FIELD

The present disclosure relates to efficiently decodable Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) code. In particular, the present disclosure relates to efficiently decodable QC- LDPC code which is based on a base matrix of an irregular QC-LDPC matrix, the base matrix being formed by columns and rows, the columns being dividable into one or more columns corresponding to punctured variable nodes (i.e. variable nodes corresponding to information bits which are used by the encoder but are not transmitted to or effectively treated as not received by the decoder) and columns corresponding to not-punctured variable nodes, and the rows being dividable into high-density rows (i.e. rows having a weight which is above a first weight) and low-density rows (i.e. rows having a weight which is below a second weight, wherein the second weight is equal to or smaller than the first weight), wherein a matrix defined by the overlap of the low-density rows and the columns corresponding to the not-punctured variable nodes is dividable into groups of orthogonal rows. BACKGROUND

Fig. 1 shows a block diagram illustrating a digital communications system 10 in which processes of the present disclosure may be implemented. The digital communications system 10 includes a transmitting side comprising an encoder 12 and a receiving side comprising a decoder 14. The input of the encoder 12 at the transmitting side is, for example, an information sequence ISi of k bits to which a redundancy sequence of r bits is added in an encoding operation performed by the encoder 12, thereby producing an encoded information sequence IS2 of k + r = n bits which may be forwarded to a modulator 16.

The modulator 16 may transform the encoded sequence IS 2 into a modulated signal vector CH_IN which is in turn transmitted through a wired or wireless channel 18 such as, for example, a conductive wire, an optical fiber, a radio channel, a microwave channel or an infrared channel. Since the channel 18 is usually subject to noisy disturbances, the channel output CH OUT may differ from the channel input CH_I .

At the receiving side, the channel output vector CH_OUT may be processed by a demodulator 20 which produces some likelihood ratio. The decoder 14 may use the redundancy in the received information sequence IS3 in a decoding operation performed by the decoder 14 to correct errors in the received information sequence IS 3 and produce a decoded information sequence IS 4 (cf. M. P. C. Fossorier et al., "Reduced Complexity Iterative Decoding of Low- Density Parity Check Codes Based on Belief Propagation", IEEE TRANSACTIONS ON COMMUNICATIONS, May 1999, Volume 47, Number 5, Pages 673-680, and J. Chen et al., "Improved min-sum decoding algorithms for irregular LDPC codes", PROCEEDINGS OF THE 2005 IEEE INTERNATIONAL SYMPOSIUM ON INFORMATION THEORY, Pages 449-453, September 2005). The decoded information sequence IS 4 is an estimate of the encoded information sequence IS2 from which (an estimate of) the information sequence ISi can be extracted. The encoding operation and the decoding operation may be governed by an LDPC code. In the general formulation of channel coding, an LDPC code may employ a generator matrix G for the encoding operation performed by the encoder 12 and a parity-check matrix H for the decoding operation performed by the decoder 14. For an LDPC code with an information sequence ISi of size 1 x k , a codeword IS2 of size I x n , and a redundancy (parity) sequence of r = (n - k) bits, the generator matrix G has size k x n and the parity-check matrix H has size r n = {n - k)x n .

The parity-check matrix H _rxn and the generator matrix G _kxn enjoy the orthogonality property, which states that for any generator matrix G _kxn with k linearly independent rows there exists a parity-check matrix H _rxn with r = (n - k) linearly independent rows. Thus, any row of the generator matrix G _kxn is orthogonal to the rows of the parity-check matrix H _rxn such that the following equation is satisfied: kxn• H n r (1)

The encoding operation can be performed by means of a multiplication between the information sequence ISi and the generator matrix G _kxn , wherein the result of the multiplication is the encoded information sequence IS 2:

IS ₂ = IS G _t (2)

At the receiving side, due to the orthogonality property between the generator matrix G _k the parity-check matrix H _rxn , the following equation should be satisfied:

where IS4 is the decoded received information sequence of size 1 x n . If the above equation is verified, the information sequence estimate IS4 may be assumed to be correct.

