DE102017206718B4 - Method for generating a class of non-binary LDPC codes - Google Patents

Abstract

Method for generating a class of non-binary LDPC codes with a predeterminable number of check nodes and with a predeterminable number of variable nodes for data transmission with PSK modulation over a non-coherent channel with the following steps - concatenating an outer code and one inner code, whereby the outer code extends over a mathematical body of order q and the inner code is designed over a mathematical ring, also of order q, and where a symbol of the outer code is mapped onto a symbol of the inner code, - specifying a protograph , which is defined by a base matrix whose number of rows is equal to the number of check nodes of the protograph and whose number of columns is equal to the number of variable nodes of the protograph and whose integer elements describe the connection of the check nodes to the variable nodes, - gradual expansion of the base matrix to create a parity check matrix describing the LDPC code to be generated, - each element of the base matrix or the matrix expanded in the previous step is replaced by a matrix whose row/column weighting is equal to the element to be replaced ,- where the expansion is carried out until the specified size of the parity check matrix is reached and all elements in the resulting matrix have either the value 1 or 0,- where the base matrix and each expanded matrix have a square rear matrix part, whose Number of rows is equal to the number of columns, and has a non-square front matrix part, - the elements of the front matrix part of the base matrix being freely selectable integers which are converted into either the value 1 or 0 in the front part of the resulting expanded matrix by expansion and are then replaced by integer elements from a mathematical body in order to generate the front part of the parity check matrix, these elements describing the symbols of the outer code, - the square rear matrix part of the base matrix only containing elements that are either 0, 1 or 2 and which are arranged in such a way that the resulting square rear matrix part of the parity check matrix has a double diagonal structure after the base matrix has been gradually expanded, the double diagonal only having elements with the value 1 and outside the double diagonal only having elements with the Value 0 exist, whereby these elements are elements from a mathematical ring and describe the symbols of the inner code, - where the elements of the front matrix part of the parity check matrix are intended for carrying out mathematical operations in body algebra and - where the elements of the rear Matrix part of the parity check matrix, intended for performing mathematical operations in ring algebra and implementing a phase accumulator for the non-coherent channel.

Description

The invention relates to a method for generating a class of non-binary LDPC codes with a predeterminable number of check nodes and with a predeterminable number of variable nodes, in particular for use for data transmission in channels with phase noise (non-coherent channels).

• - mehrfaches Kopieren des Protograph zum Erhalt von f Instanzen (wobei f entsprechend der Ziel-Blocklänge frei gewählt ist und als Expansionsfaktor bezeichnet wird),
• - Permutieren der Ränder unter den Kopien derart, dass die Umgebung jedes VN und CN beibehalten wird.

LDPC codes were originally proposed by Gallager [3] in 1963, but were forgotten for a long time. It took until the mid-1990s for their potential to be recognized. Since then, intensive research has been conducted in the field of LDPC codes with the aim of improving code performance while keeping the complexity of encoding and decoding low. This culminated in the discovery of the protograph-based LDPC codes, which are among the best known binary codes (at least for moderate to long blocks). In general, a protograph is a bipartite graph with a relatively small number of nodes, with several parallel edges between variable nodes (VN) and check nodes (CN) permitted. The Protograph serves as a blueprint for larger graphs (and thus longer codes) obtained by

• - multiple copies of the Protograph to obtain f instances (where f is freely chosen according to the target block length and is referred to as the expansion factor),
• - Permute the edges among the copies such that the environment of each VN and CN is preserved.

Apart from being represented as a bipartite graph, a protograph can also be described using its basis matrix B. For example, the protograph can be imagined with the following basic matrix: $$\mathbf{b}=\left[\begin{array}{ccc}1& 2& 0\\ 1& 0& 2\end{array}\right]$$

The corresponding graph results in the in 1 form shown, and a simple expansion can be done as in 2 be trained as shown.

Note that after the copy and permutation step, each VN of type A is coupled with a CN of types A and B as in the original basis matrix. The same applies to the remaining VNs and CNs. Thus, the environment is maintained after expansion.

Thanks to their very good performance, low encoding and decoding complexity, and low implementation complexity, Irregular Repeat Accumulate (IRA) codes are a popular choice among LDPC codes. An IRA code can be obtained by repeating the information symbols in an irregular manner, adding the repetitions and sending the result to an accumulator. In fact, IRA codes can also be used as protographs in the in 2 shown form.

