DE102011015811B3 - Device for coding k8L information-bits in e.g. universal mobile telecommunications system-applications, has coder for receiving m-symbol-parity-vector from m-symbol-vector, where elements in main-diagonals are formed as non-zero-elements - Google Patents

Abstract

The device has a memory storing parity checking matrix that comprises concatenating matrices e.g. sparely-occupied matrix and structured quadratic matrix with main-diagonals and secondary-diagonals. A vector multiplication device multiplies a k-sector-information vector with transposed sparely-occupied matrix to produce a m-symbol-vector. A recursive rate-1-coder receives a m-symbol-parity-vector from the symbol-vector. Elements in the main-diagonals are formed as non-zero-elements of Galois field, where elements on one of the secondary diagonals are formed as non-zero-elements of the field. An independent claim is also included for a method for recovering lost and/or damaged data transmitted from a transmitter to a receiver.

Description

The present invention relates to efficient coding schemes provided for non-binary Generalized Irregular Repeat Accumulate (GeIRA) codes. In particular, the invention relates to an encoder and a method for transmitting data packets by means of packet-level coding.

Packet-level coding is a simple and efficient technique for ensuring reliable transmission in a communication system. The basic idea behind packet-level coding is to further protect the information to be transmitted by coding packets of data at higher levels of the communication model. The main principles of packet-level encoding are in 1 shown.

First, the message to be transmitted is divided into k info packets and encoded into a larger number of n codeword packets. Thus, m = n-k parity packets are encoded by the packet-level encoder on the transmitter side.

Second, the n codeword packets on the physical layer are protected by error correction codes (eg, turbo, LDPC ...) or error detection codes (eg, cyclic redundancy check (CRC) codes) and then transmitted.

Third, a physical layer error correction is made on each packet on the receiver side, and residual errors are detected by the error detection code. If errors are detected, the packet is deemed lost and marked as deleted. Thus, the levels above the physical layer see the communication medium as a packet deletion channel (PEC) on which packets are either received correctly or lost with a deletion probability ε.

Finally, if a sufficient amount of packets has been received, the packet level encoder performs a recovery of the original message. For this purpose, at the receiver side, the decoder receives non-erased packets and retrieves the lost packets if enough packets are received correctly. To achieve a non-zero probability of successful recovery, P _s , at least k packets must be collected. P _s increases with the number of collected packets p depending on the designated code.

The encoding process is usually performed bitwise (character-wise) by using the encoder of a binary (non-binary) code. The decoding is then done by solving the system of equations given by the parity check matrix H of the code.

Software implementations of packet-level encoders and decoders are particularly attractive because they do not require much design and implementation effort and provide more flexibility in hardware implementations. Software modules can be easily integrated into a hardware architecture that is not specifically designed to support packet-level encoding.

Because software packet-level encoders and decoders are also intended for use on limited-power platforms (eg, on-board units for use in a spaceship, lander, rover, or orbiter in near-Earth or deep-space missions ), efficient algorithms are required.

It has also been found that erase recovery capabilities are improved when the code design is executed in non-binary arrays. Therefore, there is a need to develop efficient coding / decoding techniques even for non-binary codes.

• 1. ein Algorithmus, der auf einer ”Neuanordnung” der Paritätsprüfmatrix H durch Reihen- und Spalten-Permutationen basiert, um H in eine geeignete triangulare Matrix zu transformieren. Dann wird ein Parameter g(gap) definiert, der den Abstand von H zu einer exakt triangularen Form repräsentiert. Falls g ein kleiner Bruchteil der Code-Länge n ist, ist die Kodierkomplexität fast linear. Jedoch sind für diesen Algorithmus nur LDPC-Codes geeignet, die durch ein kleines g gekennzeichnet sind, und dies ist nicht oft der Fall [1].
• 2. zyklische oder quasi-zyklische LDPC-Codes, die durch Schaltungen niedriger Komplexität auf der Basis von Feedback-Schieberegistern kodiert werden können. Der Nachteil dieser Codes besteht darin, dass sie nur für präzise Code-Parameter n und k existieren [2].
• 3. Irregular-Repeat-Accumulate-Codes (IRA-Codes), deren Kodierschema eine lineare Komplexität aufweist. Diese unterliegen der Beschränkung, dass sie m-1 variable Knoten (Spalten der Paritätsprüfmatrix H) des Grads (Gewichts) 2 haben müssen. Folglich ist die Verteilung mit variablem Knoten-Grad etwas eingeschränkt [3].
• 4. Generalized-Irregular-Repeat-Accumulate-Codes (GeIRA-Codes) mit der gleichen niedrigen Kodierkomplexität wie bei IRA-Codes, jedoch einer höheren Flexibilität in der Wahl der Grad-Verteilungen und der Code-Parameter n und k [4].

There are four main approaches to efficient coding of binary LDPC codes in the prior art:

• 1. an algorithm based on a "rearrangement" of the parity check matrix H by row and column permutations to transform H into an appropriate triangular matrix. Then a parameter g (gap) is defined, which represents the distance of H to an exactly triangular form. If g is a small fraction of the code length n, the coding complexity is almost linear. However, only LDPC codes characterized by a small g are suitable for this algorithm and this is not often the case [1].
• 2. Cyclic or quasi-cyclic LDPC codes that can be encoded by low complexity circuits based on feedback shift registers. The disadvantage of these codes is that they only exist for precise code parameters n and k [2].
• 3. Irregular Repeat Accumulate Codes (IRA codes) whose coding scheme has a linear complexity. These are subject to the restriction that they must have m-1 variable nodes (columns of parity check matrix H) of degree (weight) 2. Consequently, the distribution with variable node degree is somewhat limited [3].
• 4. Generalized Irregular Repeat Accumulate (GeIRA) codes with the same low coding complexity as IRA codes but with greater flexibility in choice of degree distributions and code parameters n and k [4].