Once the parity-check matrix H _rxn is generated, it is possible to obtain the generator matrix G _kxn and vice versa. Accordingly, any process of determining a parity-check matrix H _{r n} may be mapped to an equivalent process of obtaining a generator matrix G _kxn and vice versa, so that any process disclosed throughout the description and claims in relation to determining a parity- check matrix H _rxn shall be understood as encompassing the equivalent process of obtaining a generator matrix G _kxn and vice versa. Moreover, it should be noted that LDPC codes having a parity-check matrix H _rxn of a particular structure such as, for example, a parity-check matrix H _rxn having a parity part of dual diagonal structure allow the encoding of the information sequence ISi using (only) the parity-check matrix H _rxn so that obtaining the generator matrix G _kxn may not be required (cf. T. J. Richardson and R. L. Urbanke, "Efficient encoding of low-density parity-check codes", IEEE TRANSACTIONS ON INFORMATION THEORY, Volume 47, Issue 2, Pages 638-656, August 2002).

A particular form of the parity-check matrix H _rxn is a regular QC-LDPC matrix ^reg Hf _x ^c _n which can be divided into quadratic submatrices l{p _j ,), i.e. circulant matrices (or "circulants" for short), which may, for example, be obtained from cyclically right-shifting an N N identity matrix /(o) by p , positions:

with N ^' = nl ' L (cf. M. P. C. Fossorier, "Quasi-Cyclic Low-Density Parity-Check Codes from Circulant Permutation Matrices", IEEE TRANSACTIONS ON INFORMATION THEORY, Volume 50, Issue 8, Pages 1788-1793, August 2004). Thus, a regular QC-LDPC matrix ^reg H _xn may be defined by a base matrix B which satisfies:

Moreover, a base matrix B of an irregular QC-LDPC matrix ' ^RREG H _N may be obtained by _ β ₀ M _MASK where " o " denotes the Hadamard product and

1,0

M 1 ,1 -l

mask (6) mj -\,0 ^m j -\A J -l.L-l

denotes a mask matrix with m _; - _(i € {0,1}· Alternatively, the base matrix B of an irregular QC- LDPC matrix ^,RREG H ^C _W may be obtained by (only) partially labelling the base matrix B with shift values ρ · ₍ 6 {0 ... N] with not labelled entries (which are sometimes represented by a value of "-1" or an asterisk "*") representing zero matrices of size N x N .

Thus, for employing a QC-LDPC code in the encoder 12 and the decoder 14, the encoder 12 and the decoder 14 may be provided with a circulant, shift values, i.e., values corresponding to the labelled entries of the base matrix B, and (optionally) a mask matrix M _MASK . For instance, an apparatus configured to choose shift values for determining a QC-LDPC matrix H may provide the shift values to the encoder 12 and/or the decoder 14. Moreover, the encoder 12 and the decoder 14 may also be provided with a mask matrix M _MMK to generate one or more irregular QC-LDPC matrices ^,MG H . Furthermore, it is to note that a QC-LDPC matrix H ^L (and more generally any LDPC code) can also be described by its equivalent bipartite graph ("Tanner graph"), wherein each edge of the Tanner graph connects one variable node of a plurality of variable nodes to one check node of a plurality of check nodes. For example, a QC-LDPC matrix H^ _n of r rows and n columns can be represented by its equivalent bipartite graph with r check nodes and n variable nodes which has edges between the check nodes and the variable nodes if there are corresponding "Is" in the QC-LDPC matrix H^ _n (cf. R. Tanner, "A Recursive Approach to Low Complexity

Codes", IEEE TRANSACTIONS IN INFORMATION THEORY, Volume 27, Issue 5, Pages 533-547, September 1981). In this regard, it is to note that the variable nodes represent codeword bits and the check nodes represent parity-check equations.

While known approaches to channel coding have proven to perform well for a wide variety of scenarios, there is still an ongoing research to provide sophisticated solutions that achieve high data throughput with decent encoding/decoding resources.