Numerous protographs (ARA-like, IRA-like or others) exist in the prior art. These protographs are usually not used for transmission over non-coherent channels, i.e. over-channels for which the phase information is not known at the receiver. Thus, the LDPC codes on such channels usually show poor performance. A solution to this problem is proposed with the protographs presented here.

A method for generating a class of non-binary LDPC codes with a predetermined number of check nodes and with a predetermined number of variable nodes is available US 2013/0013983 A1 known. In this method, a protograph is specified which is defined by a base matrix, the number of rows of which is equal to the number of check nodes of the protograph and the number of columns of which is equal to the number of variable nodes of the protograph and whose integer elements are the connection of the check nodes to the variable nodes describes. The base matrix is expanded step by step to create a parity check matrix describing the LDPC code to be generated by replacing each element of the base matrix or the matrix expanded in the previous step with a matrix whose row/column weighting is equal to the element to be replaced and the expansion is carried out until the specified size of the parity check matrix is reached and all elements in the resulting matrix have the value either 1 or 0. The base matrix and each expanded matrix have a square rear matrix part, the number of rows of which is equal to the number of columns, and a non-square front matrix part. The elements of the front matrix part of the base matrix are arbitrary integers that are converted by expansion to either the value 1 or 0 in the front part of the resulting expanded matrix, which are then replaced by integer elements from a mathematical field to form the front part to generate the parity check matrix. The square back matrix part of the base matrix contains only elements that are either 0, 1 or 2 and which are arranged in such a way that after gradually expanding the base matrix, the resulting square back matrix part of the parity Check matrix has a double diagonal structure, where the double diagonal only has elements with the value 1 and where outside the double diagonal only elements with the value 0 exist. In the known method, only body algebra operations are used for both the front and back parts of the matrix.

[2] gives an overview of different binary codes for the AWGN channel. Some of these codes contain an accumulator and can be represented as protographs with a corresponding base matrix.

In EP 2 843 844 A1 We proposed a ternary modulation for channels with phase noise in combination with spatially coupled LDPC codes. The LDPC code is preferably assumed to be ternary in Galois body arithmetic.

[7] deals with hybrid LDCP codes, i.e. LDPC codes whose symbols belong to different Galois fields (e.g. binary and non-binary). The focus here is on the AWGN channel.

In [8], accumulator-based non-binary LDPC codes for the AWGN channel are treated in the Galois field algebra. In terms of structure, this is similar to the invention report, although in the invention report the accumulator assumes ring algebra.

The object of the invention is to provide a non-binary LDPC code with little effort and low coding and decoding complexity that has good properties even for non-coherent channels.

• - Konkatenieren eines äußeren Codes und eines inneren Codes, wobei sich der äußere Code über einen mathematischen Körper der Ordnung q erstreckt und der innere Code über einen mathematischen Ring ebenfalls der Ordnung q ausgelegt ist und wobei ein Symbol des äußeren Codes auf ein Symbol des inneren Codes abgebildet wird,
• - Vorgeben eines Protographs, der durch eine Basismatrix definiert ist, deren Zeilenanzahl gleich der Anzahl an Check Nodes des Protographs und deren Spaltenanzahl gleich der Anzahl an Variable Nodes des Protographs ist und deren ganzzahlige Elemente die Verbindung der Check Nodes mit den Variable Nodes beschreibt,
• - schrittweises Expandieren der Basismatrix zur Erstellung einer den zu erzeugenden LDPC-Code beschreibenden Parity-Check Matrix,
• - wobei jedes Element der Basismatrix bzw. der in dem jeweils vorherigen Schritt expandierten Matrix durch eine Matrix ersetzt wird, deren Zeilen-/Spalten-Gewichtung gleich dem zu ersetzenden Element ist,
• - wobei die Expansion solange durchgeführt wird, bis die vorgegebene Größe der Parity-Check Matrix erreicht ist und alle Elemente in der resultierenden Matrix entweder den Wert 1 oder 0 besitzen,
• - wobei die Basismatrix und jede jeweils expandierte Matrix einen quadratischen hinteren Matrixteil, dessen Zeilenanzahl gleich der Spaltenanzahl ist, und einen nicht quadratischen vorderen Matrixteil aufweist,
• - wobei die Elemente des vorderen Matrixteils der Basismatrix frei wählbare ganze Zahlen sind, die in dem vorderen Teil der resultierenden expandierten Matrix durch Expansion entweder in den Wert 1 oder 0, umgewandelt und danach durch ganzzahlige Elemente aus einem mathematischen Körper ersetzt werden, um den vorderen Teil der Parity-Check Matrix zu erzeugen, wobei diese Elemente die Symbole des äußeren Codes beschreiben,
• - wobei der quadratische hintere Matrixteil der Basismatrix ausschließlich Elemente enthält, die entweder 0, 1 oder 2 sind und die so angeordnet sind, dass der resultierende quadratische hintere Matrixteil des Parity-Check Matrix nach schrittweisem Expandieren der Basismatrix eine Doppeldiagonalen-Struktur aufweist, wobei die Doppeldiagonale ausschließlich Elemente mit dem Wert 1 aufweist und wobei außerhalb der Doppeldiagonalen ausschließlich Elemente mit dem Wert 0 existieren, wobei diese Elemente Elemente aus einem mathematischen Ring sind und die Symbole des inneren Codes beschreiben,
• - wobei die Elemente des vorderen Matrixteils der Parity-Check Matrix, zur Durchführung von mathematischen Operationen in Körperalgebra vorgesehen sind und
• - wobei die Elemente des hinteren Matrixteils der Parity-Check Matrix, zur Durchführung von mathematischen Operationen in Ringalgebra vorgesehen sind und einen Phasenakkumulator für den nichtkohärenten Kanal implementieren.

