DE102013213778B3 - A method for transmitting a message from a transmitter to a receiver via a communication medium by means of packet-level coding - Google Patents

Abstract

A method is proposed for transmitting a message from a sender to a receiver via a communication medium by means of packet-level coding, in which a special non-binary LDPC code is used, which is defined as follows: - A number b of the variable nodes has one Grade 2, which means that two edges originate from each of these b variable nodes, where b (i) is equal to m / 2 or at most m / 2 or (ii) is the preceding integer that is less than m / 2 ; and - all of the b variable nodes are separated, which means that, with respect to all b variable nodes, the edges emanating from each of the b variable nodes are connected to respective different test nodes.

Description

The present invention relates to a method for transmitting a message from a transmitter to a receiver via a communication medium by means of packet-level coding. In particular, the invention is concerned with performance improvement of a transmission system based on 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 illustrated.

First, the message to be transmitted is divided into k info packets and encoded into a larger number of n codeword packets. Thus, from the packet-level encoder on the transmitter side, m = n-k parity packets are generated by means of a (n, k) block code defined on a Galois field of order q.

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 view 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.

Furthermore, it has been found that erase recovery capabilities are improved when the code design is executed in non-binary arrays; H. the parity check and generator matrix of the code have entries belonging to a Galois field of order q [2 and 3].

Generally, in prior art transmission schemes using packet-level encoding, the values of L, n, and k are set, and a selection is made of the Galois field where the (n, k) Block code is created. If z. For example, if a Galois field of order 256 is selected, then each packet of L bits is composed of L / 8 field symbols of GF (256) in terms of the encoder and the decoder. It should be noted that if the (n, k) block code is defined on a Galois field of order 256, its (n-k) xn parity check matrix contains symbols of GF (256). If the (n, k) code is defined on a Galois field of order 2, its (n-k) xn parity check matrix contains symbols of GF (2). It should be noted that m = n - k.

It is possible to represent an LDPC code by means of a two-part graph associated with the parity check matrix H. The two-part graph consists of a set of variable nodes (VNs), a set of check nodes (CNs), and a set of edges connecting the VNs to the CNs. Variable nodes are assigned to the columns of H, and CNs are assigned to the rows of H. An edge connects a VN to a CN if there is a nonzero entry in the corresponding position of the parity check matrix [4 and 5].

In 2 an example is shown having a parity check matrix on a Galois field (GF) of order 4, ie a Galois field containing 4 symbols (which may be represented as powers of the primitive element alpha). The two part graph associated with the parity check matrix is also in 2 shown. CNs are marked by squares, and VNs are marked by circles. 2 also shows the parity check equations that must be satisfied by a codeword vector v.

wobei λ_i und p_i die Bruchteile von Kanten sind, die mit den VNs und CNs des Grads i verbunden sind, d_v,max der maximale VN-Grad ist und d_c,max der maximale CN-Grad ist.The degree of a VN (or a CN) is defined as the number of edges emanating from it. The edge-oriented degree distribution polynomials are defined as

where λ _i and p _{i are} the fractions of edges associated with the VNs and CNs of degree i, d _{v, max is} the maximum VN degree, and d _{c, max is} the maximum CN degree.

In 3 are the degree distribution polynomials for the in 2 listed two-part graph listed. Furthermore, in 3 the definition of the code rate R is given.

In practice, the design of an LDPC code begins with the search for an appropriate set of LDPC codes which satisfies the constraints imposed by the degree distribution polynomials. In fact, sets of LDPC codes are parameterized by the degree distribution polynomials. Generally, unstructured ensembles are considered in which any permutation of the edges among the VNs and CNs is allowed, provided the constraints given by the degree-distribution polynomials are met [3].

A partially structured set of binary LDPC codes was also examined. In particular, not all edge permutations were allowed with respect to the degree distribution polynomials. In particular, the connectivity of some Grade 2 VNs was examined. Such a set was considered in [6] to improve the error floor performance of binary LDPC codes. However, to the best of the inventors' knowledge, a similar substructure has not yet been considered for nonbinary LDPC code entities.

In [7], see Chapter II, a nonbinary LDPC code with a number of variable nodes with degree 2 is described.

durchgeführt, wobei

ein Vektor ist, der die gelöschten Codewort-Pakete enthält, x_k ein Vektor ist, der die korrekt empfangenen Codewort-Pakete enthält,

die Sub-Matrix ist, die aus den Spalten in der Paritätsprüfmatrix zusammengesetzt ist, die den gelöschten Codewort-Paketen entsprechen, und H_k die Sub-Matrix ist, die aus den Spalten in der Paritätsprüfmatrix zusammengesetzt ist, die den empfangenen Codewort-Paketen entsprechen. Anzumerken ist, dass die rechte Seite des Systems in der Gleichung (1) bekannt ist und es sich bei ihr um einen die Länge (n – k) aufweisenden Vektor von Paketen mit jeweils L Bits handelt, oder äquivalent dazu jeweils mit L/log₂q Feld-Symbolen. Das System in Gleichung (1) kann mittels des Gaußschen Eliminations-Algorithmus durch Operationen gelöst werden, die an dem Galois-Feld vorgenommen werden, an dem der Code gebildet ist.At the receiver side, maximum likelihood (ML) decoding is achieved by solving the systems of linear equations

