Method and apparatus for encoding long and short block length low density parity check (LDPC) codes

RELATED APPLICATIONS

0001This application is related to, and claims the benefit of the earlier filing date under 35 U.S.C. § 119(e) of U.S. Provisional Application Serial No. 60/514,683 filed October 27, 2003, titled "Rate 1/3 and 1/4 LDPC Code," and U.S. Provisional Application Serial No. 60/518,199 filed November 7, 2003, titled "Rate 1/3, 1/4 and 2/5 LDPC Code".

FIELD OF THE INVENTION

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

BACKGROUND OF THE INVENTION

0003Communication systems employ coding to ensure reliable communication across noisy communication channels. For example, in a wireless (or radio) system, such as a satellite network, noise sources abound, from geographic and environmental factors. 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. This objective is particularly germane to bandwidth constrained satellite systems. One such class of codes that approach the Shannon limit is Low Density Parity Check (LDPC) codes.

0004Traditionally, 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.

0005From 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. Length LDPC codes, thus, require greater storage space. 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.

0006Therefore, there is a need for an 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.

SUMMARY OF THE INVENTION

0007These and other needs are addressed by the present invention, wherein an approach for encoding Low Density Parity Check (LDPC) codes is provided. An encoder generates a LDPC code having an outer Bose Chaudhuri Hocquenghem (BCH) code according to one of Tables 1-6 for transmission as the LDPC coded signal. Each of the Tables 1-6 specifies the address of parity bit accumulators. The approach advantageously provides expedient encoding as well as decoding of LDPC codes, while minimizing storage and processing resources.

0008According to another aspect of an embodiment of the present invention, the LDPC codes are represented by signals that are modulated according to a signal constellation that includes one of 8-PSK (Phase Shift Keying), 16-QAM (Quadrature Amplitude Modulation), QPSK (Quadrature Phase Shift Keying), 16-APSK (Amplitude Phase Shift Keying) and 32-APSK.

0009According to yet another aspect of an embodiment of the present invention, the modulated LDPC coded signal is transmitted over a satellite link in support of a broadband satellite application.

0010Still 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 drawings and description are to be regarded as illustrative in nature, and not as restrictive.

BRIEF DESCRIPTION OF THE DRAWINGS

0011The present invention is illustrated by way of example, and not by way of limitation, in the figures of the accompanying drawings and in which like reference numerals refer to similar elements and in which:

0012FIG. 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;

0013FIGs. 2A and 2B are diagrams of exemplary LDPC encoders deployed in the transmitter of FIG. 1;

0014FIGs. 2C and 2D are flowcharts of the encoding process of the LDPC encoder of FIG. 2B for generating, respectively, long frame length codes and short frame length LDPC codes, according to an embodiment of the present invention;

0015FIG. 3 is a diagram of an exemplary receiver in the system of FIG. 1;

0016FIG. 4 is a diagram of a sparse parity check matrix, in accordance with an embodiment of the present invention;

0017FIG. 5 is a diagram of a bipartite graph of an LDPC code of the matrix of FIG. 4;

0018FIG. 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;

0019FIG. 7 is a graph of performance of the LDPC codes at rates 1/3 and 1/4 versus repetition codes;

0020FIG. 8 is a graph of performance of the LDPC codes at the various code rates and modulation schemes supported by the transmitter of FIG. 2A;

0021FIG. 9 is a diagram of a computing system that can perform the LDPC encoding process, in accordance with embodiments of the present invention.

DESCRIPTION OF THE PREFERRED EMBODIMENT

0022A system, method, and software for efficiently encoding short frame length 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.

0023FIG. 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 in the channel 103, LDPC codes are utilized.

0024By way of example, the channel 103 is a satellite link serving satellite terminals (e.g., Very Small Aperture Terminals (VSATs)) in support of broadband satellite applications. Such applications include satellite broadcasting and interactive services (and compliant with the Digital Video Broadcast (DVB) - S2 standard). The Digital Video Broadcasting via Satellite (DVB-S) standard has been widely adopted worldwide to provide, for instance, digital satellite television programming.

0025The LDPC codes that are generated by the transmitter 101 enable 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).

0026Such 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.

0027According 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.

0028FIGs. 2A and 2B are diagrams of exemplary LDPC encoders deployed in the transmitter of FIG. 1. As seen in FIG. 2A, 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 <i>k</i> signals from a discrete alphabet, <i>X</i>. 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.

0029The encoder 203 generates signals from alphabet <i>Y</i> 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.

0030The 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 (shown in FIG. 3), as discussed below.

0031FIG. 2B shows an LDPC encoder utilized with a Bose Chaudhuri Hocquenghem (BCH) encoder and a cyclic redundancy check (CRC) encoder, according to one embodiment of the present invention. Under this scenario, the codes generated by the LDPC encoder 203, along with the CRC encoder 209 and the BCH encoder 211, have a concatenated outer BCH code and an inner low density parity check (LDPC) code. Furthermore, error detection is achieved using cyclic redundancy check (CRC) codes. The CRC encoder 209, in an exemplary embodiment, encodes using an 8-bit CRC code with generator polynomial (x⁵+x⁴+x³+x²+1)(x²+x+1)(x+1). The CRC code is output to the BCH encoder 211.

