HIGH ORDER MODULATION PROTOGRAPH CODES

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR

DEVELOPMENT:

[0001] The invention described herein was made in the performance of work under a NASA contract, and is subject to the provisions of Public Law 96-517 (35 USC 202) in which the Contractor has elected to retain title.

BACKGROUND

1. Field

[0002] The present disclosure relates to constructing low-density parity-check (LDPC) codes from small template graphs called pro to graphs. More in particular, it relates to methods for designing pro to graphs that accommodate modulation levels above simple binary coding.

2. Description of Related Art Definitions

[0003] As known to the person skilled in the art and as also mentioned in US 7,343,539 incorporated herein by reference in its entirety, a low-density parity-check (LDPC) code is a linear code determined by a sparse parity-check matrix H having a small number of Is per column. The code's parity-check matrix H can be represented by a bipartite Tanner graph wherein each column of H is represented by a transmitted variable node, each row by a check node, and each "1" in H by a graph edge connecting the variable node and check node that correspond to the column-row location of the "1". The code's Tanner graph may additionally have non-transmitted variable nodes. Each check or constraint node defines a parity check operation. Moreover, the fraction of a transmission that bears information is called the rate of the code. An LDPC code can be encoded by deriving an appropriate generator matrix G from its parity-check matrix H. An LDPC code can be decoded efficiently using a well-known iterative algorithm that passes messages along edges of the code's Tanner graph from variable nodes to check nodes and vice-versa until convergence is obtained, or a certain number of iterations is reached.

[0004] A protograph is a Tanner graph with a relatively small number of nodes, connected by a small number of edges, allowing multiple parallel edges between two nodes. Each edge is a type. Thus, the number of total edges represents the number of types that exists in the protograph. A protograph code (an equivalent LDPC code) is a larger derived graph constructed by applying a "copy-and-permutation" operation on a protograph. The protograph is copied N times, then a large LDPC code graph is obtained by permuting N variable-to-check pairs (edges), corresponding to the same edge type of the original protograph. The resulting protograph code has N times as many nodes as the protograph, but it has the same rate and the same distribution of variable and check node degrees. Thus, the degree distribution of a pro to graph -based LDPC code is the same as that of the protograph. A simple example of a protograph is shown in FIG. 1. This graph consists of 3 variable nodes and 2 check nodes that are interconnected by 7 different edge (variable-to-check) types. The derived graph is constructed by replicating the protograph 3 times, and permuting variable-to-check pairings within the same edge type of the protograph. A protograph can also be represented by a proto-matrix. For example, the proto-matrix of the protograph shown in the left portion of FIG. 1 has the form shown in Eq. 1 below:

1 1 1 λ

H proto Eq. 1

2 1 1 , where the rows and columns represent the check nodes and variable nodes in the graph respectively. The elements in the matrix are the number of parallel edges that connect the variable node and the check node associated with their positions.

[0005] One way to construct larger codes is to replace each integer j in the proto-matrix with sum of j different NxN permutation matrices. If the integer is 1, it would be replaced with 1 NxN permutation matrix. Another way is to first expand the protograph by a small factor such that the H matrix for the resulting expanded protograph does not contain any integer larger than 1. Then each integer 1 in the H matrix will be replaced with an NxN permutation matrix. The assigned NxN permutations can be any type of permutations, including circulant permutations. An NxN identity matrix can be regarded as a trivial permutation matrix. If the NxN permutation is a circulant permutation, then define matrix X as one circular shift of the identity matrix to the left. Using this notation, then all circulant permutations can be represented by X ¹ , for i = 0,1,.., (N-l). The selection and assignment of permutation matrices to construct N times larger LDPC code can be based on enlarging the smallest loop size in the expanded graph. Other optimization methods include Progressive Edge Growth (PEG) for circulant matrices (see, for example X. Hu, E. Eleftheriou, and D.M. Arnold, "Regular and irregular progressive edge-growth tanner graphs," IEEE Transactions on Information Theory, Vol. 51, issue 1, 2005, pp. 386-398, hereinafter X. Hu, et al.), or the Approximate Cycle Extrinsic message degree (ACE) algorithm (see, for example, T. Tian, C.R. Jones, J.D. Villasenor, and R.D. Wesel, "Selective avoidance of cycles in irregular LDPC code construction," IEEE Transactions on Communications, Volume 52, Issue 8, 2004, pp. 1242 - 1247, or T. Tian, C.R. Jones, J.D. Villasenor, and R.D. Wesel, "Construction of irregular LDPC codes with low error floors," IEEE International Conference on Communications, 2003, Vol. 5, pp. 3125 - 3129). So, in the case of circulant permutations, these optimization algorithms can be applied to obtain powers of X in the H matrix of the expanded protograph.