To solve this problem, the invention provides a method for generating a class of non-binary LDPC codes with a predeterminable number of check nodes and with a predeterminable number of variable nodes, in particular for use for data transmission in channels with phase noise (non- coherent channels), with the following steps proposed:

• - Concatenation of an outer code and an inner code, whereby the outer code extends over a mathematical body of order q and the inner code is designed over a mathematical ring, also of order q, and where a symbol of the outer code corresponds to a symbol of the inner code is depicted,
• - specifying a protograph that is defined by a base matrix, the number of rows of which is equal to the number of check nodes of the protograph and the number of columns of which is equal to the number of variable nodes of the protograph and whose integer elements describe the connection of the check nodes to the variable nodes,
• - gradual expansion of the base matrix to create a parity check matrix describing the LDPC code to be generated,
• - where each element of the base matrix or the matrix expanded in the previous step is replaced by a matrix whose row/column weighting is equal to the element to be replaced,
• - the expansion is carried out until the specified size of the parity check matrix is reached and all elements in the resulting matrix have either the value 1 or 0,
• - whereby the base matrix and each expanded matrix have a square rear matrix part, the number of rows of which is equal to the number of columns, and a non-square front matrix part,
• - where the elements of the front matrix part of the base matrix are freely selectable integers which are converted into the value either 1 or 0 by expansion in the front part of the resulting expanded matrix and are then replaced by integer elements from a mathematical field in order to form the front one To create part of the parity check matrix, where these elements describe the symbols of the outer code,
• - wherein the square rear matrix part of the base matrix contains exclusively elements that are either 0, 1 or 2 and which are arranged such that the resulting square rear matrix part of the parity check matrix has a double diagonal structure after gradually expanding the base matrix, whereby the Double diagonal has only elements with the value 1 and where outside the double diagonal only elements with the value 0 exist, these elements being elements from a mathematical ring and describing the symbols of the inner code,
• - wherein the elements of the front matrix part of the parity check matrix are intended for carrying out mathematical operations in body algebra and
• - wherein the elements of the rear matrix part of the parity check matrix are intended to carry out mathematical operations in ring algebra and implement a phase accumulator for the non-coherent channel.

• - niedrige Codier-Komplexität (Codierer sind durch Akkumulatoren zu realisieren),
• - niedrige Decodier-Komplexität (mindestens auf kohärenten Kanälen),
• - gute Leistungsfähigkeit (z.B. hinsichtlich der Codeword Error Rate (CER) oder der Bit Error Rate (BER)) auf kohärenten und nicht kohärenten Additive White Gaussian Noise (AWGN) Kanälen.