SUMMARY According to a first aspect of the present invention, there is provided a method, the method comprising providing entries of a base matrix of an irregular QC-LDPC code for encoding or decoding a sequence of information bits, wherein the entries represent blocks of an irregular QC-LDPC matrix and each block represents a shifted circulant matrix or a zero matrix, dividing the rows of the base matrix into a first set and a second set, wherein the rows of the first set have a higher weight than the rows of the second set, selecting a number of columns of a matrix formed by the rows of the second set, wherein rows of a submatrix formed by the selected columns are divided into different groups, each group consisting of a maximum number of orthogonal rows, wherein the selecting is based on a number of different groups, and indicating information bits corresponding to not-selected columns as punctured. Puncturing the information bits corresponding to one or more a high weight columns of the irregular QC-LDPC matrix allows for layered decoding with regard to the different groups of "remaining" orthogonal subrows (or row-vectors) in combination with flooding decoding with regard to the high weight columns, thereby achieving a high degree of parallelism during decoding while maintaining high quality code. Thus, selecting columns that are not to be punctured strives at keeping the number of different groups of orthogonal subrows (or row- vectors) as high as possible while avoiding that information bits corresponding to too many columns (e.g., more than given by a threshold) are to be punctured.

In this regard, it is noted that the term "circulant matrix" as used throughout the description and claims in particular refers to a quadratic matrix of size Nx N , e.g., the identity matrix, where each row vector is shifted one element to the right relative to the preceding row vector. Moreover, the term "circulant size" refers to the size N of the circulant. Furthermore, the term "base matrix" as used throughout the description and claims in particular refers to an array labelled with shift values. Each shift value of the base matrix gives the number of times by which the rows of the circulant, e.g., the identity matrix, are to be cyclically (right-) shifted to generate a corresponding submatrix of the QC-LDPC matrix defined by the base matrix.

Moreover, the term "weight" as used throughout the description and claims in particular refers to the number of entries in a row or column of the base matrix that are labelled with shift values, i.e. the entries in the rows or columns of the base matrix that do not represent zero matrices, which is equal to the number of "Is" in the corresponding rows and columns of the QC-LDPC matrix. In this regard, it is noted that the term "weight" as used throughout the description and claims can be interchanged by the terms "node degree" or "density" which have the same or a similar meaning. Furthermore, the term "punctured" as used throughout the description and claims in relation to information bits (or the corresponding variable nodes or the corresponding columns) in particular indicates that the information bits are only used by the encoder but are not transmitted to or effectively treated as not received by the decoder. Even further, the term "corresponding" as used throughout the description and claims in relation to columns, nodes, and information bits in particular refers to the mapping between columns and variable nodes/information bits in terms of the Tanner graph representation of the QC-LDPC matrix.

Furthermore, it is to be noted that values forming a "matrix" do not necessarily have to be physically stored or presented in matrix- (or array-) form, or used in matrix algebra throughout a process involving the matrix. Rather the term "matrix" as used throughout the description and claims may equally refer to a set of (integer) values with assigned row and column indices or to a set of (integer) values which are stored in a (logical) memory array. Moreover, if not involving matrix algebra or if respective matrix algebra routines are suitably redefined, the notion of rows and columns may even be changed or freely chosen. However, throughout the description and claims, it is adhered to the mathematical concepts and notations regularly used in the art and they shall be understood as encompassing equivalent mathematical concepts and notations.

In a first possible implementation form of the method according to the first aspect, the number of not-selected columns is one or two. Hence, there may be a relatively small number of not-selected columns which, however, have a relatively high weight (e.g., more than two times or three times the mean weight of the selected columns) thereby providing good "connectivity" of the groups of orthogonal subrows (or row vectors) while effectively allowing for a high degree of parallelism during decoding due to the relatively small number of columns which have to be "separated" (or divided into non- overlapping sets or groups) to achieve orthogonality within the groups of remaining (selected) subrows (or row vectors).

In a second possible implementation form of the method according to the first aspect as such or according to the first implementation form of the first aspect, selecting the number of columns of the matrix formed by the rows of the second set comprises ordering or grouping the columns of the matrix formed by the rows of the second set by weight and selecting columns having weights below a threshold.

Thus, groups with a large number of orthogonal subrows (or row vectors) can be achieved, which allows for a higher degree of parallelism during decoding.