Thus, GeIRA codes designed for higher order Galois fields (non-binary fields) are in the focus of the invention As is well known, a Galois field is a set of finite elements involving arithmetic operations such as additions, subtractions, Multiplications and divisions can be performed.

wobei die Leerstellen Nullen entsprechen und wobei die Koeffizienten b_j für j = 1, 2, ..., h die binären Elemente in den sekundären (Neben-)Diagonalen sind. Zu beachten ist, dass für das binäre Feld b_j-Koeffizienten, die zu der gleichen sekundären Diagonalen gehören, entweder sämtlich 1 oder sämtlich 0 betragen.The (n-k) xn check matrix of the GeIRA code is written as a concatenation of two matrices as H = [H _u | H _P ], where the (n-k) x k matrix H _{u is} the systematic part of the codeword while the m × m H _p matrix corresponds to the parity part of the codeword and is generally a multidiagonal matrix. The structure of the submatrix H _p is as follows:

where the vacancies correspond to zeros and where the coefficients b _j for j = 1, 2, ..., h are the binary elements in the secondary (minor) diagonals. Note that for the binary field, b _j coefficients belonging to the same secondary diagonal are either all 1 or all 0.

The multidiagonal structure leads to efficient coding schemes. The coder scheme for binary GeIRA designs is in 2 where D is the delay unit with transfer function z ^-1 and HT / u is the transpose of H _u .

• 1. Der k-Längen-Informationsvektor u wird mit einer niedrigdichten Matrix H T / u multipliziert. Dadurch wird der m-Längen-Vektor v generiert.
• 2. Der m-Bit-Vektor v tritt in einen rekursiven Rate-1-Kodierer ein, wobei b_j, j = 1, 2, ..., h die in H_p definierten binären Elemente sind. Somit wird der Paritäts-m-Bit-p-Vektor erzeugt. Insbesondere erhält man ausgehend von dem gegebenen ersten kodierten Paritäts-Bit p₀ die anderen Paritäts-Bits p_i mit der Rekursion p_i = v_i + Σ h / j=1 b_j p_i-j für i = 1, 2, ..., m – 1, wobei 1 ≤ h ≤ m – 1.
• 3. Das n-Bit-kodierte Codewort c wird erhalten durch Konkatenieren von u und p, was in c = [u|p] resultiert.

The binary coding scheme can be summarized in the form of the following steps:

• 1. The k-length information vector u is given a low-density matrix HT / u multiplied. This generates the m-length vector v.
• 2. The m-bit vector v enters a recursive rate 1 encoder, where b _j , j = 1, 2, ..., h are the binary elements defined in H _p . Thus, the parity m-bit p vector is generated. In particular, starting from the given first coded parity bit p _0, the other parity bits p _{i are obtained} with the recursion p _i = v _i + Σh / j = 1 b _j p _ij for i = 1, 2,..., m-1, where 1≤h≤m-1.
• 3. The n-bit coded codeword c is obtained by concatenating u and p, resulting in c = [u | p].

GeIRA codes are thus attractive due to their linear time coding complexity and their high degree of flexibility in the choice of degree distribution selection, which is of fundamental importance in balancing performance between the waterfall and error floor areas. Furthermore, the coding structure is easily configurable by merely setting coefficients b _i to zero or one, thereby reducing the feedback paths in 2 be activated / deactivated.

Non-binary LDPC designs are currently attracting a lot of attention as they are able to improve the performance of their binary counterparts. 3 illustrates the advantage of the non-binary embodiment, where performance is shown by the codeword error rate (CER) for a binary (n, k) GeIRA with n = 256 and k = 128. Two secondary diagonals are placed, especially the secondary diagonal S ₁ and the secondary diagonal S ₆₂ . The performance of the code was simulated by PEC with a Monte Carlo simulation. The packet size has been set to L = 1000 bytes, but the results in terms of CER are not dependent on the packet size. The performances are compared to the singleton barrier, which is the performance of the ideal maximum distance code (MDS) for the given (n, k) code parameters. The singleton barrier represents a lower bound on the error probability on the clear channel. It should be noted that the performance of the non-binary GeIRA has improved significantly over its binary counterparts. It may be underlined that the comparison between Galois fields of different orders is permissible since the erasure correction code is applied at the packet level. The amount of data carried in a codeword on the channel is fixed at L x n bytes, however, the way in which the data is processed depends on the bitwise approach of GF (2) e.g. B. to the bytewise approach according to GF (256) changes.

For example, if the probability of erosion is 0.4, the code at GF (256) will add an order of magnitude.

• 1. Die Paritätspüf-Matrix H wird in die systematische Form H = [P^T|I] gebracht, wobei I die Identitätsmatrix ist.
• 2. Die Generatormatrix des Codes G ist somit G = [I|P].
• 3. Das Generieren des Codeworts c erfolgt durch Multiplizieren von u und G, c = u G.

However, the focus of the study is on the decoding algorithms, and the literature (see [5] and [6]) does not provide efficient coding schemes for non-binary LDPC codes. The coding of non-binary LDPC codes is performed according to the following general algorithm:

• 1. The parity check matrix H is put into the systematic form H = [P ^T | I], where I is the identity matrix.
• 2. The generator matrix of the code G is thus G = [I | P].
• 3. The code word c is generated by multiplying u and G, c = u G.

The complexity of this approach is quadratic (O (n ² )) with the code length n if P is a high-density matrix. It should be noted that even in the case that the parity check matrix H of the code is not dense, the process in step 1 produces a dense array P which is the result of numerous row additions. To reduce the coding complexity in Galois fields of the order q> 2, efficient coding schemes must be found.