performed, where

is a vector containing the deleted codeword packets, x _{k is} a vector containing the correctly received codeword packets,

is the sub-matrix composed of the columns in the parity check matrix corresponding to the deleted codeword packets, and H _{k is} the sub-matrix composed of the columns in the parity check matrix corresponding to the received codeword packets , It should be noted that the right side of the system is known in equation (1) and is a vector of packets of L bits each having the length (n-k), or equivalently each with L / log ₂ q field symbols. The system in equation (1) can be solved by the Gaussian elimination algorithm through operations performed on the Galois field where the code is formed.

An object of the present invention is to improve the error correction and error detection performance in a packet level encoding based transmission system.

• – auf der Sender-Seite:
• – Teilen der zu übertragenden Nachricht in k Informationspakete von jeweils L Bits,
• – Kodieren der k Informationspakete in n Codewort-Pakete von jeweils L Bits, wobei n größer als k ist und
• – wobei m = n – k Paritätspakete erzeugt werden, und
• – Schützen der n Codewort-Pakete mittels eines nichtbinären LDPC-Fehler-Detektions- und Prüf-Codes,
• – wobei der LDPC-Code definiert ist durch
• – eine Anzahl n variabler Knoten, wobei diese Anzahl identisch mit der Anzahl n von Codewort-Paketen ist, wobei jeder variable Knoten einen Grad hat und die Grade der variablen Knoten gleich oder verschieden sind,
• – eine Anzahl m von Prüf-Knoten, und
• – Kanten, welche die variablen Knoten mit den Prüf-Knoten verbinden, wobei die Anzahl von Kanten, die von jedem der variablen Knoten abgehen, gleich oder verschieden ist, und die Anzahl von Kanten, die von jedem der Prüf-Knoten abgehen, gleich oder verschieden ist,
• – Übertragen der n Codewort-Pakete von dem Sender zu dem Empfänger über das Kommunikationsmedium, und
• – auf der Empfänger-Seite
• – für jedes der übertragenen Codewort-Pakete, Durchführen einer Fehlerkorrektur mittels des nichtbinären LDPC-Fehler-Korrektur-Codes und Detektieren von Restfehlern mittels des Fehler-Detektions-Codes, wobei ein Codewort, für das ein Fehler detektiert wird, als verloren angesehen und als Löschung markiert wird, und
• – falls eine hinreichende Menge an Codewort-Paketen empfangen worden ist, Dekodieren der empfangenen Codewort-Pakete zum Rückgewinnen der Nachricht, und zwar mittels einer Paritätsprüfmatrix,
• – wobei das Schützen der n Codewort-Pakete mittels eines nichtbinären LDPC-Codes vorgenommen wird, der wie folgt definiert ist:
• – eine Anzahl b der variablen Knoten hat einen Grad 2, was bedeutet, dass von jedem dieser b variablen Knoten zwei Kanten ausgehen, wobei b (i) gleich m/2 oder höchstens m/2 ist oder (ii) die jeweils vorausgehende ganze Zahl ist, welche kleiner als m/2 ist; und
• – sämtliche der b variablen Knoten sind getrennt, was bedeutet, dass hinsichtlich sämtlicher b variablen Knoten die von jedem der b variablen Knoten ausgehenden Kanten mit jeweiligen unterschiedlichen Prüf-Knoten verbunden sind, und
• – wobei das Dekodieren der empfangenen Codewort-Pakete durch Berücksichtigen derjenigen Spalten der Paritätsprüfmatrix, die den gelöschten Codewort-Paketen zugeordnet sind, und durch Lösen der folgenden Gleichung durchgeführt wird:
  
• – wobei
  
  einen Vektor bezeichnet, der die gelöschten Codewort-Pakete enthält, x_k einen Vektor bezeichnet, der die korrekt empfangenen Codewort-Pakete enthält,
  
  die Sub-Matrix ist, die aus den Spalten in der Paritätsprüfmatrix zusammengesetzt ist, die den gelöschten Codewort-Paketen entsprechen, H_k die Sub-Matrix der Paritätsprüfmatrix bezeichnet, die aus den Spalten der Paritätsprüfmatrix zusammengesetzt ist, die den korrekt empfangenen Codewort-Paketen entsprechen, und das hochgestellte T die Transponierte der jeweiligen Matrix bezeichnet.