0032FIG. 2C is a flowchart of the encoding process of the LDPC encoder of FIG. 2B for generating long frame length LDPC codes, according to an embodiment of the present invention. In step 221, information bits are received and processed to the chain of encoders 209, 211, and 203. Consequently, the LDPC encoder 203 generates LDPC codes with outer BCH codes based on the received information bits, as in step 223. The codes also contain the CRC code. Next, the LDPC codes are represented by signals that are modulated, per step 225, for transmission over the channel 103, which in an exemplary embodiment, is a satellite link to one or more satellite terminals (step 227).

0033The LDPC encoder 203 systematically encodes an information block of size <i>k</i>_<i>ldpc</i><i>, i</i> = (<i>i</i>₀ ,<i>i</i>₁ ,...,<i>i</i>_{<i>k</i><sub2><i>ldpc</i></sub2>}_-1) onto a codeword of size <i>n</i>_<i>ldpc</i><i>, c</i> = <i>i</i>₀ , <i>i</i>₁...,<i>i</i>_{<i>k</i><sub2><i>ldpc</i></sub2>}_-1,<i>p</i>₀ , <i>p</i>₁ ,... <i>p</i><maths id="math0001" num=""><math display="inline"><mrow><msub><mrow><mtext>​</mtext></mrow><mrow><mtext mathvariant="italic">n</mtext><msub><mrow><mtext>​</mtext></mrow><mrow><msub><mrow><mtext>​</mtext></mrow><mrow><mtext mathvariant="italic">ldpc</mtext></mrow></msub></mrow></msub><mtext> -</mtext><mtext mathvariant="italic">k</mtext><msub><mrow><mtext>​</mtext></mrow><mrow><msub><mrow><mtext>​</mtext></mrow><mrow><mtext mathvariant="italic">lppc</mtext></mrow></msub></mrow></msub><mtext>-1</mtext></mrow></msub></mrow></math><img file="EP1528686A1_D0001.tif" /></maths>). The transmission of the codeword starts in the given order from <i>i</i>₀ and ends with <i>p</i>_{<i>n</i><sub2><i>ldpc</i></sub2><i>-k</i><sub2><i>ldc</i></sub2>}_-1 LDPC code parameters (<i>n</i>_<i>ldpc</i>,<i>k</i>_<i>ldpc</i>).

0034The task of the LDPC encoder 203 is to determine <i>n</i>_<i>ldpc</i> -<i>k</i>_<i>ldpc</i> parity bits (<i>p</i>₀, <i>p</i>₁,....,<i>p</i>_{<i>n</i><sub2><i>ldpc</i></sub2><i>-k</i><sub2><i>ldpc</i></sub2><i>-1</i>}) for every block of <i>k</i>_<i>ldpc</i> information bits, (<i>i</i>₀,<i>i</i>₁,...,<i>i</i>_{<i>k</i><sub2><i>ldpc</i></sub2>}_-1). The procedure is as follows. First, the parity bits are initialized; <i>p</i>₀ = <i>p</i>₁ = <i>p</i>₂ =...= <i>p</i>_{<i>n</i><sub2><i>ldpc</i></sub2><i>-k</i><sub2><i>ldp</i></sub2>}_c-1 = 0. By way of example, <i>k</i>_<i>ldpc</i> bits are systematically encoded to generate <i>n</i>_<i>ldpc</i> bits. The first information bit, <i>i</i>₀, are accumulated at parity bit addresses specified in the first row of Tables 1-3. For the 1/3 rate code of Table 1, the relevant parameters are as follows: <i>q</i> = 120, <i>n</i>_<i>ldpc</i> = 64800, <i>k</i>_<i>ldpc</i> = <i>n</i>_<i>BCH</i> = 21600, <i>k</i>_<i>BCH</i> = 21408 (12 bit error correcting BCH). For the 1/ 4 rate, the LDPC code has the following relevant parameters: <i>q</i> = 135, <i>n</i>_<i>ldpc</i> = 64800, <i>k</i>_<i>ldpc</i><i> = n</i>_<i>BCH</i> = 16200, <i>k</i>_<i>BCH</i> = 16008 (12 bit error correcting BCH). For long 2/5 LDPC code, the following parameters exist: <i>q</i> = 108, <i>n</i>_<i>ldpc</i><i> =</i> 64800, <i>k</i>_<i>ldpc</i><i> = n</i>_<i>BCH</i> = 25920, <i>k</i>_<i>BCH</i> = 25728 (12 bit error correcting BCH).