In a third possible implementation form of the method according to the first aspect as such or according to the first or second implementation form of the first aspect, a matrix consisting of a subset of columns of a matrix formed by the rows of the first set has a dual diagonal or triangular structure. Hence, the high-density part of the irregular QC-LDPC matrix facilitates encoding by having a parity part with a dual diagonal or triangular structure. This also improves the susceptibility of the code to rate-adaptiveness as the number of rows (and corresponding columns if, for example, removing a row would leave an empty column) removed from the low-density part allows to change the rate of the code without, however, effectively touching on the encoding/decoding properties of the high-density part. In this regard, the method may also comprise removing a number of rows (and corresponding columns) of the second set of the irregular QC-LDPC matrix to adapt a rate of the irregular QC-LDPC code.

In a fourth possible implementation form of the method according to the first aspect as such or according to any one of the first to the third implementation forms of the first aspect, a matrix consisting of a subset of columns of the matrix formed by the rows of the second set has a triangular or identity matrix structure.

Thus, encoding may be performed in a two-step procedure comprising encoding an input sequence based on subcolumns (or column vectors) of the high-density part and encoding the encoded output sequence based on subcolumns (or column vectors) of the low-density part, thereby utilizing a raptor-like encoding process.

In a fifth possible implementation form of the method according to the first aspect as such or according to any one of the first to the fourth implementation forms of the first aspect, rows of a matrix formed by columns of the matrix formed by the rows of the first set which correspond to non-punctured information bits are divided into different groups, each group consisting of orthogonal rows.

Hence, groups (or subsets) of orthogonal subrows (or row vectors) can be formed in the high- density set and in the low-density set which allows for an even higher degree of parallelism during decoding.

In a sixth possible implementation form of the method according to the first aspect as such or according to any one of the first to the fifth implementation forms of the first aspect, the method further comprises determining a codeword corresponding to the sequence of information bits based on the provided entries of the base matrix and transmitting the codeword except for information bits that are indicated as punctured.

Thus, a rate of the QC-LDPC code can be increased.

In a seventh possible implementation form of the method according to the first aspect as such or according to any one of the first to the sixth implementation forms of the first aspect, the method further comprises decoding a received sequence of information bits based on the provided entries of the base matrix and information about which information bits are punctured, wherein the decoding comprises flooding and layered decoding operations, wherein layers correspond to the different groups. Thus, decoding convergence can be improved while maintaining high parallelism of the decoding process. In this regard, it is noted that the term "layered decoding" as used throughout the description and claims in particular refers to a decoding process where rows of a layer are processed in parallel but layers are processed (substantially) consecutively.

According to a second aspect of the present invention, there is provided a decoder, the decoder comprising a non-transient memory storing entries of a base matrix of an irregular QC-LDPC code, wherein columns of the base matrix are divided into a first set and a second set, the first set comprising one or more columns and the columns of the second set forming a matrix comprising groups of orthogonal rows, wherein the decoder is configured to decode a received sequence of information bits based on a flooding decoding process for variable nodes corresponding to the one or more columns of the first set and a layered decoding process for nodes corresponding to the columns of the second set.

Thus, decoding convergence can be improved while allowing to maintain code quality and high parallelism of the decoding process, thereby enabling high throughput at a low error rate.

In a first possible implementation form of the decoder according to the second aspect, the variable nodes corresponding to the one or more columns of the first set are indicated as punctured. Thus, a rate of the irregular QC-LDPC code can be increased.

In a second possible implementation form of the decoder according to the second aspect as such or according to the first implementation form of the second aspect, the number of columns in the first set is one or two. Hence, there may be a relatively small number of columns in the first set which, however, may have a relatively high weight (e.g., more than two times or three times the mean weight of the columns of the second set) thereby improving the "connectivity" of the layers.

In a third possible implementation form of the decoder according to the second aspect as such or according to the first or second implementation form of the second aspect, rows of the base matrix are divided into a first set and a second set, wherein the rows of the first set have a higher weight than the rows of the second set.

This makes the code even more susceptible to rate adaption by removing (or disregarding) rows of the second set (and corresponding columns of the second set) without substantially deteriorating quality of the code. In a fourth possible implementation form of the decoder according to the third implementation form of the second aspect, a matrix consisting of a subset of columns of a matrix formed by the rows of the first set has a dual diagonal or triangular structure.