• – eine Paritätsprüfmatrix, die in einem Speicher gespeichert ist und zwei konkatenierte Matrizes H_u und H_p aufweist,
• – wobei H_u eine spärlich besetzte Matrix ist und H_p eine strukturierte quadratische Matrix ist, die Elemente auf einer Haupt-Diagonalen und auf einer sekundären Diagonalen aufweist,
• – eine Vektormultiplikationsvorrichtung zum Multiplizieren eines Informationsvektors u mit der transportierten spärlich besetzen Matrix H_u ^T, um einen Vektor v zu erhalten, wobei ein Symbol des Informationsvektors u ein Element des Galois-Felds (GF(q)) ist, und
• – einen rekursiven Rate-1-Kodierer zum Erhalt eines Paritäts-Vektors p aus dem Vektor v,
• – wobei die Elemente des rekursiven Kodierers durch die Elemente der quadratischen Matrix H_p definiert sind, wobei die Elemente in der Haupt-Diagonalen der Matrix H_p Nicht-Null-Elemente des Galois-Felds (GF(q)) sind und die Elemente auf der sekundären Diagonalen der Matrix H ebenfalls Nicht-Null-Elemente des Galois-Felds (GF(q)) sind.

In [7] (see Chapter IV), an information bit coding apparatus is described in which data packets are encoded by means of a low-density parity check code in the form of a non-binary Generalized Irregular Repeat Accumulate Code a galois field (GF) (q)) of order q, where q is greater than 2, and where the Galois field (GF (g)) contains a number q of elements. This known device has among others:

• A parity check matrix which is stored in a memory and has two concatenated matrices H _u and H _p ,
• Where H _{u is} a sparsely populated matrix and H _{p is} a structured square matrix having elements on a major diagonal and on a secondary diagonal,
• A vector multiplication device for multiplying an information vector u with the transported sparse matrix H _u ^T to obtain a vector v, wherein a symbol of the information vector u is an element of the Galois field (GF (q)), and
• A recursive rate-1 coder for obtaining a parity vector p from the vector v,
• - wherein the elements of the recursive encoder are defined by the elements of the square matrix H _p , where the elements in the main diagonal of the matrix H _{p are} non-zero elements of the Galois field (GF (q)) and the elements of the secondary diagonals of the matrix H are also non-zero elements of the Galois field (GF (q)).

Furthermore, an apparatus for coding information bits is also known in [8] (see Chapter I.A) by encoding data packets by means of a low-density parity check code in the form of a non-binary generalized irregular repeat accumulate code indicated on a Galois field (GF (q)) of order q, where q is greater than 2, and where the Galois field (FG ((q)) contains a number q of elements, the device having a Parity check matrix stored in a memory and comprising two concatenated matrices H _u and H _p .

• – eine Paritätsprüfmatrix, die in einem Speicher gespeichert ist und zwei konkatenierte Matrizes H_u und H_p aufweist,
• – wobei H_u eine spärlich besetzte Matrix ist und H_p eine strukturierte quadratische Matrix ist, die Elemente auf einer Haupt-Diagonalen und auf einer sekundären Diagonalen aufweist,
• – wobei die Elemente in der Haupt-Diagonalen der Matrix H_p Nicht-Null-Elemente des Galois-Feldes (GF(q)) sind und die Elemente auf der sekundären Diagonalen der Matrix H_p ebenfalls Nicht-Null-Elemente des Galois-Feldes (GF(q)) sind.

Finally, [9] (see Chapter II) also discloses an information bit coding apparatus by encoding data packets by means of a low-density parity check code in the form of a non-binary Generalized Irregular Repeat Accumulate Code which is stored on a Galois field (GF (q)) of the order q, where q is greater than 2, and where the Galois field (GF (q)) contains a number q of elements. This known device has, inter alia:

• A parity check matrix which is stored in a memory and has two concatenated matrices H _u and H _p ,
• Where H _{u is} a sparsely populated matrix and H _{p is} a structured square matrix having elements on a major diagonal and on a secondary diagonal,
• - where the elements in the main diagonal of the matrix H _{p are} non-zero elements of the Galois field (GF (q)) and the elements on the secondary diagonal of the matrix H _{p are} also non-zero elements of the Galois field (GF (q)).

It is an object of the present invention to reduce complexity for the packet-level encoding of non-binary low density parity check (LDPC) codes. In particular, the task is to use the structure of GeIRA codes also for non-binary fields in order to create an efficient coding scheme. As far as is known, the prior art does not provide an efficient coding scheme for a particular class of non-binary LDPC codes, namely non-binary GeIRA codes.

The invention provides an efficient coding scheme (Scheme 1 in FIG 4 ) for non-binary GeIRA codes, which is applied at the packet level. Furthermore, an improvement of the proposed scheme is presented (Scheme 2 in 2 ).

• – eine Paritätsprüfmatrix, die in einem Speicher gespeichert ist und zwei konkatenierte Matrizes H_u und H_p aufweist,
• – wobei H_u eine spärlich besetzte Matrix ist und H_p eine strukturierte quadratische Matrix ist, die Elemente auf einer Haupt-Diagonalen und auf einer Gruppe von mindestens zwei sekundären Diagonalen aufweist,
• – eine Vektormultiplikationsvorrichtung zum Multiplizieren eines k-Symbol-Informationsvektors u mit der transponierten spärlich besetzten Matrix H T / u , um einen m-Symbol-Vektor v zu erhalten, wobei ein Symbol des Informationsvektors u ein Element des Galois-Felds ist, und
• – einen rekursiven Rate-1-Kodierer zum Erhalt eines m-Symbol-Paritäts-Vektors p aus dem m-Symbol-Vektor v,
• – wobei die Elemente des rekursiven Kodierers durch die Elemente der quadratischen Matrix H_p definiert sind, wobei die Elemente in der Haupt-Diagonalen der Matrix H_p Nicht-Null-Elemente des Galois-Felds sind und die Elemente auf jeder der mindestens zwei sekundären Diagonalen der Gruppe sekundärer Diagonalen der Matrix H_p ebenfalls Nicht-Null-Elemente des Galois-Felds sind.