0035For example, for rate 1/3 (Table 1), the following results for <i>i</i>₀:<maths id="math0002" num=""><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>34903</mtext></mrow></msub><mtext> = </mtext><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>34903</mtext></mrow></msub><mtext> ⊕ </mtext><msub><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext>0</mtext></mrow></msub></mrow></math><img file="EP1528686A1_D0002.tif" /></maths><maths id="math0003" num=""><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>20927</mtext></mrow></msub><mtext> = </mtext><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>20927</mtext></mrow></msub><mtext> ⊕ </mtext><msub><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext>0</mtext></mrow></msub></mrow></math><img file="EP1528686A1_D0003.tif" /></maths><maths id="math0004" num=""><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>32093</mtext></mrow></msub><mtext> = </mtext><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>32093</mtext></mrow></msub><mtext> ⊕ </mtext><msub><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext>0</mtext></mrow></msub></mrow></math><img file="EP1528686A1_D0004.tif" /></maths><maths id="math0005" num=""><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>1052</mtext></mrow></msub><mtext> = </mtext><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>1052</mtext></mrow></msub><mtext> ⊕ </mtext><msub><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext>0</mtext></mrow></msub></mrow></math><img file="EP1528686A1_D0005.tif" /></maths><maths id="math0006" num=""><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>25611</mtext></mrow></msub><mtext> = </mtext><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>25611</mtext></mrow></msub><mtext> ⊕ </mtext><msub><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext>0</mtext></mrow></msub></mrow></math><img file="EP1528686A1_D0006.tif" /></maths><maths id="math0007" num=""><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>16093</mtext></mrow></msub><mtext> = </mtext><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>16093</mtext></mrow></msub><mtext> ⊕ </mtext><msub><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext>0</mtext></mrow></msub></mrow></math><img file="EP1528686A1_D0007.tif" /></maths><maths id="math0008" num=""><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>16454</mtext></mrow></msub><mtext> = </mtext><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>16454</mtext></mrow></msub><mtext> ⊕ </mtext><msub><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext>0</mtext></mrow></msub></mrow></math><img file="EP1528686A1_D0008.tif" /></maths><maths id="math0009" num=""><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>5520</mtext></mrow></msub><mtext> = </mtext><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>5520</mtext></mrow></msub><mtext> ⊕ </mtext><msub><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext>0</mtext></mrow></msub></mrow></math><img file="EP1528686A1_D0009.tif" /></maths><maths id="math0010" num=""><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>506</mtext></mrow></msub><mtext> = </mtext><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>506</mtext></mrow></msub><mtext> ⊕ </mtext><msub><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext>0</mtext></mrow></msub></mrow></math><img file="EP1528686A1_D0010.tif" /></maths><maths id="math0011" num=""><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>37399</mtext></mrow></msub><msub><mrow><mtext> = P</mtext></mrow><mrow><mtext>37399</mtext></mrow></msub><mtext> ⊕ </mtext><msub><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext>0</mtext></mrow></msub></mrow></math><img file="EP1528686A1_D0011.tif" /></maths><maths id="math0012" num=""><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>18518</mtext></mrow></msub><mtext> = </mtext><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>18518</mtext></mrow></msub><mtext> ⊕ </mtext><msub><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext>0</mtext></mrow></msub></mrow></math><img file="EP1528686A1_D0012.tif" /></maths><maths id="math0013" num=""><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>21120</mtext></mrow></msub><mtext> = </mtext><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>21120</mtext></mrow></msub><mtext> ⊕ </mtext><msub><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext>0</mtext></mrow></msub></mrow></math><img file="EP1528686A1_D0013.tif" /></maths> In the above equations, the additions are in GF(2)).

0036Then, for the next 359 information bits, <i>i</i>_<i>m</i><i>,m</i> = 1,2,...,359 , these bits are accumulated at parity bit addresses {<i>x</i> + <i>m</i> mod 360 × <i>q</i>} <i>mod</i>(<i>n</i>_<i>ldpc</i> - <i>k</i>_<i>ldpc</i>), where <i>x</i> denotes the address of the parity bit accumulator corresponding to the first bit <i>i</i>₀, and <i>q</i> is a code rate dependent constant. Continuing with the example with <i>q</i> = 120 for rate 1/3, for information bit <i>i</i>₁, the following operations are performed:<maths id="math0014" num=""><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>35023</mtext></mrow></msub><mtext> = </mtext><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>35023</mtext></mrow></msub><mtext> ⊕</mtext><msub><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext>1</mtext></mrow></msub></mrow></math><img file="EP1528686A1_D0014.tif" /></maths><maths id="math0015" num=""><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>21047</mtext></mrow></msub><mtext> = </mtext><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>21047</mtext></mrow></msub><mtext> ⊕</mtext><msub><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext>1</mtext></mrow></msub></mrow></math><img file="EP1528686A1_D0015.tif" /></maths><maths id="math0016" num=""><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>32213</mtext></mrow></msub><mtext> = </mtext><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>32213</mtext></mrow></msub><mtext> ⊕</mtext><msub><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext>1</mtext></mrow></msub></mrow></math><img file="EP1528686A1_D0016.tif" /></maths><maths id="math0017" num=""><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>1172</mtext></mrow></msub><mtext> = </mtext><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>1172</mtext></mrow></msub><mtext> ⊕</mtext><msub><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext>1</mtext></mrow></msub></mrow></math><img file="EP1528686A1_D0017.tif" /></maths><maths id="math0018" num=""><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>25731</mtext></mrow></msub><mtext> = </mtext><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>25731</mtext></mrow></msub><mtext> ⊕</mtext><msub><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext>1</mtext></mrow></msub></mrow></math><img file="EP1528686A1_D0018.tif" /></maths><maths id="math0019" num=""><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>16574</mtext></mrow></msub><mtext> = </mtext><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>16574</mtext></mrow></msub><mtext> ⊕</mtext><msub><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext>1</mtext></mrow></msub></mrow></math><img file="EP1528686A1_D0019.tif" /></maths><maths id="math0020" num=""><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>16574</mtext></mrow></msub><mtext> = </mtext><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>16574</mtext></mrow></msub><mtext> ⊕</mtext><msub><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext>1</mtext></mrow></msub></mrow></math><img file="EP1528686A1_D0020.tif" /></maths><maths id="math0021" num=""><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>5640</mtext></mrow></msub><mtext> = </mtext><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>5640</mtext></mrow></msub><mtext> ⊕</mtext><msub><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext>1</mtext></mrow></msub></mrow></math><img file="EP1528686A1_D0021.tif" /></maths><maths id="math0022" num=""><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>626</mtext></mrow></msub><mtext> = </mtext><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>626</mtext></mrow></msub><mtext> ⊕</mtext><msub><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext>1</mtext></mrow></msub></mrow></math><img file="EP1528686A1_D0022.tif" /></maths><maths id="math0023" num=""><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>37519</mtext></mrow></msub><mtext> = </mtext><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>37519</mtext></mrow></msub><mtext> ⊕</mtext><msub><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext>1</mtext></mrow></msub></mrow></math><img file="EP1528686A1_D0023.tif" /></maths><maths id="math0024" num=""><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>18638</mtext></mrow></msub><mtext> = </mtext><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>18638</mtext></mrow></msub><mtext> ⊕</mtext><msub><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext>1</mtext></mrow></msub></mrow></math><img file="EP1528686A1_D0024.tif" /></maths><maths id="math0025" num=""><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>21240</mtext></mrow></msub><mtext> = </mtext><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>21240</mtext></mrow></msub><mtext> ⊕</mtext><msub><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext>1</mtext></mrow></msub></mrow></math><img file="EP1528686A1_D0025.tif" /></maths>