[0006] The rate of a protograph is defined to be the lowest (and typical) rate of any LDPC code constructed from that protograph. All LDPC codes constructed from a given protograph have the same rate except for possible check constraint degeneracies, which can increase (but never decrease) this rate and typically occur only for very small codes. Since the protograph serves as a blueprint for the Tanner graph representing any LDPC code expandable from that protograph, it also serves as a blueprint for the routing of messages in the iterative algorithm used to decode such expanded codes through a permutation per each edge type. The rate of a protograph code is computed as shown below in Eq. 2:

where n and m are number of variable nodes and check nodes respectively in the protograph, no is number of punctured (untransmitted) variable nodes. [0007] Excluding check nodes connected to degree- 1 variable nodes, applicants have proved that the number of degree-2 nodes should be at most one less than the number of check nodes provided that no loop exists in the graph between degree-2 nodes and the checks connected to these nodes for a protograph to have the linear minimum distance property. A given protograph is said to have the linear minimum distance property if the typical minimum distance of a random ensemble of arbitrarily large LDPC codes built from that protograph grows linearly with the size of the code, with linearity coefficient which can be denoted by 6 _m i _n > 0 (see S. Abu- Surra, D. Divsalar and W. Ryan, "On the existence of typical minimum distance for protograph- based LDPC codes," in Information Theory and Applications Workshop (ITA), Jan 2010, pp. 1- 7).

[0008] The iterative decoding threshold of a given protograph is similarly defined with respect to this random ensemble of LDPC codes as the lowest value of signal-to-noise ratio for which an LDPC decoder's iterative decoding algorithm will find the correct codeword with probability approaching one as the size of an LDPC code built from that protograph is made arbitrarily large. Iterative decoding thresholds can be calculated by using a reciprocal channel approximation (D. Divsalar, S. Dolinar, C. R. Jones, and K. Andrews, "Capacity approaching protograph codes," IEEE J. Select. Areas Communication, vol. 27, no. 6, pp. 876-888, Aug. 2009, hereinafter D. Divsalar, et al.) or the PEXIT method (see, for example, G. Liva and M. Chiani, "Protograph LDPC codes design based on ΕΧΓΓ analysis," in Proc. IEEE GLOBECOM, Nov. 2007, pp. 3250-3254, hereinafter G. Liva, et al.). Thresholds can be lowered either by using precoding (a subgraph of the protograph with degree- 1 nodes, check nodes connected to these degree- 1 nodes, and all edges connected to these checks and other variable nodes with at least one punctured node) or through the use of at least one very high-degree node in the base protograph. A protograph is said to have a low iterative decoding threshold if its threshold is close to the capacity limit for its rate.

[0009] A family of protographs of different rates is said to be rate-compatible with embedding or embedded rate-compatible if the protographs for different rates produce embedded codewords with the same information block-length. In other words, with the same input, the codeword of a high-rate code is embedded into the codeword of a lower-rate code. This property makes this family of protographs suitable for Hybrid Automatic Repeat Request (HARQ) applications. [0010] Coded modulation is a method of communication where the messages are encoded with an error correcting code, and the resulting coded bits are mapped to modulation symbols such as BPSK, M-PSK, or M-QAM modulations. The overall efficiency of the coded modulation is measured by spectral efficiency, whose unit is bits/seconds/Hz.