The invention proposes a device for coding (k × 8 × L) information bits by dividing data packets having a size of L bytes by means of a low-density parity check code in the form of a non-binary generalized irregular repeat accumulator. Encoded on a Galois field of order q, where q is greater than 2, and wherein the Galois field contains a number q of elements, the device comprising:

• A parity check matrix which is stored in a memory and has two concatenated matrices H _u and H _p ,
• Where H _{u is} a sparsely populated matrix and H _{p is} a structured square matrix having elements on a major diagonal and on a group of at least two secondary diagonals,
• A vector multiplication device for multiplying a k-symbol information vector u by the transposed sparse matrix HT / u to obtain an m-symbol vector v, wherein a symbol of the information vector u is an element of the Galois field, and
• A recursive rate-1 encoder for obtaining an m-symbol parity vector p from the m-symbol vector v,
• - wherein the elements of the recursive encoder are defined by the elements of the square matrix H _p , the elements in the main diagonal of the matrix H _{p being} non-zero elements of the Galois field and the elements on each of the at least two secondary diagonals of the group of secondary diagonals of the matrix H _{p are} also non-zero elements of the Galois field.

• – eine Paritätsprüfmatrix, die in einem Speicher gespeichert ist und zwei konkatenierte Matrizes H_u und H_p aufweist,
• – wobei H_u eine spärlich besetzte Matrix ist und H_p eine strukturierte quadratische Matrix ist, die Elemente auf einer Haupt-Diagonalen und auf einer Gruppe von mindestens zwei sekundären Diagonalen aufweist,
• – eine Vektormultiplikationsvorrichtung zum Multiplizieren des k-Symbol-Informationsvektors u mit der transponierten spärlich besetzten Matrix H T / u , um einen m-Symbol-Vektor v zu erhalten, wobei ein Symbol des Informationsvektors u ein Element des Galois-Felds (GF(q)) ist, und
• – einen rekursiven Rate-1-Kodierer zum Erhalt eines m-Symbol-Paritäts-Vektors p aus dem m-Symbol-Vektor v,
• – wobei die Elemente des rekursiven Kodierers durch die Elemente der quadratischen Matrix H_p definiert sind, wobei die Elemente in der Haupt-Diagonalen der Matrix H_p sämtlich 1 sind und die Elemente auf jeder der mindestens zwei sekundären Diagonalen der Gruppe sekundärer Diagonalen der Matrix H_p Nicht-Null-Elemente des Galois-Felds sind.

According to an alternative, the invention proposes a device for encoding (k × 8 × L) information bits by providing data packets having a size of L bytes by means of a low-density parity check code in the form of a non-binary generalized irregular repeat Acumulate codes denoted on a Galois field of order q, where q is greater than 2, and wherein the Galois field (GF (q)) contains a number q of elements, the device comprising:

• A parity check matrix which is stored in a memory and has two concatenated matrices H _u and H _p ,
• Where H _{u is} a sparsely populated matrix and H _{p is} a structured square matrix having elements on a major diagonal and on a group of at least two secondary diagonals,
• A vector multiplication device for multiplying the k-symbol information vector u by the transposed sparse matrix HT / u to obtain an m-symbol vector v, where a symbol of the information vector u is an element of the Galois field (GF (q)), and
• A recursive rate-1 encoder for obtaining an m-symbol parity vector p from the m-symbol vector v,
• In which the elements of the recursive coder are defined by the elements of the square matrix H _p , where the elements in the main diagonal of the matrix H _{p are} all 1 and the elements on each of the at least two secondary diagonals of the group of secondary diagonals H _{p are} non-zero elements of the Galois field.

• – Kodieren der Daten mittels eines senderseitigen Kodierers,
• – Senden der kodierten Daten von dem Sender zu dem Empfänger über ein Übertragungssystem, und
• – Dekodieren der empfangenen Daten mittels eines empfängerseitigen Dekodierers, wobei verlorene und/oder beschädigte Daten während des Dekodierens wiederhergestellt werden,
• – wobei das Kodieren mittels eines nichtbinären Generalized-Irregular-Repeat-Accumulate-Codes durchgeführt wird, der auf einem Galois-Feld der Ordnung q bezeichnet ist, wobei q größer als 2 ist, und wobei das Galois-Feld (GF(q)) eine Anzahl q von Elementen enthält,
• – wobei das Dekodieren durch Auflösen des durch die Dekodier-Matrix des Codes definierten Gleichungssystems durchgeführt wird, und
• – wobei das Kodieren der Daten mittels einer Paritätsprüfmatrix H durchgeführt wird, die zwei konkatenierte Matrizes H_u und H_p aufweist, wobei H_u eine spärlich besetzte Matrix ist und H_p eine strukturierte quadratische Matrix ist, die Elemente auf einer Haupt-Diagonalen und eine Gruppe von mindestens zwei sekundären Diagonalen aufweist, wobei die Elemente in der Haupt-Diagonalen der Matrix H_p Nicht-Null-Elemente des Galois-Felds sind und die Elemente auf jeder der mindestens zwei sekundären Diagonalen der Gruppe sekundärer Diagonalen der Matrix H_p Nicht-Null-Elemente des Galois-Felds sind, auf dem der Code basiert.

Furthermore, the invention proposes a method for recovering lost and / or damaged data that is sent from a sender to a recipient, the method comprising the following steps:

• Encoding the data by means of a transmitter-side encoder,
• Sending the coded data from the transmitter to the receiver via a transmission system, and
• Decoding the received data by means of a receiver-side decoder, wherein lost and / or damaged data is recovered during decoding,
• Wherein the coding is performed by means of a non-binary Generalized Irregular Repeat Accumulate Code denoted on a Galois field of order q, where q is greater than 2, and where the Galois Field (GF (q)) contains a number q of elements,
• - wherein the decoding is performed by resolving the equation system defined by the decoding matrix of the code, and
• Wherein the coding of the data is performed by means of a parity check matrix H having two concatenated matrices H _u and H _p , where H _{u is} a sparse matrix and H _{p is} a structured square matrix comprising elements on a major diagonal and a Group of at least two secondary diagonals, the elements in the main diagonal of the matrix H _{p being} non-zero elements of the Galois field, and the elements on each of the at least two secondary diagonals of the group of secondary diagonals of the matrix H _p non-zero. Are zero elements of the Galois field on which the code is based.