0037For the 361^st information bit <i>i</i>₃₆₀, the addresses of the parity bit accumulators are given in the second rows of Tables 1-3. In a similar manner the addresses of the parity bit accumulators for the following 359 information bits <i>i</i>_<i>m</i><i>, m</i> = 361,362,...,719 are obtained using the formula {<i>x</i> + <i>m</i> mod 360 × <i>q</i>}mod(<i>n</i>_<i>ldpc</i> - <i>k</i>_<i>ldpc</i>), where <i>x</i> denotes the address of the parity bit accumulator corresponding to the information bit <i>i</i>₃₆₀, i.e., the entries in the second row of the Tables 1-3. In a similar manner, for every group of 360 new information bits, a new row from Tables 1 - 3 are used to find the addresses of the parity bit accumulators. <tables id="tabl0001" num="0001"><img file="EP1528686A1_D0026.tif" /></tables><tables id="tabl0002" num="0002"><img file="EP1528686A1_D0027.tif" /></tables><tables id="tabl0003" num="0003"><img file="EP1528686A1_D0028.tif" /></tables><tables id="tabl0004" num="0004"><img file="EP1528686A1_D0029.tif" /></tables><tables id="tabl0005" num="0005"><img file="EP1528686A1_D0030.tif" /></tables><tables id="tabl0006" num="0006"><img file="EP1528686A1_D0031.tif" /></tables><tables id="tabl0007" num="0007"><img file="EP1528686A1_D0032.tif" /></tables>

0038After all of the information bits are exhausted, the final parity bits are obtained as follows. First, the following operations are performed, starting with <i>i</i> = 1<maths id="math0026" num=""><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext></mrow></msub><mtext> = </mtext><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext></mrow></msub><mtext> ⊕ </mtext><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext><mtext>-1</mtext></mrow></msub><mtext>,   </mtext><mtext mathvariant="italic">i</mtext><mtext> = 1,2,...,</mtext><msub><mrow><mtext mathvariant="italic">n</mtext></mrow><mrow><mtext mathvariant="italic">ldpc</mtext></mrow></msub><mtext> - </mtext><msub><mrow><mtext mathvariant="italic">k</mtext></mrow><mrow><mtext mathvariant="italic">ldpc</mtext></mrow></msub><mtext> - 1.</mtext></mrow></math><img file="EP1528686A1_D0033.tif" /></maths>    Final content of <i>p</i>_<i>i</i><i> , i</i> = 0,1,..,<i>n</i>_<i>ldpc</i> - <i>k</i>_<i>ldpc</i> - 1 is equal to the parity bit <i>p</i>_<i>i</i>.

0039After all of the information bits are exhausted, the final parity bits are obtained as follows. First, the following operations are performed, starting with <i>i</i> = 1<maths id="math0027" num=""><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext></mrow></msub><mtext> = </mtext><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext></mrow></msub><mtext> ⊕ </mtext><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext><mtext>-1</mtext></mrow></msub><mtext>,   </mtext><mtext mathvariant="italic">i</mtext><mtext> = 1,2,...,</mtext><msub><mrow><mtext mathvariant="italic">n</mtext></mrow><mrow><mtext mathvariant="italic">ldpc</mtext></mrow></msub><mtext> - </mtext><msub><mrow><mtext mathvariant="italic">k</mtext></mrow><mrow><mtext mathvariant="italic">ldpc</mtext></mrow></msub><mtext> - 1.</mtext></mrow></math><img file="EP1528686A1_D0034.tif" /></maths>    Final content of <i>p</i>_<i>i</i>, <i>i</i> = 0,1,..,<i>n</i>_<i>ldpc</i> - <i>k</i>_<i>ldpc</i> - 1 is equal to the parity bit <i>p</i>_<i>i</i>.