[0011] Bit-interleaved coded modulation (BICM) is a technique that allows relatively simple design of bandwidth-efficient coded modulation systems. A general BICM system is shown in FIG. 2. The information sequence is encoded by an LDPC encoder to get a coded bit sequence. Depending on the fading regime, the coded sequence may be bit-interleaved before being sent to a modulator. The M-ary modulator maps m = log ₂ M coded bits at a time to a complex symbol chosen from an M-ary constellation^. The discrete-time baseband channel model can be written as shown in Eq. 3 below: y _t = hx _t + w _t Eq. 3 where t is the discrete time index, y _t is the received signal, x _t is the transmitted symbol, h = h _j + ih _Q is the zero mean complex Gaussian distributed fading coefficient with the variance of 1/2 in each dimension and w _t is a complex white Gaussian noise sample with the zero mean and the variance per dimension of σ ² = N( 2. It is assumed that the channel state information is available at the receiver, i.e. the receiver can estimate the channel coefficient h.

[0012] The MAP symbol-to-bit metric calculator will compute the symbol-to-bit metrics based on the received symbol y from the channel. These bit metrics are passed to a decoder, which employs the iterative belief propagation algorithm. When an interleaver is used in the transmitter, the bit metrics in the receiver are de-interleaved before being passed to the LDPC decoder. Let ^ be the subset of all the signal points X G χ whole label has value b e {0,1} in position i. The symbol-to-bit metric is computed by the MAP calculator at each time to feed into the iterative decoder is given as shown below in Eq. 4:

for i = 1, . . . , m, where M=2 ^m . The BICM capacity with perfect CSI and uniform inputs is given by Eq. 5 as shown below:

[0013] BICM capacity is the sum of mutual information of m parallel channels as shown in Eq. 6 below: m

C =∑/(Z,,b _; ) Eq. 6

i=l

This mutual information I(Lj,bi) can be computed by using a Monte Carlo simulation as shown in Eq. 7 and Eq. 8 below:

I(L _{ ,b _{ ) = 1 - E[log ₂ (l + exp(- (1 - 2b, )L _t ))] Eq. 7

I(L _i ,b _i ) = l (l + exp(- (l - 2b _n )L _i Eq. 8 where N is the number of coded modulation symbols, b, and L, are the bit and log-likelihood random variables respectively for the level i of coded modulation, and ¾„ and L _i>n are respectively the bit and log-likelihood values at level i for the n-t symbol.

[0014] The BICM approach has proven to be powerful and capacity-approaching for fading channels with applications of low-density parity-check (LDPC) codes. However, many of the previous works design a particular code with a specific modulation scheme. It is desirable to find a general scheme that can support multiple rates and multiple modulation schemes within the context of BICM.

SUMMARY

[0015] Described herein are embodiments that provide for digital communication coding methods for designing protograph-based BICM that is general and applies to any modulation. The iterative decoding thresholds of the protograph codes while mapped to higher order modulations are calculated. This general coding framework can support not only multiple rates but also adaptive modulation. Certain families of protograph codes are shown to achieve a threshold within a gap of approximately 0.2 - 0.3 dB of BICM capacity limit across a wide range of rates and modulations.

[0016] The description below presents digital communication coding methods for designing pro to graph-based BICM that is general and applies to any modulation. The general coding framework can support not only multiple rates but also adaptive modulation. The method is a two stage lifting approach. In the first stage, an original protograph is lifted to a slightly larger intermediate protograph. The intermediate protograph is then lifted via a circulant matrix to the expected codeword length to form a protograph-based LDPC code.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

[0017] FIG. 1 shows an example of a protograph and demonstrates a copy-and-permute operation on the protograph to obtain a larger graph. The protograph also can be represented by a protomatrix whose elements present the number of edges connecting a variable node (column) to a check node (row) in the protograph.