• – Kodieren der Daten mittels eines senderseitigen Kodierers,
• – Senden der kodierten Daten von dem Sender zu dem Empfänger über ein Übertragungssystem, und
• – Dekodieren der empfangenen Daten mittels eines empfängerseitigen Dekodierers, wobei verlorene und/oder beschädigte Daten während des Dekodierens wiederhergestellt werden,
• – wobei das Kodieren mittels eines nichtbinären Generalized-Irregular-Repeat-Accumulate-Codes durchgeführt wird, der auf einem Galois-Feld der Ordnung q bezeichnet ist, wobei q größer als 2 ist, und wobei das Galois-Feld (GF(q)) eine Anzahl q von Elementen enthält,
• – wobei das Dekodieren durch Auflösen des durch die Dekodier-Matrix des Codes definierten Gleichungssystems durchgeführt wird, und
• – wobei das Kodieren der Daten mittels einer Paritätsprüfmatrix H durchgeführt wird, die zwei konkatenierte Matrizes H_u und H_p aufweist, wobei H_u eine spärlich besetzte Matrix ist und H_p eine strukturierte quadratische Matrix ist, die Elemente auf einer Haupt-Diagonalen und auf einer Gruppe von mindestens zwei sekundären Diagonalen aufweist, wobei die Elemente in der Haupt-Diagonalen der Matrix H_p sämtlich 1 sind und die Elemente auf jeder von mindestens zwei sekundären Diagonalen der Matrix H_p Nicht-Null-Elemente des Galois-Felds sind, auf dem der Code basiert.

According to an alternative of this method, the invention proposes a method for recovering lost and / or damaged data sent from a sender to a recipient, the method comprising the following steps:

• Encoding the data by means of a transmitter-side encoder,
• Sending the coded data from the transmitter to the receiver via a transmission system, and
• Decoding the received data by means of a receiver-side decoder, wherein lost and / or damaged data is recovered during decoding,
• Wherein the coding is performed by means of a non-binary Generalized Irregular Repeat Accumulate Code denoted on a Galois field of order q, where q is greater than 2, and where the Galois Field (GF (q)) contains a number q of elements,
• - wherein the decoding is performed by resolving the equation system defined by the decoding matrix of the code, and
• Wherein the coding of the data is performed by means of a parity check matrix H having two concatenated matrices H _u and H _p , where H _{u is} a sparse matrix and H _{p is} a structured square matrix containing elements on a major diagonal and a group of at least two secondary diagonals, wherein the elements in the main diagonal of the matrix H _{p are} all 1 and the elements on each of at least two secondary diagonals of the matrix H _{p are} non-zero elements of the Galois field the code is based.

The invention can be used in a wireless or wired transmission system in a broadcasting or multicasting scenario, in particular for the multimedia broadcast multicast service MBMS.

It should be noted that the elements on the main diagonal and on the at least two secondary diagonals of the matrix H _{p are} each generally different (when viewing the individual diagonals), but it is possible that these elements repeat. This depends on the value of the parameters n and k of the designated GeIRA code and the order q of the Galois field. These elements differ from each other (when viewing the individual diagonals) or can be repeated. For example, if an order 4 Galois field is used, there are only three nonzero elements {1, 2, 3}, and if there are six elements on the main diagonal (n - k = m = 6) the diagonal as its elements z. B. 1, 3, 2, 1, 3, 2, or 3, 1, 1, 2, 3, 2, or z. 3, 1, 3, 2, 1, 1, or any other combination of elements of the group of non-zero elements of the Galois field. Thus, the elements may not all be different from each other. The same can also be the case for the elements of each of the at least two secondary diagonals. As long as the at least two chosen secondary diagonals of the matrix H have _p elements in a number less than the number of non-zero elements of the Galois field, it is possible that the elements of the respective secondary diagonals are different from those of the other differ. Again, however, this is not necessarily required.

In the following, preferred embodiments of the invention are explained in more detail with reference to the drawings. Show it:

1 a schematic representation of the packet-level coding scenario,

2 a coding scheme for binary GeIRA codes,

3 a diagram showing the performance of the (256, 128) GeIRA in Galois fields of different orders,

4 a coding scheme for non-binary GeIRA codes according to a first embodiment of the invention and

5 a coding scheme for non-binary GeIRA codes according to a second embodiment of the invention.

wobei die Einträge der Matrix Nicht-Null-Einträge in GF(q) sind, die beliebig gegeben sind. Anzumerken ist, dass Elemente

auf der Haupt-Diagonalen generell für verschiedene i, i = j, ..., m – 1 verschieden sind und auch die Elemente

auf den sekundären Diagonalen für verschiedene i = j, ..., m – 1 verschieden sind, jedoch auch wiederholt werden können.Binary and non-binary GeIRA codes are completely specified by means of the parity check matrix H. The design of non-binary codes is easily accomplished by constructing the parity check matrix of the code for the binary field and then replacing the non-zero entries in GF (2) with any non-zero entries in GF (q). Thus, the multi-diagonal part of the parity check matrix for a non-binary GeIRA code has the following structure:

where the entries of the matrix are non-zero entries in GF (q) given arbitrarily. It should be noted that elements

on the main diagonal are generally different for different i, i = j, ..., m - 1 and also the elements

on the secondary diagonal for different i = j, ..., m - 1 are different, but can also be repeated.