The subject of the invention is a novel code design for non-binary LDPC codes, which is based on mathematical operations in ring and field algebra. The codes can be described by a design of a protograph [1], which makes it possible to obtain forward error correction (FEC) codes with practically freely selectable bit lengths through a certain expansion (lifting) and puncturing. When expanded correctly, the resulting codes resemble punctured Irregular Repeat Accumulate (IRA) codes [2]. However, according to the invention, all operations performed on the double diagonal part of the parity check matrix of the code are the subject of calculations over a ring, while all calculations affecting the information part of the parity check matrix are the subject of operations over a field (ie about a body or a ring). The double diagonal part is designed to implement a phase accumulator for non-coherent channels with PSK modulations. Due to their special design, the codes allow

• - low coding complexity (encoders can be implemented using accumulators),
• - low decoding complexity (at least on coherent channels),
• - good performance (e.g. in terms of Codeword Error Rate (CER) or Bit Error Rate (BER)) on coherent and non-coherent Additive White Gaussian Noise (AWGN) channels.

wobei I eine k x k -Identitätsmatrix, D eine (m x m)-Matrix mit einem Reihen-/Spalten-Gewicht von 2 und C eine (m-k x k)-Matrix ist, deren Einträge von dem Code-Designer abhängen.Let the number of rows of a photographer be denoted by m and the number of columns by n, where n>=m. Assume that nm=k. The following set of protographs is specified by their base matrices: $$\mathbf{b}=\left[\begin{array}{c}\text{I}\\ \text{C}\end{array}\text{D}\right],$$

where I is a kxk identity matrix, D is an (mxm) matrix with a row/column weight of 2, and C is a (mk xk) matrix whose entries depend on the code designer.

wobei e eine ganze Zahl außer Null ist, die vom Code-Designer zu spezifieren ist, und $${\mathbf{B}}_2=\left[\begin{array}{cccccc}1& 0& 2& 0& 0& 0\\ 0& 1& 0& 2& 0& 0\\ a& b& 0& 0& 2& 0\\ c& d& 0& 0& 0& 2\end{array}\right],$$

wobei die Parameter {a, b, c, d} dem Designer belassen sind und nicht negative, ganzzahlige Werte annehmen sollen. Die Werte für n, m, k sind n=3, m=2, k=1 und n=6, m=4, k=2 für B₁ bzw. B₂. Größere Matrizen werden aus B durch Expansion mit dem Expansionsfaktor f erhalten, was die Matrix E ergibt. Die Matrix E ist eine Zwischen-Matrix zwischen der Protograph-Matrix B und der Final Parity-Check Matrix H. Sie weist nur einzelne Verbindungen zwischen Variable und Check Nodes auf (Deshalb handelt es sich bei den einzigen Nicht-Null-Elementen, die sie enthält, um Einsen). Die Anzahl der Spalten beträgt N=f*n, und die Anzahl der Reihen beträgt M=f*m. Die Expansion kann in mehreren aufeinanderfolgenden Schritten vorgenommen werden, d.h. die Matrix E kann nach mehreren Expansionsschritten erhalten werden.Next, two possible protographs derived from B are exemplified. Be it $${\mathbf{b}}_1=\left[\begin{array}{ccc}1& 2& 0\\ \text{e}& 0& 2\end{array}\right]$$

where e is an integer other than zero to be specified by the code designer, and $${\mathbf{b}}_2=\left[\begin{array}{cccccc}1& 0& 2& 0& 0& 0\\ 0& 1& 0& 2& 0& 0\\ a& b& 0& 0& 2& 0\\ c& d& 0& 0& 0& 2\end{array}\right],$$

where the parameters {a, b, c, d} are left to the designer and should assume non-negative, integer values. The values for n, m, k are n=3, m=2, k=1 and n=6, m=4, k=2 for B ₁ and B ₂ , respectively. Larger matrices are obtained from B by expansion with the expansion factor f, yielding the matrix E. The matrix E is an intermediate matrix between the protograph matrix B and the final parity check matrix H. It has only single connections between variables and check nodes (which is why these are the only non-zero elements it contains to ones). The number of columns is N=f*n, and the number of rows is M=f*m. The expansion can be carried out in several successive steps, that is, the matrix E can be obtained after several expansion steps.

• - Die ganz rechts angeordneten m Spalten der Basismatrix B, d.h. die Matrix D, sollen derart expandiert werden, dass sich nach der Expansion eine doppelte diagonale Matrix ergibt, die den ganz rechts angeordneten M=m*f Spalten in E entspricht.
• - Die Submatrix E, die aus den ersten K=k*f Spalten besteht, wird weiter verarbeitet. Jeder einzelne Eintrag wird durch ein Nicht-Null-Feldelement aus dem Galois-Feld (GF) der Ordnung q ersetzt, wobei q ein Design-Parameter ist. Nach diesem letzten Schritt besteht das Ergebnis in der Parity-Check-Matrix H eines Codes. Beide oben genannten Fälle (B₁ und B₂) können verwendet werden, um spezielle derartige Parity-Check-Matrizen zu erhalten.