0040As regards BCH encoding, according to one embodiment of the present invention, the generator polynomial of the BCH code utilized by the BCH encoder 211 is as follows:<maths id="math0028" num=""><math display="block"><mrow><mtext mathvariant="italic">g</mtext><mtext>(</mtext><mtext mathvariant="italic">x</mtext><mtext>)=(1+</mtext><mtext mathvariant="italic">x</mtext><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>3</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>5</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>14</mtext></mrow></msup><mtext>)×(1+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>6</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>8</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>11</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>14</mtext></mrow></msup><mtext>)</mtext><mtext mathvariant="italic">x</mtext><mtext>(1+</mtext><msup><mrow><mtext mathvariant="italic">x+x</mtext></mrow><mrow><mtext>2</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>6</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>9</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>10</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>14</mtext></mrow></msup><mtext>)×</mtext><mspace linebreak="newline" /><mtext> (1+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>4</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>7</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>8</mtext></mrow></msup><msup><mrow><mtext>+x</mtext></mrow><mrow><mtext>10</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>12</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>14</mtext></mrow></msup><mtext>)×(1+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>2</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>4</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>6</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>8</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>9</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>11</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>13</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>14</mtext></mrow></msup><mtext>)×</mtext><mspace linebreak="newline" /><mtext> (1+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>3</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>7</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>8</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>9</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>13</mtext></mrow></msup><msup><mrow><mtext>+x</mtext></mrow><mrow><mtext>14</mtext></mrow></msup><mtext>)×(1+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>2</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>5</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>6</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>7</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>10</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>11</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>13</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>14</mtext></mrow></msup><mtext>)×</mtext><mspace linebreak="newline" /><mtext> (1+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>5</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>8</mtext></mrow></msup><msup><mrow><mtext mathvariant="italic">+x</mtext></mrow><mrow><mtext>9</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>10</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>11</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>14</mtext></mrow></msup><mtext>)</mtext><mtext mathvariant="italic">×</mtext><mtext>(1+</mtext><msup><mrow><mtext mathvariant="italic">x+x</mtext></mrow><mrow><mtext>2</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>3</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>9</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>10</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>14</mtext></mrow></msup><mtext>)</mtext><mtext mathvariant="italic">x</mtext><mtext>(1+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>3</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>6</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>9</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>11</mtext></mrow></msup><msup><mrow><mtext mathvariant="italic">+x</mtext></mrow><mrow><mtext>12</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>14</mtext></mrow></msup><mtext>)×</mtext><mspace linebreak="newline" /><mtext> (1+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>4</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>11</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>12</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>14</mtext></mrow></msup><mtext>)</mtext><mtext mathvariant="italic">x</mtext><mtext>(1+</mtext><mtext mathvariant="italic">x</mtext><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>2</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>3</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>5</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>6</mtext></mrow></msup><msup><mrow><mtext mathvariant="italic">+x</mtext></mrow><mrow><mtext>7</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>8</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>10</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>13</mtext></mrow></msup><mtext>+</mtext><msup><mrow><mtext mathvariant="italic">x</mtext></mrow><mrow><mtext>14</mtext></mrow></msup><mtext>).</mtext></mrow></math><img file="EP1528686A1_D0035.tif" /></maths>

0041BCH encoding of information bits <i><b>m</b></i> = (<i>m</i>_{<i>k</i><sub2><i>bch</i></sub2>}_-1,<i>m</i>_{<i>k</i><sub2><i>bch</i></sub2>}_-2,...,<i>m</i>₁,<i>m</i>₀) onto a codeword <i>c</i> = (<i>m</i>_{<i>k</i><sub2><i>bch</i></sub2>}_-1,<i>m</i>_{<i>k</i><sub2><i>bch</i></sub2>}_-2,...,<i>m</i>₁,<i>m</i>₀,<i>d</i>_{<i>n</i><sub2><i>dch</i></sub2><i>-k</i><sub2><i>bch</i></sub2>}_-1,<i>d</i>_{<i>nbch-kbch</i>}_-2,...,<i>d</i>₁,<i>d</i>₀) is achieved as follows. The message polynomial <i>m</i>(<i>x</i>) = <i>m</i>_{<i>k</i><sub2><i>bch</i></sub2>}_-1<i>x</i>^{<i>k</i><sub2><i>bch</i></sub2>}^-1 + <i>m</i>_{<i>k</i><sub2><i>bch</i></sub2>-2}<i>x</i>^{<i>k</i><sub2><i>bchc</i></sub2>-2} +...+ <i>m</i>₁<i>x</i> + <i>m</i>₀ is multiplied by <i>x</i>^{<i>n</i><sub2><i>bch</i></sub2><i>-k</i><sub2><i>bch</i></sub2>}. Next, <i>x</i>^{<i>n</i><sub2><i>bch</i></sub2><i>-k</i><sub2><i>bch</i></sub2>}<i>m</i>(<i>x</i>) divided by <i>g</i>(<i>x</i>). With <i>d</i>(<i>x</i>) = <i>d</i>_{<i>n</i><sub2><i>bch</i></sub2>-<i>k</i><sub2><i>bch</i></sub2>-1}<i>x</i>^{<i>n</i><sub2><i>bch</i></sub2><i>-k</i><sub2><i>bch</i></sub2><i>-1</i>} +...+ <i>d</i>₁<i>x</i>+<i>d</i>₀ as the remainder, the codeword polynomial is set as follows: <i>c</i>(<i>x</i>) = <i>x</i>^{<i>n</i><sub2><i>bch</i></sub2><i>-k</i><sub2><i>bch</i></sub2>} m(<i>x</i>) + <i>d</i>(<i>x</i>).