mit i = 0, ..., m – 1 sind die Invers-Multiplikative von GF(q) des entsprechenden Elements auf der Haupt-Diagonalen von H_p, und h erfüllt 1 ≤ h ≤ m – 1. Dieses Schema lässt sich durch die folgenden Schritte beschreiben (wobei jedes Codewort-Symbol ein Element von GF(q) ist, das mit log₂(q)-Bits ausgedrückt wird):

• 1. Der k-Symbol-Längen-Vektor u wird mit einer Matrix H T / u niedriger Dichte multipliziert, wobei Additionen und Multiplikationen an GF(q) vorgenommen werden. Dadurch wird der m-Symbol-Längen-Vektor v erzeugt.
• 2. Der m-Symbol-Vektor v tritt in einen rekursiven Rate-1-Kodierer ein, wobei
  
  i = 0, ..., m – 1 die in der Haupt-Diagonalen von H_p definierten nichtbinären Elemente sind und
  
  ..., h und i = j, ..., m – 1 die nichtbinären Elemente in den sekundären Diagonalen von H_p sind. Es wird der Paritäts-m-Bit-p-Vektor erzeugt. Insbesondere erhält man ausgehend von dem gegebenen ersten kodierten Paritäts-Symbol
  
  die anderen Paritäts-Symbole p_i durch die Rekursion
  
  für i = 1, 2, ..., m – 1, wobei 1 ≤ h ≤ m – 1. Wiederum werden algebraische Funktionen in einem Galois-Feld der Ordnung q ausgeführt.
• 3. Das n-symbol-kodierte Codewort c wird erhalten durch Konkatenieren von u und p, was zu c = [u|p] führt.

The coding scheme is in 4 shown. The coefficients

with i = 0, ..., m - 1 are the inverse multiplicative of GF (q) of the corresponding element on the main diagonal of H _p , and h satisfies 1 ≤ h ≤ m - 1. This scheme can be understood by describe the following steps (where each codeword symbol is an element of GF (q) expressed with log ₂ (q) bits):

• 1. The k-symbol length vector u is given a matrix H T / u multiplied by low density, with additions and multiplications made to GF (q). This generates the m symbol length vector v.
• 2. The m symbol vector v enters a recursive rate 1 encoder, where
  
  i = 0, ..., m-1 are the non-binary elements defined in the main diagonal of H _p , and
  
  ..., h and i = j, ..., m - 1 are the non-binary elements in the secondary diagonal of H _p . The parity m-bit p vector is generated. In particular, one obtains from the given first coded parity symbol
  
  the other parity symbols p _i by the recursion
  
  for i = 1, 2, ..., m - 1, where 1 ≤ h ≤ m - 1. Again, algebraic functions are performed in a Galois field of order q.
• 3. The n-symbol coded codeword c is obtained by concatenating u and p, resulting in c = [u | p].

für i = 0, ..., m – 1 multipliziert. Folglich weist der Teil von H, der entsprechend dem Paritätsteil des Codeworts multipliziert wird, nun die folgende Struktur auf:

The scheme according to 4 can be improved by taking care that multiplying a row of the parity check matrix H for an element does not affect the relation c H ^T = 0 which must be satisfied by a generic codeword c. Thus, each row of H can be multiplied by a generic element β pertaining to GF (q) without changing the characteristics of the code. This operation is also known as elementary row operation. Thus, after the non-binary parity check matrix, the i-th row of H is present

multiplied by i = 0, ..., m - 1. Thus, the part of H multiplied according to the parity part of the codeword now has the following structure:

j = 1, ..., h und i = j, ..., m – 1, wobei 1 ≤ h ≤ m – 1. Die Leistungsfähigkeit des Codes wird nach diesen elementaren Zeilen-Operationen nicht verändert, doch das Kodierschema ist vereinfacht. Dieses Kodierschema ist in 5 gezeigt.Here, elements on the main diagonal are normalized to 1, and the α coefficients are non-binary elements in GF (q), which are obtained as

j = 1, ..., h and i = j, ..., m-1, where 1≤h≤m-1. The performance of the code is not changed after these elementary line operations, but the coding scheme is simplified , This coding scheme is in 5 shown.

• 1. Der k-Symbol-Längen-Vektor u wird mit einer Matrix H T / u niedriger Dichte multipliziert, wobei Additionen und Multiplikationen an GF(q) vorgenommen werden. Dadurch wird der m-Symbol-Längen-Vektor v erzeugt.
• 2. Der m-Symbol-Vektor v tritt in einen rekursiven Rate-1-Kodierer ein, wobei
  
  j = 1, ..., h und i = j, ..., m – 1 die nichtbinären Elemente in den sekundären Diagonalen von H_p sind. Es wird der Paritäts-m-Symbol-p-Vektor erzeugt. Insbesondere erhält man ausgehend von dem gegebenen ersten kodierten Paritäts-Symbol p₀ = v₀ die anderen Paritäts-Symbole p_i mit der Rekursion
  
  für i = 1, 2, ..., m – 1 wobei 1 ≤ h ≤ m – 1. Wiederum werden algebraische Funktionen in einem Galois-Feld der Ordnung q ausgeführt.
• 3. Das n-symbol-kodierte Codewort c wird erhalten durch Konkatenieren von u und p, was zu c = [u|p] führt.

The new scheme can be summarized by the following steps:

• 1. The k-symbol length vector u is given a matrix HT / u multiplied by low density, with additions and multiplications made to GF (q). This generates the m symbol length vector v.
• 2. The m symbol vector v enters a recursive rate 1 encoder, where
  
  j = 1, ..., h and i = j, ..., m - 1 are the non-binary elements in the secondary diagonal of H _p . The parity m symbol p vector is generated. In particular, starting from the given first coded parity symbol p ₀ = v _0, the other parity symbols p _{i are obtained} with the recursion
  
  for i = 1, 2, ..., m - 1 where 1 ≤ h ≤ m - 1. Again, algebraic functions are performed in a Galois field of order q.
• 3. The n-symbol coded codeword c is obtained by concatenating u and p, resulting in c = [u | p].

