374/314
355/0	176/230	367/63	73/343	393/88	174/202	259/362	249/256
360/0	241/60	202/21	348/66	328/351	139/144	94/258	384/41
365/0	26/374	262/54	303/391	153/132	254/145	209/307	74/126
370/0	91/25	382/139	238/269	128/309	239/56	24/260	369/279
375/0	151/213	317/133	253/161	188/92	399/371	194/116	39/302
380/0	296/140	307/14	93/216	148/311	389/87	334/264	109/335
385/0	286/76	222/17	33/116	8/191	304/360	19/282	379/2
390/0	106/217	212/188	103/68	248/264	29/48	44/174	349/274
395/0	281/269	162/333	3/243	88/320	159/75	59/300	229/136
0/0	346/176	157/302
5/0	171/47	42/1
10/0	86/124	267/2
15/0	321/291	197/8
20/0	236/149	147/50
25/0	1/168	347/191
30/0	386/257	252/12
35/0	301/64	397/176
40/0	341/340	272/97
45/0	101/201	122/134
50/0	136/201	57/343
55/0	131/169	292/299
60/0	166/389	352/216
65/0	76/132	297/33
70/0	211/261	167/45
75/0	161/323	12/150
80/0	116/93	107/105
85/0	246/180	322/244
90/0	261/190	342/297
95/0	366/385	177/103
100/0	391/240	77/328
105/0	266/327	182/182
110/0	181/73	47/322
115/0	191/126	72/135
120/0	251/115	227/161
125/0	276/85	172/213
130/0	371/17	327/236
135/0	66/326	62/109
140/0	141/232	357/107
145/0	271/218	207/370
150/0	56/252	127/20
155/0	61/143	97/305
160/0	11/383	237/214
165/0	376/359	337/112
170/0	186/149	277/321
175/0	206/79	92/316
180/0	291/315	362/135
185/0	51/93	232/326
190/0	6/197	102/103
195/0	331/142	187/122
200/0	361/363	27/368
205/0	326/15	152/187
210/0	111/82	52/214
215/0	41/385	312/150
220/0	356/387	137/254
225/0	81/175	302/84
230/0	351/11	17/303
235/0	231/55	117/265

[0100]