Special (final) expansion rules are imposed:

• - The rightmost m columns of the base matrix B, ie the matrix D, should be expanded in such a way that after the expansion a double diagonal matrix results, which corresponds to the rightmost M=m*f columns in E.
• - The submatrix E, which consists of the first K=k*f columns, is further processed. Each individual entry is replaced by a non-zero field element from the Galois field (GF) of order q, where q is a design parameter. After this last step, the result is the parity check matrix H of a code. Both cases mentioned above (B ₁ and B ₂ ) can be used to obtain special such parity check matrices.

The matrix H obtained after the expansion is intended to describe a serial concatenation of two codes, an outer code c ₀ and an inner code c _l . The following restrictions are imposed. The outer code c ₀ is a code that extends over a field of order q. The inner code c _l should be designed over a ring of order q. A field symbol of the outer code should be mapped to a ring symbol of the inner code. The parity check matrices H for c _o and c _l can be derived from H. Therefore, the following structure is introduced in which H is divided into two sub-matrices: H = [A|B].

• - Die Informationssymbole des inneren Codes können von dem Codierer weitergegeben werden (systematischer Code) oder punktiert werden (nicht systematischer Code). Bei nicht kohärenten Kanälen wird angenommen, dass die Info-Symbole punktiert werden.
• - Das Ausgangssignal des äußeren Codes kann auf der Basis eines Punktiermusters punktiert werden, um die Rate einzustellen.
• - Es wird ein invertierbares Abbilden zwischen den Symbolen des äußeren Codes und des inneren Codes durchgeführt. Dieses Abbilden obliegt dem Code-Designer.

Then the parity check matrix of the outer code is H ₀ = [A|0] and that of the inner one Codes H _I = [0|B], where 0 is an (m × 1) matrix. Please note the following:

• - The information symbols of the inner code can be passed on by the encoder (systematic code) or punctured (non-systematic code). For non-coherent channels it is assumed that the info symbols are dotted.
• - The output signal of the outer code can be punctured based on a dot pattern to adjust the rate.
• - An invertible mapping is performed between the symbols of the outer code and the inner code. This mapping is the responsibility of the code designer.

3 shows a visual interpretation of the code as a serial concatenation of two codes.

After decoding, the encoded symbols are mapped to complex modulation symbols. The choice of modulation falls under the design of the physical layer. For non-coherent channels, a restriction to PSK modulations is made. Consequently, the outer code can be viewed as a phase accumulator. Thus, the original information consists of the phase difference between consecutive symbols. Such a scheme is well suited for channels with phase noise and non-coherent receivers.

To illustrate this, assume that the channel causes a codeword constant and an unknown phase shift and that the codeword consists of the m output symbols of the outer code (the information symbols are dotted). Since the two adjacent symbols experience the same phase offset, the offset is canceled when its difference is calculated, thus resulting in the original information symbol.

• - Die Parity-Check Matrix erlaubt sowohl Körper- als auch Ring-Operationen
• - Gute Leistung sowohl für kohärente als auch für nicht kohärente Kanäle,
• - Ermöglichung geringer Kodier-Komplexität
• - Ermöglichung geringer Kodier-Komplexität bei kohärenten Kanälen
• - Die Dekodier-Komplexität bei nicht kohärenten Kanälen ist abhängig von dem Dekodier-Algorithmus für den äußeren Code (Detektor)
• - Geringe Implementierungs-Komplexität

Essential features of the invention are:

• - The parity check matrix allows both body and ring operations
• - Good performance for both coherent and non-coherent channels,
• - Enabling low coding complexity
• - Enabling low coding complexity with coherent channels
• - The decoding complexity for non-coherent channels depends on the decoding algorithm for the outer code (detector)
• - Low implementation complexity