This scheme requires fewer operations than the previous one. In particular, m multiplications on GF (q) are saved for coding a codeword.

Multiplikationen in GF(q) eingespart.In the foregoing, two novel coding schemes for non-binary GeIRA codes designed according to the invention have been described. The described schemes allow the use of the flexibility of GeIRA codes even for non-binary fields. In particular, Scheme 2 allows, as compared to Scheme 1, the saving of m multiplications to GF (q) for each coded codeword. The savings are even greater given that the target application is packet-level encoding. In fact, the encoding process is at the packet level in the 1 in which k packets are coded from L bytes to n packets of L size bytes: Compared to Scheme 1

Saved multiplications in GF (q).

A detailed description of the coding schemes has already been made. The two schemes have been tested by software implementation as part of LDPC codes for the packet clear channel.

mal laufen muss. Zu beachten ist, dass

die Anzahl von Symbolen von GF(q) ist, die in einem Paket von Bytes der Größe L enthalten ist, da log₂q die Menge an Bits ist, die zum Ausdrücken von Symbolen von GF(q) erforderlich ist.The described GeIRA code was designed and simulated. Codewords were coded according to the two schemes described. The complexity of coding a codeword depends on the packet size L, since one codeword is a block of Lxn bytes and the coder is per codeword

have to walk. It should be noted that

is the number of symbols of GF (q) contained in a packet of L size bytes, since log ₂ q is the amount of bits required to express symbols of GF (q).

(Packet-Level-Kodierung). Tabelle 1. Kodier-Komplexität für den (256, 128) GeIRA-Code für F(256). Standard-Kodierschema Schema 1 Schema 2 Anzahl von Additionen für GF(256) {128 – 1)·128·(255/256)= 16193 {62*{8 – 1) + (128 – 62)·(7 – 1)} + {(62 – 1) + (66·2)} = 1023 {62·(8 – 1) + (128 – 62)·(7 – 1)} + {(62 – 1) + (66·2)} = 1023 Anzahl von Multiplikationen für GF(256) 128·128·(255/256) = 16320 {[62·8 + (128 – 62)·7]}+ {(128 – 1)+ (128 – 62) + 128} = 1279 {[62·8 + (128 – 62}·7]} + {(128 – 1) + (128 – 62)} = 1151 Table 1 shows a comparison of the complexity of the two proposed coding schemes with the complexity of the general non-binary coding scheme (multiplication for the generator matrix G). The complexity is given by the necessary elementary operations (additions and multiplications) to encode the designated (256, 128) GeTRA code for GF (256). The results listed in Table 1 are divided by

(Packet-level coding). Table 1. Coding complexity for the (256, 128) GeIRA code for F (256). Standard coding Scheme 1 Scheme 2 Number of additions for GF (256) {128 - 1) · 128 · (255/256) = 16193 {62 * {8 - 1) + (128 - 62) · (7 - 1)} + {(62 - 1) + (66 · 2)} = 1023 {62 · (8 - 1) + (128 - 62) · (7 - 1)} + {(62 - 1) + (66 · 2)} = 1023 Number of multiplications for GF (256) 128 · 128 · (255/256) = 16320 {[62 · 8 + (128 - 62) · 7]} + {(128 - 1) + (128 - 62) + 128} = 1279 {[62 · 8 + (128 - 62} · 7]} + {(128 - 1) + (128 - 62)} = 1151

Scheme 1 saves about 94% additions and 92% multiplies over a standard coding scheme.

Scheme 2 saves about 94% additions and 93% multiplies over a standard coding scheme.

Multiplikationen gegenüber GF(q) einspart.It should also be noted here that, relative to Scheme 1, Scheme 2

Saves multiplication to GF (q).

The two coding schemes according to the present invention are applicable to all types of commercial wireless and wired transmission systems. As shown, the proposed algorithms allow reducing the complexity of a packet-level coder so that they can also be applied to low-billboard platforms.

List of abbreviations used

• PEC

  Packet erasure channel

  LDPC

  Parity check (code) of low density

  GF

  Galois field

  ML

  highest probability

  CRC

  Cyclic redundancy check

  IRA

  Irregular Repeat Accumulate (Code)

  Geira

  Generalized-Irregular-Repeat-Accumulate (code)

literature

• [1] T. Richardson and R. Urbanke, "Efficient encoding of low density parity check codes", IEEE Trans. Inform. Theory, Vol. 47, pp. 638-656, Feb. 2001.
• [2] Y. Kou, S. Lin, and M.P. Fossorier, "Low density parity-check codes based on finite geometries: a rediscovery and new results", IEEE Trans. Inform. Theory, vol. 47, Nov. 2001.
• [3] H. Jin, A. Khandekar, and R. McEliece, "Irregular repeat-accumulate codes," in Proc. 2 ^nd Intern. Symp. On Turbo Codes and Related Topics, Sept. 2000, pp. 1-8.
• [4] G. Liva, E. Paolini, M. Chiani, "Simple Reconfigurable Low-Density Parity-Check Codes", IEEE Communications Letters, Vol. 9, No. 3, pp. 258-260, March 2005.
• [5] E. Paolini, G. Liva, B. Matuz, M. Chiani, "Generalized IRA Erasure Correcting Codes for Hybrid Iterative / Maximum Likelihood Decoding," IEEE Communications Letters, Vol. 12, No. 6, p. 450- 452, June 2008.
• [6] G. Liva, E. Paolini, M. Chiani, "Packet loss recovery in the telemetry downlink via maximum-likelihood LDPC decoding", 10 ^th International workshop on signal processing for space communications, Oct. 2008.
• [7] Garramone, G .; Matuz, B .: Short Erasure Correcting LDPC IRA Codes over GF (q). In: IEEE Global Telecommunication Conference, 2010, p. 1-5. - ISSN 1930-529X
• [8] Chiu, M.-C .: Bandwidth-Efficient Modulation Codes Based on Nonbinary Irregular Repeat-Accumulate Codes. In: IEEE Transactions on Information Theory, Vol. 56, 2010, no. 1, pp. 152-167. - ISSN 0018-9448
• [9] Lin, W. [et al]: Design of-ary Irregular Repeat-Accumulate Codes. In: International Conferences on Advanced Information Networking and Applications, 2009, pp. 201-206. - ISSN 1550-445X