0042As discussed, <i>k</i>_<i>ldpc</i> bits are systematically encoded to generate <i>n</i>_<i>ldpc</i> bits. According to one embodiment of the present invention, <i>n</i>_<i>ldpc</i> is 16200 bits, which is a short block length. Given the relatively short length of such codes, LDPC codes having approximate lengths of 16200 bits or less are deemed "short" block length codes.

0043In accordance with an embodiment of the present invention, some of the short blocks codes (e.g., rate 1/5 of Table 5) are generated by shortening versions of slightly longer (<i>k</i>_<i>m</i>,<i>n</i>_<i>m</i>) "mother" codes of block size <i>n</i>_<i>m</i> > 16200. As shown in FIG. 2D, <i>k</i>_<i>ldpc</i> of the BCH encoded bits are preceded by <i>k</i>_<i>m</i> - <i>k</i>_<i>ldpc</i> dummy zeros (per step 251). The resulting <i>k</i>_<i>m</i> bits are systematically encoded to generate <i>n</i>_<i>m</i> bits, as in step 253. The first <i>k</i>_<i>m</i> - <i>k</i>_<i>ldPc</i> dummy zeros are then deleted, as in step 255, and the resulting <i>n</i>_<i>ldpc</i> =16200 bits will be transmitted (step 257). It is noted that <i>k</i>_<i>m</i> - <i>k</i>_<i>ldpc</i> = <i>n</i>_<i>m</i> - <i>n</i>_<i>ldpc</i>. Tables 4-6 provide other exemplary short code rates, 1/3, 1/5 and 2/5 (<i>n</i>_<i>ldpc</i> of 16200 bits): <tables id="tabl0008" num="0008"><img file="EP1528686A1_D0036.tif" /></tables><tables id="tabl0009" num="0009"><img file="EP1528686A1_D0037.tif" /></tables><tables id="tabl0010" num="0010"><img file="EP1528686A1_D0038.tif" /></tables>

0044The above approach to designing LDPC codes, as provided in Tables 1-6, advantageously permits storage and retrieval of relevant information regarding partitioned groups of bit nodes and check nodes to be always placed in contiguous memory locations within memory (e.g., Random Access Memory (RAM)). Consequently, multiple code rates can be supported without employing different decoders. Further, the design enables use of a single RAM bank, thereby minimizing size of the integrated circuit. As mentioned, the above LDPC codes, in an exemplary embodiment, can be used to variety of digital video applications, such as MPEG (Motion Pictures Expert Group) packet transmission.

0045FIG. 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 LDPC decoder 305, which attempts to reconstruct the original source messages by generating messages, <i>X</i>', in conjunction with a bit metric generator 307. The bit metric generator 307 may exchange information with the decoder 305 back and forth (iteratively) during the decoding process. These decoding approaches are more fully described in co-pending application, entitled "Method and System for Routing in Low Density Parity Check (LDPC) Decoders," filed July 3, 2003 (Serial No. 10/613,824; Attorney Docket PD-203009). It is noted that LDPC codes, as constructed per Tables 1-6, can support multiple code rates (consequently, multiple data rates) using a common decoding architecture; in contrast, convolutional codes, for example, require puncturing techniques to achieve intermediate rates.

0046To further appreciate the advantages offered by the present invention, it is instructive to examine how LDPC codes are generated, as discussed in FIG. 4.

0047FIG. 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 <i>H</i>₍_<i>n</i>_-_<i>k</i>₎_<i>xn</i>. Typically the block length, <i>n</i>, 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 1/2 is shown in FIG. 4. The same code can be equivalently represented by the bipartite graph, per FIG. 5.

0048FIG. 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 <i>m</i>₁, the following expression exists <i>n</i>₁, + <i>n</i>₄ + <i>n</i>₅ + <i>n</i>₈ = 0 with respect to the bit nodes.

0049Returning to the receiver 300, 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.

0050From 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 <i>n</i>₄, <i>n</i>₅ and <i>n</i>₈ "looks like" 0 to <i>m</i>₁, then <i>m</i>_<i>1</i> would indicate to <i>n</i>₁ that the value of <i>n</i>₁ is believed to be 0 (since <i>n</i>₁ + <i>n</i>₄ +<i>n</i>₅ + <i>n</i>₈ = 0); otherwise <i>m</i>₁ indicate to <i>n</i>₁ that the value of <i>n</i>₁ is believed to be 1. Additionally, for soft decision decoding, a reliability measure is added.

0051From 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 <i>n</i>₁ has only two adjacent check nodes <i>m</i>₁ and <i>m</i>₃. If the feedback coming from <i>m</i>₃ to <i>n</i>₁ indicates that the value of <i>n</i>₁ is probably 0, then <i>n</i>₁ would notify <i>m</i>₁ that an estimate of <i>n</i>₁'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.