[0018] FIG. 2 shows a block diagram of a bit-interleaved coded modulation system.

[0019] FIG. 3 shows the AR4JA family of protographs with rates 1/2 and higher.

[0020] FIG. 4 shows a rate 1/2 protograph which proves a better iterative decoding threshold.

[0021] FIG. 5 shows the protomatrix corresponding to the protograph shown in FIG. 4.

[0022] FIG. 6 shows the protomatrices of high-rate (>l/2) protographs that are extended from the rate- 1/2 protomatrix shown in FIG. 5.

[0023] FIG. 7 shows the protomatrices of rate-compatible family built from extending the rate-3/4 protograph whose protomatrix is shown in FIG. 6.

[0024] FIG. 8 shows a rate- 1/2 AR4JA protograph. [0025] FIG. 9 shows a rate- 1/2 AR4JA protograph after lifting by a factor of 4 with a mapping for 16 QAM.

[0026] FIG. 10 shows the iterative decoding thresholds from methods described herein with iterative coding thresholds obtained from other mapping methods with rate- 1/2 AR4JA protograph and 16QAM in AWGN channel.

[0027] FIG. 11 shows the iterative coding thresholds of AR4JA codes shown in FIG. 3 utilizing the described BICM methods in Rayleigh faded channels.

[0028] FIG. 12 shows the iterative coding thresholds of another family of codes whose protomatrices are shown in FIG. 5 and FIG. 6 utilizing the described BICM methods in Rayleigh faded channels.

[0029] FIG. 13 shows the iterative coding thresholds of rate-compatible family of codes whose protomatrices are shown in FIG. 7 utilizing the described BICM methods in Rayleigh faded channels.

[0030] FIG. 14 shows 8PSK with Gray labeling. [0031] FIG. 15 shows 16QAM with Gray labeling.

[0032] FIG. 16 shows the performance of rate- 1/2 AR4JA protograph and the rate- 1/2 protograph whose proto matrix is shown in FIG. 7 with 16QAM in Rayleigh channel and 16k information bits.

DETAILED DESCRIPTION

[0033] The exemplary embodiments described below show the application of three families of protograph codes with embodiments of the present invention. Note, however, that the present invention is not limited to the protograph codes presented below. Descriptions of these protograph codes are presented to aid in understanding of the invention. Those skilled in the art will understand that other protograph codes are within the scope of the invention. The families of protograph codes presented below include: AR4JA codes, discussed in more detailed in D. Divsalar, et al., and two other families of protograph codes presented in T. V. Nguyen, A. Nosratinia, and D. Divsalar, "The design of rate compatible protograph LDPC codes," IEEE Trans. Commun., 2011, hereinafter T. V. Nguyen, et al.. The AR4JA codes have been recommended for the space communication for the code rate- 1/2 or higher. Based on the same AR4JA protograph structure, T. V. Nguyen, et al. provide a design procedure that gave a family of new codes with an improvement of about 0.25 dB in terms of iterative decoding thresholds in AWGN channels compared with that of AR4JA protograph codes. The third family that consists of rate-compatible codes is built from extending a rate-3/4 code in the second family. This third family of codes gives rise to coded modulations that can support hybrid ARQ applications, among others.

[0034] The family of AR4JA protograph codes is plotted in FIG. 3. In FIG. 3, the dark circles represent transmitted variable nodes, the white circle is a punctured node and the circles with a plus sign are parity check nodes. The graph contains 4+2n transmitted variable nodes and 3 n + l

check nodes that is equivalent to code rate R , with n = 0, 1, . . ..