Hence, the high-density part of the irregular QC-LDPC matrix facilitates encoding by having a parity part with a dual diagonal or triangular structure. In a fifth possible implementation form of the decoder according to the third or fourth implementation form of the second aspect, a matrix consisting of a subset of columns of the matrix formed by the rows of the second set has a triangular or identity matrix structure. Thus, encoding may be performed utilizing a raptor-like encoding process which reduces or obviates the need for requesting retransmissions by the decoder.

In a sixth possible implementation form of the decoder according to any one of the third to fifth implementation forms of the second aspect, rows of a matrix formed by overlapping entries of the columns of the second set and the rows of the first set are divided into different groups, each group consisting of orthogonal rows.

Hence, a higher degree of parallelism during decoding can be achieved.

According to a third aspect of the present invention, there is provided a non-transient computer- readable medium storing instructions which, when carried out by a computer cause the computer to provide a base matrix of an irregular QC-LDPC matrix, the base matrix being formed by columns and rows, the columns being dividable into one or more columns corresponding to punctured variable nodes and columns corresponding to not-punctured variable nodes, and the rows being dividable into first rows having a weight which is above a first weight and second rows having a weight which is below a second weight, wherein the second weight is equal to or smaller than the first weight, wherein an overlap of the second rows and the columns corresponding to the not-punctured variable nodes is dividable into groups of orthogonal row-vectors.

Puncturing the variable nodes corresponding to one or more a high weight columns of the base matrix of the irregular QC-LDPC matrix allows for layered decoding with regard to the different groups of "remaining" orthogonal subrows (or row-vectors) in combination with flooding decoding with regard to the high weight columns, thereby achieving a high degree of parallelism during decoding while maintaining high quality code.

BRIEF DESCRIPTION OF THE DRAWINGS

Fig. 1 shows a schematic illustration of a digital communication system; Fig. 2 shows a flow chart of a process of providing an irregular QC-LDPC code for encoding or decoding a sequence of information bits;

Fig. 3 shows a structure of a base matrix of an irregular QC-LDPC code;

Fig. 4 shows a base matrix of an irregular QC-LDPC code; Fig. 5 shows a first part of a flow chart of a decoding process;

Fig. 6 shows a second part of the flow chart of the decoding process;

Fig. 7 shows a third part of the flow chart of the decoding process;

Fig. 8 shows a schedule of a first hardware implementation of the decoding process; and

Fig. 9 shows a schedule of a second hardware implementation of the decoding process. DETAILED DESCRIPTION

Fig. 2 shows a flow chart of a process 22 of providing an irregular QC-LDPC code for encoding or decoding a sequence of information bits, such as information sequence ISi and Z¾, respectively. The process 22 may, for example, be computer-implemented. For instance, the process 22 may be implemented by persistently stored computer-readable instructions which, if executed by a computer, cause the computer to perform the process 22. The provided base matrix B of the irregular QC-LDPC code may, for example, be provided to the encoder 12 and the decoder 14 of the digital communication system 10 and used for encoding or decoding operations performed by the encoder 12 and the decoder 14, respectively, i.e., for encoding or decoding the sequence of information bits. The process 22 of providing an irregular QC-LDPC code for encoding or decoding a sequence of information bits may start at step 24 with providing entries of a base matrix B of an irregular QC-LDPC code, wherein the entries represent blocks of an irregular QC-LDPC matrix and each block represents a shifted circulant matrix or a zero matrix. A possible structure of the base matrix B is shown in Fig. 3. It comprises a "core" base matrix in the high-density part (indicated in grey on the upper left of Fig. 3). The core base matrix has a parity part with a dual diagonal structure for easy encoding. If a highest rate is required, an information sequence will be encoded using only the shift values of the core base matrix. If lower rates are acceptable, additional rows and columns can be appended to the base matrix. As shown in Fig. 3, an overlap between the additional rows and columns may form an identity matrix although a lower triangular form would also be possible. The additional rows typically have a lower weight than the rows of the core base matrix and provide (in combination with the added 'corresponding' columns) for additional parity bits in the codeword to be transmitted.