0052FIG. 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 FIGs. 2A and 2B) 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:<maths id="math0029" num=""><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">H</mtext></mrow><mrow><mtext>(</mtext></mrow></msub><msub><mrow><mtext>​</mtext></mrow><mrow><mtext mathvariant="italic">n</mtext></mrow></msub><msub><mrow><mtext>​</mtext></mrow><mrow><mtext>-</mtext></mrow></msub><msub><mrow><mtext>​</mtext></mrow><mrow><mtext mathvariant="italic">k</mtext></mrow></msub><msub><mrow><mtext>​</mtext></mrow><mrow><mtext>)</mtext></mrow></msub><msub><mrow><mtext>​</mtext></mrow><mrow><mtext mathvariant="italic">xn</mtext></mrow></msub><mtext> = [</mtext><msub><mrow><mtext mathvariant="italic">A</mtext></mrow><mrow><mtext>(</mtext></mrow></msub><msub><mrow><mtext>​</mtext></mrow><mrow><mtext mathvariant="italic">n</mtext></mrow></msub><msub><mrow><mtext>​</mtext></mrow><mrow><mtext>-</mtext></mrow></msub><msub><mrow><mtext>​</mtext></mrow><mrow><mtext mathvariant="italic">k</mtext></mrow></msub><msub><mrow><mtext>​</mtext></mrow><mrow><mtext>)</mtext></mrow></msub><msub><mrow><mtext>​</mtext></mrow><mrow><mtext mathvariant="italic">xk</mtext></mrow></msub><msub><mrow><mtext mathvariant="italic"> B</mtext></mrow><mrow><mtext>(</mtext></mrow></msub><msub><mrow><mtext>​</mtext></mrow><mrow><mtext mathvariant="italic">n</mtext></mrow></msub><msub><mrow><mtext>​</mtext></mrow><mrow><mtext>-</mtext></mrow></msub><msub><mrow><mtext>​</mtext></mrow><mrow><mtext mathvariant="italic">k</mtext></mrow></msub><msub><mrow><mtext>​</mtext></mrow><mrow><mtext>)</mtext></mrow></msub><msub><mrow><mtext>​</mtext></mrow><mrow><mtext mathvariant="italic">x</mtext></mrow></msub><msub><mrow><mtext>​</mtext></mrow><mrow><mtext>(</mtext></mrow></msub><msub><mrow><mtext>​</mtext></mrow><mrow><mtext mathvariant="italic">n</mtext></mrow></msub><msub><mrow><mtext>​</mtext></mrow><mrow><mtext>-</mtext></mrow></msub><msub><mrow><mtext>​</mtext></mrow><mrow><mtext mathvariant="italic">k</mtext></mrow></msub><msub><mrow><mtext>​</mtext></mrow><mrow><mtext>)</mtext></mrow></msub><mtext>]</mtext></mrow></math><img file="EP1528686A1_D0039.tif" /></maths> , where <i>B</i> is lower triangular.

0053Any information block <i>i</i> = (<i>i</i>₀,<i>i</i>₁,..,<i>i</i>_<i>k</i>_-1) is encoded to a codeword <i>c</i> = (<i>i</i>₀,<i>i</i>₁,...,<i>i</i>_<i>k</i>_-1,<i>p</i>₀,<i>p</i>₁,...<i>p</i>_n-_<i>k</i>_-1) using <i>H</i>c^T = 0, and recursively solving for parity bits; for example,<maths id="math0030" num=""><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">a</mtext></mrow><mrow><mtext>00</mtext></mrow></msub><msub><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext>0</mtext></mrow></msub><mtext> + </mtext><msub><mrow><mtext mathvariant="italic">a</mtext></mrow><mrow><mtext>01</mtext></mrow></msub><msub><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext>1</mtext></mrow></msub><mtext> +...+ </mtext><msub><mrow><mtext mathvariant="italic">a</mtext></mrow><mrow><mtext>0,</mtext></mrow></msub><msub><mrow><mtext>​</mtext></mrow><mrow><mtext mathvariant="italic">k</mtext></mrow></msub><msub><mrow><mtext>​</mtext></mrow><mrow><mtext>-1</mtext></mrow></msub><msub><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext mathvariant="italic">k</mtext></mrow></msub><msub><mrow><mtext>​</mtext></mrow><mrow><mtext>-1</mtext></mrow></msub><mtext> + </mtext><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>0</mtext></mrow></msub><mtext> = 0 ⇒ </mtext><msub><mrow><mtext mathvariant="italic">Solve p</mtext></mrow><mrow><mtext>0</mtext></mrow></msub><mtext>,</mtext></mrow></math><img file="EP1528686A1_D0040.tif" /></maths><maths id="math0031" num=""><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">a</mtext></mrow><mrow><mtext>10</mtext></mrow></msub><msub><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext>0</mtext></mrow></msub><mtext> + </mtext><msub><mrow><mtext mathvariant="italic">a</mtext></mrow><mrow><mtext>11</mtext></mrow></msub><msub><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext>1</mtext></mrow></msub><mtext> +...+ </mtext><msub><mrow><mtext mathvariant="italic">a</mtext></mrow><mrow><mtext>1,</mtext></mrow></msub><msub><mrow><mtext>​</mtext></mrow><mrow><mtext mathvariant="italic">k</mtext></mrow></msub><msub><mrow><mtext>​</mtext></mrow><mrow><mtext>-1</mtext></mrow></msub><msub><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext mathvariant="italic">k</mtext></mrow></msub><msub><mrow><mtext>​</mtext></mrow><mrow><mtext>-1</mtext></mrow></msub><mtext> + </mtext><msub><mrow><mtext mathvariant="italic">b</mtext></mrow><mrow><mtext>10</mtext></mrow></msub><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>0</mtext></mrow></msub><mtext> + </mtext><msub><mrow><mtext mathvariant="italic">p</mtext></mrow><mrow><mtext>1</mtext></mrow></msub><mtext> = 0 ⇒ </mtext><msub><mrow><mtext mathvariant="italic">Solve p</mtext></mrow><mrow><mtext>1</mtext></mrow></msub></mrow></math><img file="EP1528686A1_D0041.tif" /></maths>    and similarly for <i>p</i>₂, <i>p</i>₃,...,<i>p</i>_n-k-1.