n + 2

[0035] A protograph can also be represented by a protomatrix. For example, the rate- 1/2 AR4JA protograph (n = 0) has a protomatrix in the form shown in Eq. 9 below:

where the rows and columns represent the check nodes and variable nodes in the graph respectively. The elements in the matrix are the number of parallel edges that connect the variable node and the check node associated with their positions. The family of AR4JA codes n + l

with rate R has a protomatrix in the form shown in Eq. 10 below:

n + 2

n + l

[0036] where n = 1, 2, . . .. The embedded codes with the rate of R are built by n + 2

extending the rate -1/2 code with the protomatrix of Eq. l.The structure of the rate- 1/2 AR4JA code has 5 variable nodes (one punctured node) and 3 check nodes. For the same rate- 1/2 structure with a slightly larger graph which contains 7 variable nodes (one punctured node) and 4 check nodes, the pro to graph of a better code in terms of iterative decoding threshold is shown in FIG. 4. Its corresponding protomatrix is shown in FIG. 5. The threshold of this code is 0.395 dB as shown in T. V. Nguyen, et al, , which shows a gap of 0.208 dB of capacity, less than that of the rate 1/2 AR4JA code of Eq. 9 as reported in D. Divsalar, et al. Other high rate protographs which are also built by extending this rate- 1/2 protograph have protomatrices shown in FIG. 6.

[0037] The third family of rate-compatible protograph codes has proto-matrices that are shown in FIG. 7. This family is built from adding equal number of variable (column) and check (row) nodes to the rate-3/4 protograph whose protomatrix is shown in FIG. 6. The protograph codes in this family have the same information block-lengths, thus are suitable for Hybrid ARQ applications.

[0038] An iterative decoding threshold of a protograph is the minimum channel quality that supports reliable iterative decoding of asymptotically large LDPC codes built from the protograph. T. J. Richardson, M. A. Shokrollahi, and R. L. Urbanke, "Design of capacity- approaching irregular low-density parity-check codes," IEEE Trans. Inform. Theory, vol. 47, no. 2, pp. 619-637, Feb. 2001, describes computation of thresholds of LDPC code ensembles by using density evolution. S. ten Brink, "Convergence behavior of iteratively decoded parallel concatenated codes," IEEE Trans. Commun., vol. 49, no. 10, pp. 1727- 1737, Oct 2001, describe computation of thresholds of LDPC code ensembles by extrinsic information transfer (EXIT) chart techniques. However, the general EXIT chart cannot be applied to compute the thresholds of protograph codes due to its inability of computing thresholds of graphs that have degree- 1 variable, punctured variable nodes or protographs with same degree distributions and different thresholds. See, for example, G. Liva, et al. To solve these problems, G. Liva, et al. proposed the ΡΕΧΓΓ method that is simple and allows for the computation of thresholds of protographs even more accurately than the method of reciprocal channel approximation (RCA) described in D. Divsalar, et al., cited above. Thresholds of the AR4JA family of protograph codes plotted in FIG. 3 and the other family of protographs were discussed in T. V. Nguyen, et al., cited above.

[0039] As discussed above, protograph codes can be represented by a small graph with only a few variable nodes and check nodes. For the purposes of BICM, the binary variable nodes of the code must be mapped to the bit-levels of the modulation. The iterative decoding threshold of the overall coded modulation scheme naturally depends in part on the protograph code, but also on the mapping between the code and the modulation levels.

[0040] In D. Divsalar and C. Jones, "Protograph based low error floor LDPC coded modulation," in Proc. IEEE MILCOM, Oct. 2005, pp. 378 -385 Vol. 1, hereinafter Divsalar & Jones,, Divsalar and Jones proposed a mapping algorithm based on Variable Degree Matched Mapping (VDMM). The idea is to directly assign protograph variable nodes in proportion to mutual information of coded bits in the modulation symbol. This is a well-known water-filling problem. However, in Divsalar & Jones, variable nodes in a protograph are directly assigned to the bits of a modulation symbol. This limits the coded modulation that can be designed with this and similar method to only one particular coded modulation for each protograph. For example, FIG. 8 shows the rate- 1/2 AR4JA protograph with transmitted variable nodes indexed by Vj, i=l,..., 4. For a specific 16 QAM modulation, the method of Divsalar & Jones is proposed with the mapping {b0, bl, b2, b3} = {V2, V4, VI, V3}.