• - Das erfindungsgemäße Verfahren gilt für nicht-binäre Codes, also für non-binary LDPC-Codes.
• - Der Protograph ist ein bipartiter Graph, der auch multiple Verbindungen zwischen Check Nodes und Variable Nodes erlaubt. Der wird durch B beschrieben. Der Tanner Graph ist auch ein bipartiter Graph, der aber viel größer ist als der Protograph ist, jedoch lediglich einzelne Verbindungen zwischen Check Nodes und Variable Nodes erlaubt. Der Tanner Graph wird durch die Parity-Check Matrix H beschrieben.
• - Die Matrix E ist eine Zwischen-Matrix auf dem Weg vom Protograph zur Parity-Check Matrix H. Genauer gesagt ist E eine Matrix, die entweder 1 oder 0 als Elemente hat und genauso groß ist wie die endgültige Parity-Check Matrix H. Der einzige Unterschied zwischen E und H ist, dass E noch keine Körperelemente als Einträge aufweist, während H diese Körperelemente bereits beinhaltet.
• - Der Expansions-Prozess endet dann, wenn die Blocklänge des gewünschten Codes erreicht wurde.

In general, the following statements can be made with regard to the invention:

• - The method according to the invention applies to non-binary codes, i.e. to non-binary LDPC codes.
• - The Protograph is a bipartite graph that also allows multiple connections between check nodes and variable nodes. It is described by B. The Tanner Graph is also a bipartite graph, which is much larger than the Protograph, but only allows individual connections between check nodes and variable nodes. The Tanner graph is described by the parity check matrix H.
• - The matrix E is an intermediate matrix on the way from the protograph to the parity-check matrix H. More precisely, E is a matrix that has either 1 or 0 as elements and is the same size as the final parity-check matrix H. The The only difference between E and H is that E does not yet have any body elements as entries, while H already contains these body elements.
• - The expansion process ends when the block length of the desired code has been reached.

• 1 eine graphische Darstellung eines Graphen mit Check Nodes (quadratisch eingerahmt) und Variable Nodes (eingekreist),
• 2 eine graphische Darstellung eines erweiterten Graphen, und zwar basierend auf dem Graph gemäß 1,
• 3 ein Blockschaltbild eines Codes, der durch Reihenkonkatenation zweier Codes entstanden ist,
• 4 ein Diagramm zur Beschreibung der Eigenschaften eines Beispiels eines erfindungsgemäßen Codes,
• 5 die Struktur des Encoders für die Verwendung beim erfindungsgemä-ßen LDPC-Code und
• 6 die Struktur des Decoders für die Verwendung beim erfindungsgemä-ßen LDPC-Code.

The invention is explained in more detail below using an exemplary embodiment of an LDPC code. In detail we show:

• 1 a graphical representation of a graph with check nodes (framed square) and variable nodes (circled),
• 2 a graphical representation of an extended graph based on the graph according to 1 ,
• 3 a block diagram of a code that was created by concatenating two codes in series,
• 4 a diagram describing the properties of an example of a code according to the invention,
• 5 the structure of the encoder for use in the LDPC code according to the invention and
• 6 the structure of the decoder for use in the LDPC code according to the invention.

The invention will be briefly outlined again below.

Protograph constructions and accumulator-based LDPC codes (e.g. codes with double diagonals) are known in the literature and also in US 2013/0013983 A1 listed. The invention concerns a special code construction based on a hybrid structure. One part of the code (systemic part) is used in encoding and decoding the data for calculations based on a field (Galois Field), while the other part of the code (parity part) is used in encoding and decoding the data for calculations a ring is used. This is not found in the prior art.

For any type of coding that uses body algebra or ring algebra, the associative and commutative laws must be satisfied. Furthermore, the existence of a neutral element must be guaranteed. To transmit the data, it is encoded using the code and decoded after reception. In both cases (encoding and decoding), the code is used to perform calculations that are partly based on body algebra and partly on ring algebra. This special code structure requires a special adaptation of the encoder and decoder as follows.

For better understanding, it should be said that the code can be split into two parts (outer and inner code). The parity check matrix after the protograph expansion according to the invention is referred to as $$H=\left[G^T|W\right]$$

Starting with this parity check matrix, the encoder and decoder are adjusted as follows:

ein Körperelement beschreibt. Der ℤ_m-Akkumulator entspricht den Operationen, die von der Matrix W durchgeführt werden.

1. 1) Äußerer Code: Encodieren der Informationssymbole mit Hilfe der Körperalgebra. Hier wird die Matrix G aus US 2013/0013983 A1 verwendet: $$\underset{\_}{u}G=\underset{\_}{v}$$
   

The structure of the encoder is in 5 shown, where ℤ _m is a ring element and ${\mathbb{F}}_m$

describes a body element. The ℤ _m accumulator corresponds to the operations performed by the matrix W.