TABLE 5
Row Index/Starting Column Index (Rate 3/4)
0/0	113/334	100/308	423/175	493/163	32/370	116/20	467/48	243/275	370/284	356/114	77/201	7/214
14/0	29/350	44/366	185/335	3/40	494/155	144/324	383/185	229/96	230/376	188/182	427/304	385/269
28/0	435/215	366/165	101/329	17/221	46/276	74/130	341/4	313/169	314/11	272/267	21/376	273/122
42/0	155/306	240/253	353/325	451/355	312/33	88/27	47/23	327/90	286/87	34/201	483/221	175/39
56/0	197/263	492/185	283/223	367/316	60/241	228/91	145/175	439/3	454/168	202/98	133/214	203/82
70/0	463/384	352/298	269/9	129/294	256/303	214/387	5/316	285/257	90/282	48/376	399/317	329/102
84/0	1/159	170/317	409/245	255/173	270/11	438/179	271/224	89/131	300/144	328/199	343/321	231/338
98/0	211/266	450/256	199/279	171/358	242/192	466/378	187/100	19/70	62/98	384/313	35/382	245/164
112/0	141/386	128/357	87/172	465/64	424/35	354/238	117/300	257/174	146/154	496/182	161/232	91/355
126/0	15/196	296/183	395/218	73/356	452/367	158/342	173/70	131/251	258/268	76/176	455/172	119/109
140/0	57/265	58/45	437/175	59/369	284/357	102/53	103/286	33/318	412/49	160/25	105/120	371/188
154/0	323/272	198/11	31/140	227/330	410/150	298/113	61/249	495/207	244/190	426/233	63/30	189/283
168/0	477/41	408/85	311/63	45/301	326/13	200/292	159/218	481/99	20/171	174/192	217/102	315/178
182/0	43/340	212/289	381/152	115/273	172/111	368/2	75/34	369/291	132/92	482/375	413/195	301/219
196/0	225/338	436/232	479/161	339/50	340/372	396/293	355/218	397/80	468/212	342/375	497/351	259/314
210/0	253/84	30/254	297/89	241/165	382/65	18/60	299/186	425/104	440/255	398/62	441/191	469/14
224/0	449/109	478/333	325/82	143/94	186/39	130/44	453/22	411/329	6/168	118/357	287/119	357/258
238/0	169/152	310/308	213/159	157/365	480/361	4/64	201/245	215/92	104/185	216/189	147/125	49/310
252/0	267/180	380/44
266/0	71/132	184/228
280/0	281/48	268/91
294/0	393/59	254/241
308/0	379/129	86/21
322/0	127/319	114/57
336/0	85/227	282/298
350/0	491/101	324/74
364/0	309/378	226/317
378/0	239/220	2/201
392/0	407/135	156/221
406/0	365/360	394/114
420/0	183/335	422/129
434/0	421/105	464/120
448/0	295/245	142/160
462/0	351/37	338/29
476/0	337/16	72/305
490/0	99/220	16/347
8/0	23/384	346/305
22/0	415/118	444/373
36/0	65/28	24/211
50/0	261/130	80/113
64/0	275/316	220/366
78/0	205/109	206/255
92/0	51/110	38/74
106/0	373/262	304/363
120/0	345/250	472/134
134/0	37/173	388/301
148/0	359/272	430/234
162/0	429/71	150/189
176/0	93/332	94/299
190/0	163/385	416/307
204/0	289/144	164/50
218/0	457/140	290/145
232/0	79/42	360/26
246/0	387/59	262/196
260/0	233/93	66/21
274/0	303/116	136/28
288/0	121/176	276/279
302/0	485/235	332/69
316/0	443/336	178/353
330/0	499/298	458/45
344/0	191/240	234/244
358/0	9/13	248/94
372/0	219/36	402/112
386/0	331/192	52/58
400/0	471/294	500/144
414/0	317/186	318/150
428/0	107/69	108/346
442/0	149/139	486/346
456/0	135/170	192/65
470/0	401/2	10/281
484/0	177/326	374/2
498/0	247/143	122/242
11/0	474/49	433/281
25/0	292/134	335/294
39/0	404/29	265/296
53/0	320/345	111/194
67/0	208/221	13/84
81/0	264/133	419/95
95/0	54/157	83/51
109/0	166/363	195/303
123/0	194/389	377/15
137/0	460/36	447/169
151/0	306/23	279/311
165/0	40/133	153/233
179/0	236/53	27/257
193/0	446/121	293/259
207/0	250/350	167/310
221/0	68/104	209/119
235/0	334/224	251/323
249/0	432/83	503/117
263/0	180/192	125/201
277/0	362/183	391/267
291/0	110/347	405/288
305/0	222/22	223/10
319/0	502/80	489/249
333/0	12/100	139/370
347/0	390/229	321/44
361/0	376/295	97/70
375/0	124/166	69/108
389/0	418/73	349/223
403/0	96/321	237/242
417/0	26/23	181/237
431/0	82/7	55/264
445/0	348/347	461/381
459/0	138/244	41/239
473/0	488/356	475/320
487/0	278/80	307/248
501/0	152/153	363/334

[0101] With the organization shown in FIGS. 15A and 15B, speed of memory access is greatly enhanced during LDPC coding.

[0102] FIG. 16 illustrates a computer system upon which an embodiment according to the present invention can be implemented. The computer system 1600 includes a bus 1601 or other communication mechanism for communicating information, and a processor 1603 coupled to the bus 1601 for processing information. The computer system 1600 also includes main memory 1605, such as a random access memory (RAM) or other dynamic storage device, coupled to the bus 1601 for storing information and instructions to be executed by the processor 1603. Main memory 1605 can also be used for storing temporary variables or other intermediate information during execution of instructions to be executed by the processor 1603. The computer system 1600 further includes a read only memory (ROM) 1607 or other static storage device coupled to the bus 1601 for storing static information and instructions for the processor 1603. A storage device 1609, such as a magnetic disk or optical disk, is additionally coupled to the bus 1601 for storing information and instructions.

[0103] The computer system 1600 may be coupled via the bus 1601 to a display 1611, such as a cathode ray tube (CRT), liquid crystal display, active matrix display, or plasma display, for displaying information to a computer user. An input device 1613, such as a keyboard including alphanumeric and other keys, is coupled to the bus 1601 for communicating information and command selections to the processor 1603. Another type of user input device is cursor control 1615, such as a mouse, a trackball, or cursor direction keys for communicating direction information and command selections to the processor 1603 and for controlling cursor movement on the display 1611.

[0104] According to one embodiment of the invention, generation of LDPC codes is provided by the computer system 1600 in response to the processor 1603 executing an arrangement of instructions contained in main memory 1605. Such instructions can be read into main memory 1605 from another computer-readable medium, such as the storage device 1609. Execution of the arrangement of instructions contained in main memory 1605 causes the processor 1603 to perform the process steps described herein. One or more processors in a multi-processing arrangement may also be employed to execute the instructions contained in main memory 1605. In alternative embodiments, hard-wired circuitry may be used in place of or in combination with software instructions to implement the embodiment of the present invention. Thus, embodiments of the present invention are not limited to any specific combination of hardware circuitry and software.