[0041] An exemplary embodiment of the present invention provides a general method that can work for any protograph structure and modulation level and can achieve BICM-liked performance. More specifically, this exemplary embodiment provides a method to map any protograph to any modulation level, unlike the previous art. Exemplary embodiments use a two- stage lifting approach. Assuming that the desired modulation is -ary and the original protograph has n transmitted variable nodes, the first step is to start with a smaller lifting of the original protograph by a factor of m = log ₂ M to a slightly larger protograph. This protograph transformation may be accomplished using the PEG algorithm disclosed in X. Hu, et al., or other such techniques. An example is shown in FIG. 9, where an intermediate protograph is constructed from m=4 original AR4JA protographs. This intermediate protograph is then mapped to n = 4 modulation symbols as follows: each labeling bit position from all the n modulation symbols is mapped to the variable nodes of one of the planes in the intermediate protograph. For example, four bo bits from four modulation symbols are mapped to the top plane, four bi bits are mapped to the next plane, etc.

[0042] Now, the mapping between coding and modulation is complete within a relatively small "intermediate" protograph. The advantage of this intermediate protograph is that it is small enough to allow optimizations, but it also has enough degrees of freedom to provide a good mapping. Now this intermediate protograph is lifted (via a circulant matrix) to the expected codeword length to form a protograph-based LDPC code.

[0043] Embodiments of the present invention may be contrasted with VDMM in terms of flexibility. In VDMM as it is proposed, the coded bits in each protograph correspond to the symbol bits in one transmitted symbol. Thus, for example, a 4-variable node protograph naturally corresponds to 16-QAM. With the proposed framework, however, it is very easy to use any protograph together with any modulation. All that is needed is to produce the right intermediate protograph.

[0044] FIG. 10 compares the iterative decoding thresholds of embodiments according to the present invention compared with Divsalar & Jones. By necessity this table is small, as Divsalar & Jones as mentioned above, naturally correspond to only one type of modulation. The iterative decoding threshold of embodiments according to the present invention is slightly better than that of Divsalar & Jones. Note, however, that embodiments according to the present invention are not simply directed at producing the absolutely smallest threshold for the specific case of 16-QAM. Embodiments according to the present invention expand the horizon of available designs, as previous designs only allowed the use of modulations whose bit-levels were the same as the number of nodes in a protograph, for example, a protograph with four transmitted nodes could previously be mapped only to a 16-QAM or 16-PSK modulation. As discussed below, the threshold gap to capacity provided by embodiments according to the present invention is very good over a large spectrum of rates and modulations.

[0045] Presented below are the results obtained by using the PEXIT algorithm briefly discussed above to compute iterative decoding thresholds. Computation of iterative decoding thresholds for protograph coded modulation using the PEXIT algorithm is similar to that discussed in G. Liva, et al., for AWGN channel, except the initialization step. Instead of using the exact equation, as in the initialization step discussed in G. Liva, et al., the mutual information for coded bits is computed using the Monte Carlo method shown in Eqs. 7 and 8 above.

[0046] The iterative decoding thresholds of protograph-based LDPC coded BICM embodiments according to the present invention are shown tables presented in FIG. 11, FIG. 12 and FIG. 13. From FIG. 11, AR4JA protograph-based coded BICM can operate within about 0.7 dB to the BICM capacity limits. On the other hand, from FIG. 12, the family of protograph codes reported in T. V. Nguyen, et al. can operate within about 0.2-0.3 dB to the BICM capacity limit, which shows an improvement of about 0.4 dB compared with that of the AR4JA family. From FIG. 13, the family of rate-compatible protographs reported in T. V. Nguyen, et al., can operate within about 0.1-0.2 dB to the BICM capacity limit. These observations match with the results reported in T. V. Nguyen, et al. which provided the coding thresholds in AWGN channel. Discussed below is the generation of protograph codes for three modulation schemes with Gray labeling, i.e. Quadrature Phase Shift Keying (QPSK), 8-ary Phase Shift Keying (8PSK) and 16- ary Quadrature Amplitude Modulation (16QAM) and the calculated the iterative decoding thresholds for those protographs. FIG. 14 shows 8PSK with Gray labeling and FIG. 15 shows 16QAM with Gray labeling.