The extension part comprises one, two, three, or more high-weight columns which typically have a substantially higher weight than all other columns of the extension part. For example, one, two, or all high-weight columns may have no empty cells, i.e. no entries representing the zero matrix. As shown in Fig. 3, the variable nodes corresponding to two high- weight columns are indicated as punctured and the "remaining" subrows are grouped into (non-overlapping) layers of orthogonal subrows (or row vectors).

Fig. 4 shows a numeric example of provided entries of a base matrix B of size 19x35 wherein labelled entries (cells) of the base matrix B are indicated by the corresponding shift values and not-labelled entries (corresponding to zero matrices) are left blank. As shown in Fig. 4, the rows of the base matrix B can be divided into an upper part having a weight of above 17 and a lower part having a weight of below 9, i.e. less than half the weight of the rows of the upper part. Thus, the base matrix B shown in Fig. 4 can be divided into a high-density part comprising rows 1 to 3 and a low-density part comprising rows 4 to 19 as indicated at step 26 of the process 22 shown in Fig. 2.

Moreover, as shown in Fig. 4, the rows of the submatrix formed by the overlap of columns 2 to 35 and the low density-part can be divided into layers (or groups) of orthogonal rows, wherein each layer comprises about the same number of cells. Furthermore, the high-density part comprises a dual diagonal submatrix allowing to easily encode a sequence of information bits based on the non-zero columns of the high-density part. Moreover, the low-density part provides a raptor-like extension with a parity part which has a lower triangular form which allows for easy encoding of the codeword.

As further indicated at step 28 of the process 22 illustrated in Fig. 2, column 1 of the base matrix B is punctured. After encoding a sequence of information bits by the encoder 12 based on the provided entries of the base matrix B and transmitting the corresponding codeword (except for the information bits corresponding to the punctured nodes) via the channel 18 to the decoder 14, the encoder 14 may iteratively decode the received information bits using a normalized Min-Sum decoding process combining flooding and layered decoding steps as illustrated by steps A, B, and C of Fig. 5, Fig. 6, and Fig. 7. Moreover, in the decoding process, it can be taken advantage of the fact that the shift values of the punctured column that correspond to the extended part of the matrix (shown in dark grey in Fig. 4) can be set to zeros using row and column shifting operations. Furthermore, parallelism of the decoding operation can be increased by providing for groups of orthogonal rows in the portion of the dense part to which layered decoding steps are applied (i.e., the part corresponding to the not-punctured columns).

As indicated by step A shown in Fig. 5, the log-likelihood-ratios (llr-s) of the punctured columns are calculated using the following formulas with llr, sgO _j , sg _j , miri _j , submiri _j , col _j , v2c _Jt csg _j , c2v _j , nsgO _j , cmiri _j , psg _Jt pmin _j , psubmiri _j , pcol _j , nsg _j , nmiri _j . nsubmin _j , ncolj, and nc2v _j being vectors of length N and alpha denoting the scale parameter of the normalized Min-Sum decoding process:

• llr denotes the vector of llr-s of the punctured node, which may be stored in application- specific integrated circuit (ASIC) registers.

• sgO _j denotes the signs of the variable to check messages (v2c) of the punctured nodes and the y ^' -th row inside a group, which may be stored in random access memory (RAM). · sg _j , miri _j , submiri _j , col _j denote multiplications of signs, minimums, sub-minimums and zero-based argMinimums in the 7-th row of the given orthogonality group with 1 < = j < = n. All these values may be calculated before starting the decoding process and stored in memory. • P ^s gJ> prnirij, psubmirij, pcolj denote updated multiplications of signs, minimums, sub- minimums and argMinimums in the y ^' -th row of current orthogonality group. These values are to be determined for each column but the punctured one.

First, csgj, \c2v _j \, and c2v are calculated for / < = j < = n by: · csgj = sg _j * sgO _j ,

• \c2vj\ = (colj ==0)? submin/. min _} , and

• c2v ^~ csg*\c2v\.