[0105] The computer system 1600 also includes a communication interface 1617 coupled to bus 1601. The communication interface 1617 provides a two-way data communication coupling to a network link 1619 connected to a local network 1621. For example, the communication interface 1617 may be a digital subscriber line (DSL) card or modem, an integrated services digital network (ISDN) card, a cable modem, or a telephone modem to provide a data communication connection to a corresponding type of telephone line. As another example, communication interface 1617 may be a local area network (LAN) card (e.g. for Ethernet™ or an Asynchronous Transfer Model (ATM) network) to provide a data communication connection to a compatible LAN. Wireless links can also be implemented. In any such implementation, communication interface 1617 sends and receives electrical, electromagnetic, or optical signals that carry digital data streams representing various types of information. Further, the communication interface 1617 can include peripheral interface devices, such as a Universal Serial Bus (USB) interface, a PCMCIA (Personal Computer Memory Card International Association) interface, etc.

[0106] The network link 1619 typically provides data communication through one or more networks to other data devices. For example, the network link 1619 may provide a connection through local network 1621 to a host computer 1623, which has connectivity to a network 1625 (e.g. a wide area network (WAN) or the global packet data communication network now commonly referred to as the “Internet”) or to data equipment operated by service provider. The local network 1621 and network 1625 both use electrical, electromagnetic, or optical signals to convey information and instructions. The signals through the various networks and the signals on network link 1619 and through communication interface 1617, which communicate digital data with computer system 1600, are exemplary forms of carrier waves bearing the information and instructions.

[0107] The computer system 1600 can send messages and receive data, including program code, through the network(s), network link 1619, and communication interface 1617. In the Internet example, a server (not shown) might transmit requested code belonging to an application program for implementing an embodiment of the present invention through the network 1625, local network 1621 and communication interface 1617. The processor 1603 may execute the transmitted code while being received and/or store the code in storage device 169, or other non-volatile storage for later execution. In this manner, computer system 1600 may obtain application code in the form of a carrier wave.

[0108] The term “computer-readable medium” as used herein refers to any medium that participates in providing instructions to the processor 1603 for execution. Such a medium may take many forms, including but not limited to non-volatile media, volatile media, and transmission media. Non-volatile media include, for example, optical or magnetic disks, such as storage device 1609. Volatile media include dynamic memory, such as main memory 1605. Transmission media include coaxial cables, copper wire and fiber optics, including the wires that comprise bus 1601. Transmission media can also take the form of acoustic, optical, or electromagnetic waves, such as those generated during radio frequency (RF) and infrared (IR) data communications. Common forms of computer-readable media include, for example, a floppy disk, a flexible disk, hard disk, magnetic tape, any other magnetic medium, a CD-ROM, CDRW, DVD, any other optical medium, punch cards, paper tape, optical mark sheets, any other physical medium with patterns of holes or other optically recognizable indicia, a RAM, a PROM, and EPROM, a FLASH-EPROM, any other memory chip or cartridge, a carrier wave, or any other medium from which a computer can read.

[0109] Various forms of computer-readable media may be involved in providing instructions to a processor for execution. For example, the instructions for carrying out at least part of the present invention may initially be borne on a magnetic disk of a remote computer. In such a scenario, the remote computer loads the instructions into main memory and sends the instructions over a telephone line using a modem. A modem of a local computer system receives the data on the telephone line and uses an infrared transmitter to convert the data to an infrared signal and transmit the infrared signal to a portable computing device, such as a personal digital assistance (PDA) and a laptop. An infrared detector on the portable computing device receives the information and instructions borne by the infrared signal and places the data on a bus. The bus conveys the data to main memory, from which a processor retrieves and executes the instructions. The instructions received by main memory may optionally be stored on storage device either before or after execution by processor.

[0110] Accordingly, the various embodiments of the present invention provide an approach for generating structured Low Density Parity Check (LDPC) codes, as to simplify the encoder and decoder. Structure of the LDPC codes is provided by restricting the parity check matrix to be lower triangular. Also, the approach can advantageously exploit the unequal error protecting capability of LDPC codes on transmitted bits to provide extra error protection to more vulnerable bits of high order modulation constellations (such as 8-PSK (Phase Shift Keying)). The decoding process involves iteratively regenerating signal constellation bit metrics into an LDPC decoder after each decoder iteration or several decoder iterations. The above approach advantageously yields reduced complexity without sacrificing performance.

[0111] While the present invention has been described in connection with a number of embodiments and implementations, the present invention is not so limited but covers various obvious modifications and equivalent arrangements, which fall within the purview of the appended claims.