1. 1) External code: Encoding the information symbols using body algebra. Here the matrix G becomes US 2013/0013983 A1 used: $$\underset{\_}{u}G=\underset{\_}{v}$$
   

The result is intermediate symbols v.

2) Mapping the intermediate symbols v to ring symbols a using the reversible function f. $$\underset{\_}{a}=f\left(\underset{\_}{v}\right)$$

wobei c die Parity-Symbole des hybriden Codes darstellt. Im Gegensatz zum in US 2013/0013983 A1 verfolgten Ansatz erfordert die Berechnung von a die Anwendung von Körperalgebra sowie eine Mappingfunktion. Die Berechnung von c erfordert reine Ringalgebra. In dem Fall, dass W eine niedrige Dichte an 1-er Bits besitzt (engl. sparse), lässt sich die Matrixmultiplikation effizient mittels Akkumulation rekursiv berechnen, wie in 5 zu sehen ist. Erfindungsgemäß wird die Akkumulation mittels Ringalgebra durchgeführt und erfordert zuvor die Mappingfunktion.3) Inner code: Encoding the ring symbols a using ring algebra. That makes $$\underset{\_}{a}+W\ast \underset{\_}{c}=\underset{\_}{0},$$

where c represents the parity symbols of the hybrid code. In contrast to the in US 2013/0013983 A1 According to the approach followed, the calculation of a requires the use of body algebra and a mapping function. Calculating c requires pure ring algebra. In the case that W has a low density of 1's bits (sparse), the matrix multiplication can be efficiently calculated recursively using accumulation, as in 5 you can see. According to the invention, the accumulation is carried out using ring algebra and requires the mapping function beforehand.

If the code symbols are PSK modulated after encoding, it can be shown that the inner code and the PSK modulator perform the function of a DPSK modulator (and only if the inner code is based on a ring).

1. 1) Innerer Code: Decodieren der (phasen-)verrauschten Ringsymbole mittels eines speziellen Algorithmus, beispielsweise DP Algorithmus [5], welche für Ringalgebra ausgelegt sind. Diese Algorithmen unterscheiden sich erheblich von denen welche für Körperalgebra verwendet werden!
2. 2) Mapping der Wahrscheinlichkeitsfunktionen (probability mass functions) mittels f^-1(·).
3. 3) Äußerer Code: Decodieren der Zwischen-Symbolen unter Verwendung von Körperalgebra, z.B. durch „non-binary belief propagation“ für LDPC Codes wie in [6].

Decoder (see also 6 ):

1. 1) Inner code: Decoding the (phase-)noisy ring symbols using a special algorithm, for example DP algorithm [5], which is designed for ring algebra. These algorithms differ significantly from those used for body algebra!
2. 2) Mapping the probability mass functions using f ^-1 (·).
3. 3) Outer code: Decoding the intermediate symbols using body algebra, e.g. through “non-binary belief propagation” for LDPC codes as in [6].

This encoder and decoder structure listed is not known in the prior art, but has advantages.

The motivation for this hybrid code structure lies in the applicability of the code for channels with phase noise (non-coherent channels). PSK modulations, especially DPSK modulations, are usually used here. A DPSK modulator performs ring algebra operations. The inner decoder tries to eliminate phase noise using special algorithms. Since the code according to the invention is also partly based on ring algebra, the code structure allows such algorithms to be used to decode the code. The construction according to US 2013/0013983 A1 , which uses only body algebra, does not allow this and is not intended for channels with phase noise.

According to an embodiment of the invention, a B-based code with k=100 symbols, n=300 symbols and m=200 was designed. There was a puncture of all information symbols, which resulted in a final block length n _punc =200 symbols. The outer code was based on an order 8 field and the inner code was based on an order 8 ring. The modulation was 8PSK, with the modulation order matched to the ring order. In terms of bits, the present code had a codeword length of 600 and an information length of 300, thus a rate of ½. A standard AWGN channel was adopted for transmission and the codeword error rate was measured against E _b /N ₀ . The corresponding arbitrary coding limit was evaluated as a benchmark. The results are in 4 shown.

The method proposed here can be used in all types of commercial wireless and wired transmission systems.