Then, nsgOj, cmiri _j , and v2c _j , are calculated by:

• nsgOj = sign(v2c ),

· cmiri _j = \ v2c _j |* alpha,

Now, as indicated by step B shown in Fig. 6, new minimums, sub-minimums and argMinimums, as well as signs of the punctured column in each row of the current orthogonality group are calculated for 7 < =j < = n by: · sgO _j = nsgOj

• nsgj = nsgOj * psgj,

• ncol _j = (cmirij > prniri _j ) ? pcolj : 0,

• nmirij = (cmirij > pminj) ? prnirij : cmiri _j , and

• nsubmiri _j - (cmiri _j > prniri _j ) ? (( cmiri _j > psubmiri _j ) ? psubmiri _j : cmiri _j ) : prniri _j . These values are stored in memory where sgj, miri _j , submirt _j , colj, and sgO _j replace the currently stored values for the next decoding iteration.

Finally, psum, c2vsum, and the new llr-s of the punctured node are calculated as indicated by step C shown in Fig. 7 by: • psum = psgi * pmini +...+ psg _n * pmin _n ,

• c2vsum = c2vi +...+ c2v _n , and

• llr = llr - c2vsum + psum.

Moreover, the proposed scheme can be efficiently implemented in hardware as will become apparent from the following example in which each group of orthogonal rows is processed in 3 clock cycles, all llr-s are stored in registers, the number of available processors equals the number of columns of the QC-LDPC matrix and sgj, mirij, submirij, col _} are loaded to registers before the 1 ^st clock cycle begins.

Processing of the non-punctured columns is done using the same scheme, and processing of the punctured column is done according to the above described formulas:

Clock cycle 1. For all columns but the punctured one, c2v messages are calculated. The calculated c2v messages are subtracted from llr-s, so that v2c messages are obtained which are stored in the same registers in which the llr-s were stored. The obtained v2c messages are used to determine nsgO and cmin which are used for determining partial minimums. Also, at the clock cycle 1 , llr-s of the punctured columns of the previous group are obtained according. After that, these llr-s are shifted and stored on registers. As a result, llr-s of the punctured columns are calculated 1 clock cycle later that llr-s of other columns, but they are also used 1 clock cycle later (at the second clock cycle).

Clock cycle 2. Minimums (i.e. psgj, pminj, psubmin _Jt pcol _j ) are calculated from the partial minimums. Also, nsgO, cmin and c2v are calculated for the punctured column. After that, values of nsg _j , nminj, nsubmin _j , ncol _j are calculated and stored in memory.

Clock cycle 3. From the obtained values of nsg _j , nmin _j , nsubmin _j , ncol _j for all columns but the punctured ones, new c2v messages can be calculated. After that, these values are summed up with v2c-messages and llr-s of all non-punctured columns are obtained. The obtained llr-s are then stored in registers. Also, at this clock cycle, psum and c2vsum are calculated. If more than one column of the base matrix is indicated as punctured, the above modified Min- Sum decoding process is to be extended accordingly. For example, processing of two punctured columns can be done according to the following scheme:

Clock cycle la) For all non-punctured columns c2v messages are calculated. c2v messages are subtracted from llr-s. As a result, v2c messages are obtained that can be stored on the same registers as llr-s. These v2c messages can be used to get values of nsgO and cmin. These values are used to get partial minimums.

Clock cycle lb) Also at the 1 ^st clock cycle llr-s of the 2 punctured columns can be obtained for the previous orthogonality group. After that, these llr-s can be shifted and stored in registers. As a result, llr-s for the punctured columns can be obtained 1 clock later but causes no problem because they are needed 1 clock later.

Clock cycle 2a) Minimums are collected from partial minimums, i.e. values for psgj, pmiri _j , psubmirij, and pcol _j are obtained.

Clock cycle 2b) At this clock cycle nsgO, cmin and c2v are calculated for 2 punctured columns. After that, the values of nsgj, nmirij, nsubmirij, and ncol _j are calculated and stored in memory.

Clock cycle 3a) From the obtained values of nsgj, nmin nsubminj, ncol _j for all non-punctured columns, messages c2v are calculated, which after being summed up with v2c messages, llr-s of all non-punctured columns are given. The llr-s are shifted and stored in registers.