Claims (7)

An apparatus for encoding (k × 8 × L) information bits by encoding L byte data packets by means of a low density parity check code in the form of a non-binary Generalized Irregular Repeat Accumulate Code which is encoded on a Galois field (GF (q)) of order q, where q is greater than 2, and where the Galois field (GF (q)) contains a number q of elements, the device comprising: a parity check matrix, which is stored in a memory and has two concatenated matrices H _u and H _p , Where H _{u is} a sparsely populated matrix and H _{p is} a structured square matrix having elements on a major diagonal and on a group of at least two secondary diagonals, a vector multiplier for multiplying a k symbol information vector u by the one transposed sparse matrix HT / u to obtain an m-symbol vector v, where a symbol of the information vector u is an element of the Galois field (GF (q)), and - a recursive rate-1 encoder to obtain an m-symbol parity Vector p from the m-symbol vector v, - where the elements of the recursive encoder are defined by the elements of the square matrix H _p , where the elements in the main diagonal of the matrix H _p non-zero elements of the Galois field (GF (q)) and the elements on each of the at least two secondary diagonals of the group of secondary diagonals of the matrix H _{p are} also non-zero elements of the Galois field (GF (q)). An apparatus for encoding (k × 8 × L) information bits by encoding L byte data packets by means of a low density parity check code in the form of a non-binary Generalized Irregular Repeat Accumulate Code which is encoded on a Galois field (GF (q)) of order q, where q is greater than 2, and where the Galois field (GF (q)) contains a number q of elements, the device comprising: a parity check matrix, which is stored in a memory and has two concatenated matrices H _u and H _p , where H _{u is} a sparse matrix and H _{p is} a structured square matrix containing elements on a major diagonal and on a group of at least two secondary Diagonal, a vector multiplication device for multiplying a k-symbol information vector u by the transposed sparse matrix HT / u to obtain an m-symbol vector v, where a symbol of the information vector u is an element of the Galois field (GF (q)), and - a recursive rate-1 encoder to obtain an m-symbol parity Vector p from the m symbol vector v, - wherein the elements of the recursive encoder are defined by the elements of the square matrix H _p , where the elements in the main diagonal of the matrix H _{p are} all 1 and the elements on each of the at least two secondary diagonals of the group of secondary diagonals of the matrix H _{p are} non-zero elements of the Galois field (GF (q)). Apparatus according to claim 1 or 2, characterized in that the elements within each of the main diagonals and within at least two secondary diagonals of the group of secondary diagonals each differ from each other or can be repeated or partially equal. A method of recovering lost and / or damaged data sent from a transmitter to a receiver, the method comprising the steps of: - encoding the data by means of a transmitter-side encoder, - transmitting the encoded data from the transmitter to the receiver via a transmission system and decoding the received data by means of a receiver-side decoder, wherein lost and / or corrupted data is recovered during decoding, the encoding being performed by means of a non-binary generalized irregular repeat accumulate code generated on a Galois field (GF (q)) of the order q, where q is greater than 2, and wherein the Galois field (GF (q)) contains a number q of elements, and - wherein the decoding by dissolving the decoding by the decoder Matrix of the code defined equation system is performed, characterized in that - the coding of the data by means of a Parity check matrix H having two concatenated matrices H _u and H _p , where H _{u is} a sparsely populated matrix and H _{p is} a structured square matrix having elements on a major diagonal and on a group of at least two secondary diagonals , where the elements in the main diagonal of the matrix H _{p are} non-zero elements of the Galois field (GF (q)) and the elements on each of the at least two secondary diagonals of the group of secondary diagonals of the matrix H _{p are} also non-zero elements. Zero elements of the Galois field (GF (q)) are. A method of recovering lost and / or damaged data sent from a transmitter to a receiver, the method comprising the steps of: encoding the data by means of a transmitter-side encoder, - transmitting the encoded data from the transmitter to the receiver via a transmission system, and decoding the received data by means of a receiver-side decoder, wherein lost and / or corrupted data is recovered during decoding, the encoding being performed by means of a non-binary generalized irregular repeat accumulate code which is applied to a Galois field of the Order q, where q is greater than 2, and where the Galois field (GF (q)) contains a number q of elements, the decoding being performed by resolving the system of equations defined by the decoding matrix of the code, characterized in that - coding the data by means of a parity check m trix H having two concatenated matrices H _u and H _p , where H _{u is} a sparsely populated matrix and H _{p is} a structured square matrix having elements on a major diagonal and on a group of at least two secondary diagonals where the elements in the main diagonal of the matrix H _{p are} all 1 and the elements on each of the at least two secondary diagonals of the secondary diagonal group of the matrix H _{p are} non-zero elements of the Galois field (GF (q)) , A method according to claim 4 or 5, characterized in that the elements within each of the main diagonals and within at least two secondary diagonals of the group of secondary diagonals are each different or repeating each other or partially equal to each other. Use of the device according to one of claims 1 to 3 and / or the method according to one of claims 4 to 6 in a wireless or wired transmission system in a broadcasting or Multicasting scenario, in particular for a UMTS application, in particular for the multimedia broadcast multicast service MBMS.

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