0054FIGs. 7and 8 are graphs of performance of the LDPC codes at the various code rates and modulation schemes supported by the transmitter 200 of FIG. 2A. Specifically, the graph of FIG. 7 shows a performance comparison between rate 1/3 and 1/4 LDPC codes and repetition rate 2/3 and repetition rate 1/2 LDPC codes, respectively. Under this scenario, performance is considered in the case of rate 1/3 and 1/4 for QPSK LDPC codes on an Additive White Gaussian Noise (AWGN) channel. As seen, the rate 1/3 code offers about 1.3 dB gain with respect to a repetition rate 2/3 LDPC code, whereas rate 1/4 LDPC code offers about 0.4 dB gain with respect to a repetition rate 1/2 LDPC code. Repetition codes are constructed by repeating the source information bits as the set of parity bits. Therefore rate 1/3 and 1/4 LDPC codes would offer performance advantage with respect to repetition codes when used in low priority branch of backward compatible hierarchical modulation schemes.

0055FIG. 8 illustrates the performance of short frame length codes of rates 1/5, 1/3 and 2/5 along with the long frame length codes of rates 1/4, 1/3 and 2/5. As shown, the short codes for rates 1/3 and 2/5 exhibit nearly comparable performance to their counterpart long codes (within 0.2 dB).

0056The LDPC encoding processes as detailed above can be executed through a variety of hardware and/or software configurations. In fact, this approach also can be readily deployed solely through a software change, thereby eliminating costly hardware modifications.

0057FIG. 9 illustrates exemplary hardware upon which an embodiment according to the present invention can be implemented. A computing system 900 includes a bus 901 or other communication mechanism for communicating information and a processor 903 coupled to the bus 901 for processing information. The computing system 900 also includes main memory 905, such as a random access memory (RAM) or other dynamic storage device, coupled to the bus 901 for storing information and instructions to be executed by the processor 903. Main memory 905 can also be used for storing temporary variables or other intermediate information during execution of instructions by the processor 903. The computing system 900 may further include a read only memory (ROM) 907 or other static storage device coupled to the bus 901 for storing static information and instructions for the processor 903. A storage device 909, such as a magnetic disk or optical disk, is coupled to the bus 901 for persistently storing information and instructions.

0058The computing system 900 may be coupled via the bus 901 to a display 911, such as a liquid crystal display, or active matrix display, for displaying information to a user. An input device 913, such as a keyboard including alphanumeric and other keys, may be coupled to the bus 901 for communicating information and command selections to the processor 903. The input device 913 can include a cursor control, such as a mouse, a trackball, or cursor direction keys, for communicating direction information and command selections to the processor 903 and for controlling cursor movement on the display 911.

0059According to one embodiment of the invention, the processes of FIGs. 2C and 2D can be provided by the computing system 900 in response to the processor 903 executing an arrangement of instructions contained in main memory 905. Such instructions can be read into main memory 905 from another computer-readable medium, such as the storage device 909. Execution of the arrangement of instructions contained in main memory 905 causes the processor 903 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 905. 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. In another example, reconfigurable hardware such as Field Programmable Gate Arrays (FPGAs) can be used, in which the functionality and connection topology of its logic gates are customizable at run-time, typically by programming memory look up tables. Thus, embodiments of the present invention are not limited to any specific combination of hardware circuitry and software.

0060The computing system 900 also includes at least one communication interface 915 coupled to bus 901. The communication interface 915 provides a two-way data communication coupling to a network link (not shown). The communication interface 915 sends and receives electrical, electromagnetic, or optical signals that carry digital data streams representing various types of information. Further, the communication interface 915 can include peripheral interface devices, such as a Universal Serial Bus (USB) interface, a PCMCIA (Personal Computer Memory Card International Association) interface, etc.

0061The processor 903 may execute code that is being received over the communication interface 915 and/or store the code in the storage device 909, or other non-volatile storage for later execution. In this manner, the computing system 900 may obtain application code in the form of a carrier wave.

0062The term "computer-readable medium" as used herein refers to any medium that participates in providing instructions to the processor 903 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 the storage device 909. Volatile media include dynamic memory, such as main memory 905. Transmission media include coaxial cables, copper wire and fiber optics, including the wires that comprise the bus 901. 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.

0063Various 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 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 assistant (PDA) or 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 can optionally be stored on storage device either before or after execution by processor.

0064Accordingly, the various embodiments of the present invention provide an LDPC encoder generates a LDPC code having an outer Bose Chaudhuri Hocquenghem (BCH) code according to one of Tables 1-6 for transmission as a LDPC coded signal. Each of the Tables 1-6 specifies the address of parity bit accumulators. The above approach advantageously yields reduced complexity without sacrificing performance.

0065While 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.