Clock cycle 3b). At this clock cycle, messages nc2v and sums c2vsum and nc2vsum are calculated.

If using above described scheme, a special processor for puncture nodes may be needed for steps la), 2a), 3a), and also a processor may be needed for each not-punctured column which performs operations lb), 2b), 3b). If the QC-LDPC matrix has m groups of orthogonal subrows, 3*m Clocks per iteration may be required. Every sub-processor la), 2a), 3a), l b), 2b), 3b) will have a stall for 2 clocks from 3 available. Another even more memory efficient scheme may be used with 4 clocks per processor but less processors (1 processor for the punctured columns and one processors for four not-punctured columns).

Clock la) Calculations from la) are performed for a first quarter of the non-punctured columns. Clock lb) llr-s of punctured columns of the first row of the previous group are calculated. After that, the llr-s are shifted and stored in registers.

Clock Ila) Actions of 2a) are performed for the non-punctured columns and actions of 1 a) are performed for a second quarter of the non-punctured columns.

Clock lib) nsgO, cmin and c2v are calculated for the punctured columns and nsg _j , nmin _} , nsubmirij, and ncol _j are obtained and stored in registers.

Clock Ilia) Actions of 3 a) are performed for the first quarter of not-punctured. Actions of 2a) are performed for the second quarter of the not-punctured columns. Actions of la) are performed for a third quarter of not-punctured columns.

Clock Illb) Values of nsgj, nmirij, nsubmirij, and ncolj are determined and stored in memory. Also, c2vsum sums are calculated.

Clock IVa) Actions of 3 a) are performed for the second quarter of the non-punctured columns. Actions of 2a) are performed for the third quarter of the non-punctured columns. Actions of l a) are performed for the fourth quarter of the non-punctured columns.

Clock IVb) Messages nc2v and sums nc2vsum are calculated. A generalization of this approach is possible, if, for example, the number of processors is decreased and the throughput is decreased correspondingly. When denoting the processing steps with letters A-I:

• A - Calculating c2v for non-punctured nodes. Determining and storing v2c in registers replacing llr-s. Calculating nsgO and cmin. Calculating partial minimums.

• B - Getting minimums from partial minimums, i.e. psgj, pmiri _j , psubmiri _j , pcol _j .

• C - Calculating new c2v from nsgj, nmiri _j , nsubmirij, ncol _j for all non-punctured nodes, summing them up with v2c to obtain llr-s of these columns, shifting them and storing in registers.

• D - Calculating nsgO, cmin and c2v for punctured nodes.

• E - Determining nsgj, nminj, nsubmin _j , and ncol _j .

• F - Calculating c2vsum.

• G - Calculating nc2vsum.

• H - Calculating nc2v.

• I - Calculating llr-s of punctured columns. Shifting and storing them in registers. and having, for example, three groups of orthogonal subrows (in the high-density and the low- density part) each comprising two orthogonal rows which are denoted as:

• 1 -2 - 1 ^st group

• 3-4 - 2 ^nd group

• 5-6 - 3 ^rd group and twelve not-punctured columns with:

• j. l - first half of non-empty cells in the y ^' -th row, 1 <=j <= 6 and

• j.l - second half of non-empty cells of the y ^' -th row, 1 <=j <= 6, wherein actions A, B, and C are performed for non-punctured columns and actions D, E, F, G, H, and I are performed for punctured columns, Fig. 8 and 9 depict the schedule for the 3-clock cycle scheme and the schedule for the 4-clock cycle scheme. As can be seen from Fig. 8 and 9, high parallelism can be achieved. W 201

Moreover, it is to be noted that in addition to enabling the encoder 12 and the decoder 14 to perform encoding and decoding operations on basis of the provided base matrix B, the encoder 12 and the decoder 14 may also use the provided base matrix B to derive irregular QC-LDPC child codes of different rates in accordance with different transmission scenarios, e.g., transmission scenarios which differ from each other in view channel quality and/or throughput requirements, by, for instance, removing (or neglecting) rows of the low-density part and/or columns of the parity part of the provided base matrix B.