LIST OF ABBREVIATIONS

ERA

Accumulate-Repeat-Accumulate

CERIUM

Codeword Error Rate

CN

Check node

BER

Bit Error Rate

LDPC

Low-density parity check (code)

I.T

Iterative (decoder)

VN

Variable node

DP

Discretized phase

PSK

Phase shift keying

DPSK

Differential phase shift keying

BIBLIOGRAPHY

1. [1] Thorpe, J.; Andrews, K.; Dolinar, S., “Methodologies for designing LDPC codes using protographs and circulants,” Information Theory, 2004. ISIT 2004. Proceedings. International Symposium on , vol., no., pp.238" 27 June-2 July 2004
2. [2] G. Liva, S. Song, L. Lan, Y. Zhang, S. Lin, and W.E. Ryan, “Design of LDPC Codes: A Survey and New Results,” Journal of Communication Software and Systems (JCOMSS). Special issue on channel coding in wireless systems, September 2006. Invited paper
3. [3] R. G. Gallager, Low Density Parity Check Codes. Cambridge, MA: MIT Press, 1963.
4. [4] Abbasfar, A.; Divsalar, D.; Yao, K., “Accumulate-Repeat-Accumulate Codes,” Communications, IEEE Transactions on , vol. 55, no. 4, pp. 692,702, April 2007
5. [5] G. Colavolpe, A. Barbieri, and G. Caire, “Algorithms for iterative decoding in the presence of strong phase noise,” IEEE J. Sel. Areas Commun., vol. 23, no. 9, pp. 1748-1757, Sep. 2005
6. [6] M. Davey and D. MacKay, “Low density parity check codes over GF(q),” IEEE Commun. Lett., vol. 2, no. 6, pp. 70-71, Jun. 998 .
7. [7] Sassatelli, L.; Declercq, D.: Analysis of Non-binary Hybrid LDPC Codes. In: IEEE International Symposium on Information Theory, Nice, France, 2007, pp. 2261-2265 , DOI: 10.1109/ISIT.2007.4557556 - URL: https://ieeexplore.ieee.org/document/4557 [accessed on February 16, 2023]
8. [8th] GARRAMONNE, G.; MATUZ, B.: Short Erasure Correcting LDPC IRA Codes over GF (q). In: IEEE Global Telecommunications Conference GLOBECOM 2010, Miami, FL, USA, 2010, pp. 1-5 , DOI; 10.1109/GLOCOM.2010.5683889 - URL: https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5683889 [accessed on February 16, 2023]

• US 2013/0013983 A1
• EP 2 843 844 A1

Claims (2)

Method for generating a class of non-binary LDPC codes with a predeterminable number of check nodes and with a predeterminable number of variable nodes for data transmission with PSK modulation over a non-coherent channel with the following steps - concatenating an outer code and one inner code, whereby the outer code extends over a mathematical body of order q and the inner code is designed over a mathematical ring, also of order q, and where a symbol of the outer code is mapped onto a symbol of the inner code, - specifying a protograph , which is defined by a base matrix whose number of rows is equal to the number of check nodes of the protograph and whose number of columns is equal to the number of variable nodes of the protograph and whose integer elements describe the connection of the check nodes to the variable nodes, - gradual expansion of the base matrix to create a parity check matrix describing the LDPC code to be generated, - whereby each element of the base matrix or the matrix expanded in the previous step is replaced by a matrix whose row/column weighting is equal to the element to be replaced , - whereby the expansion is carried out until the specified size of the parity check matrix is reached and all elements in the resulting Matrix have either the value 1 or 0, - where the base matrix and each expanded matrix have a square rear matrix part, the number of rows of which is equal to the number of columns, and a non-square front matrix part, - where the elements of the front matrix part of the base matrix are freely selectable wholes are numbers that are converted in the front part of the resulting expanded matrix by expansion to either the value 1 or 0, and then replaced by integer elements from a mathematical field to produce the front part of the parity-check matrix, these elements describe the symbols of the outer code, - where the square rear matrix part of the base matrix contains exclusively elements that are either 0, 1 or 2 and which are arranged in such a way that the resulting square rear matrix part of the parity check matrix after gradually expanding the base matrix is a Has a double diagonal structure, whereby the double diagonal only has elements with the value 1 and where outside the double diagonal there are only elements with the value 0, these elements being elements from a mathematical ring and describing the symbols of the inner code, - where the elements of front matrix part of the parity check matrix, are intended for carrying out mathematical operations in body algebra and - wherein the elements of the rear matrix part of the parity check matrix are intended for carrying out mathematical operations in ring algebra and implement a phase accumulator for the non-coherent channel. Procedure according to Claim 1 , characterized in that the gradual expansion ends when the length of the code to be generated is reached.

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