[0047] The protograph codes are built from protographs discussed above in two lifting steps. First, the protograph is lifted by a small factor in order to accommodate all modulation schemes. In the case of three above modulation schemes, the protograph is lifted by a factor of 12 which is a common divisor of 2, 3 and 4 bits. This first lifting procedure may be implemented by using the progressive edge growth (PEG) algorithm (such as discussed in X. Hu, et al.) in order to remove all multiple parallel edges. Other protograph lifting procedures known in the art may also be used. Secondly, the intermediate protograph is further circulantly lifted to an expected codeword length depending on different applications.

[0048] In order to support multiple rates, one can begin with the highest-rate (4/5) protograph. Since the protographs discussed above and shown in FIG. 3, FIG. 4, FIG. 5 and FIG. 6 are embedded, the parity-check matrix of lower rate can be obtained by removing certain columns from codes of higher rate. To decode the lower-rate codewords, the missing coded bits are replaced by erasure at the decoder. Thus the family of protograph codes can be implemented within a common encoder/decoder structure and at the same time being able to support adaptive modulation schemes as well. In order to support rate-compatible codes whose protomatrices are shown in FIG. 7, one can begin with the lowest rate (0.45) protograph. Other higher-rate codes are decoded by replacing missing parity bits by erasures at the decoder, therefore the same decoder can be used for the rate-compatible codes. [0049] The performance of the rate- 1/2 protograph code reported in T. V. Nguyen, et al., whose protomatrix is shown in FIG. 7 as well as that of rate- 1/2 AR4JA code transmitted with 16QAM in Rayleigh channel is plotted in FIG. 16 with the information block- length of 16k. Both these coded modulations perform within 1 to 1.2 dB of their capacity limit at 10 ^~6 FER.

[0050] The foregoing Detailed Description of exemplary and preferred embodiments is presented for purposes of illustration and disclosure in accordance with the requirements of the law. It is not intended to be exhaustive nor to limit the invention to the precise form or forms described, but only to enable others skilled in the art to understand how the invention may be suited for a particular use or implementation. The possibility of modifications and variations will be apparent to practitioners skilled in the art.

[0051] No limitation is intended by the description of exemplary embodiments which may have included tolerances, feature dimensions, specific operating conditions, engineering specifications, or the like, and which may vary between implementations or with changes to the state of the art, and no limitation should be implied therefrom. In particular it is to be understood that the disclosures are not limited to particular compositions or biological systems, which can, of course, vary. This disclosure has been made with respect to the current state of the art, but also contemplates advancements and that adaptations in the future may take into consideration of those advancements, namely in accordance with the then current state of the art. It is intended that the scope of the invention be defined by the Claims as written and equivalents as applicable. It is also to be understood that the terminology used herein is for the purpose of describing particular embodiments only, and is not intended to be limiting. Reference to a claim element in the singular is not intended to mean "one and only one" unless explicitly so stated. As used in this specification and the appended claims, the singular forms "a," "an," and "the" include plural referents unless the content clearly dictates otherwise. The term "several" includes two or more referents unless the content clearly dictates otherwise. Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which the disclosure pertains.

[0052] Moreover, no element, component, nor method or process step in this disclosure is intended to be dedicated to the public regardless of whether the element, component, or step is explicitly recited in the Claims. No claim element herein is to be construed under the provisions of 35 U.S.C. Sec. 112, sixth paragraph, unless the element is expressly recited using the phrase "means for . . . " and no method or process step herein is to be construed under those provisions unless the step, or steps, are expressly recited using the phrase "comprising step(s) for . . . "

[0053] A number of embodiments of the disclosure have been described. Nevertheless, it will be understood that various modifications may be made without departing from the spirit and scope of the present disclosure. Accordingly, other embodiments are within the scope of the following claims .