To this end, according to the invention, a method is proposed for transmitting a message from a sender to a receiver via a communication medium by means of packet-level coding, comprising:

• - on the sender page:
• Dividing the message to be transmitted into k information packets of L bits each,
• Encoding the k information packets into n codeword packets of L bits each, where n is greater than k and
• - where m = n - k parity packets are generated, and
• Protecting the n codeword packets by means of a non-binary LDPC error detection and test code,
• - where the LDPC code is defined by
• A number n of variable nodes, this number being identical to the number n of codeword packets, each variable node having one degree and the grades of the variable nodes being the same or different,
• - a number m of check nodes, and
• Edges connecting the variable nodes to the check nodes, the number of edges going off each of the variable nodes being the same or different, and the number of edges going off each of the check nodes being equal to or different is different
• Transmitting the n codeword packets from the transmitter to the receiver via the communication medium, and
• - on the receiver side
• For each of the transmitted codeword packets, performing error correction by means of the non-binary LDPC error correction code and detecting residual errors by means of the error detection code, a codeword for which an error is detected being considered lost and Deletion is marked, and
• If a sufficient number of codeword packets have been received, decoding the received codeword packets to recover the message by means of a parity check matrix,
• - wherein the protection of the n codeword packets is performed by means of a non-binary LDPC code, which is defined as follows:
• A number b of the variable nodes has a degree 2, which means that two edges emanate from each of these b variable nodes, where b (i) equals m / 2 or at most m / 2 or (ii) the respective preceding integer is less than m / 2; and
• All of the b variable nodes are separated, which means that with respect to all the b variable nodes, the edges originating from each of the b variable nodes are connected to respective different test nodes, and
• Wherein the decoding of the received codeword packets is performed by considering those columns of the parity check matrix associated with the deleted codeword packets and by solving the following equation:
  
• - in which
  
  denotes a vector containing the deleted codeword packets, x _k denotes a vector containing the correctly received codeword packets,
  
  is the sub-matrix composed of the columns in the parity-check matrix corresponding to the deleted codeword packets, H _k denotes the sub-matrix of the parity-check matrix composed of the columns of the parity-check matrix corresponding to the correctly received codeword packets and the superscript T denotes the transpose of the respective matrix.

According to the invention, a specific use of a particular class of non-binary LDPC codes is proposed which offers high performance in terms of codeword error rate (CER) and low error floor.

• – Eine Anzahl b der variablen Knoten hat einen Grad 2, was bedeutet, dass von jedem dieser b variablen Knoten zwei Kanten ausgehen, wobei b (i) gleich m/2 oder höchstens m/2 ist oder (ii) die jeweils vorausgehende ganze Zahl ist, welche kleiner als m/2 ist (d. h. durch Verwenden der mathematischen Abrundungsfunktion von m/2;
• – sämtliche der b variablen Knoten sind getrennt, was bedeutet, dass hinsichtlich sämtlicher b variablen Knoten die von jedem der b variablen Knoten ausgehenden Kanten mit jeweiligen unterschiedlichen Prüf-Knoten verbunden sind;
• – folglich ist jede Kanten-Permutation für Kanten erlaubt, die von VNs eines sich von 2 unterscheidenden Grads ausgehen;
• – folglich bedeutet eine Kante, die einen VN mit einem CN verbindet, dass in der entsprechenden Position der Paritätsprüfmatrix H kein Nicht-Null-Symbol des Galois-Felds existiert, an dem der Code gebildet ist; und
• – die Anzahl der VNs sowie der CNs kann gerade oder ungerade sein.

In accordance with the invention, a partially structured set of non-binary packet-level LDPC codes is used, using separate degree -2 degree variable nodes (VNs). In particular, according to the invention, the nonbinary LDPC code used is defined as follows:

• A number b of the variable nodes has a degree 2, which means that two edges originate from each of these b variable nodes, where b (i) is equal to m / 2 or at most m / 2 or (ii) the respective preceding integer which is smaller than m / 2 (that is, by using the mathematical rounding function of m / 2;
• All of the b variable nodes are separated, which means that, with respect to all the b variable nodes, the edges originating from each of the b variable nodes are connected to respective different check nodes;
• Thus, any edge permutation is allowed for edges starting from VNs of a degree different from 2;
• Consequently, an edge connecting a VN to a CN means that in the corresponding position of the parity check matrix H there is no non-zero symbol of the Galois field on which the code is formed; and
• - The number of VNs and CNs can be even or odd.

According to a preferred embodiment, for the variable nodes other than said b variable nodes having a degree 2, each edge permutation is allowed. These other variable nodes, too, may not have Grade 2, but may or may not be separate.

In the following the invention will be described in more detail with reference to the drawings. Show it:

1 the scenario of packet-level coding,

2 an example of a non-binary parity check matrix of a non-binary LDPC code and its associated two-part graph;

3 shows the degree distribution polynomial for the code example according to FIG

2 , and the definition of the code rate R,

4 shows the partially structured LDPC composite proposed according to the invention,

5 shows the degree distribution polynomials for the designated and simulated codes according to the invention, and

6 shows examples of the performance of two non-binary LDPC codes designated according to the invention, also showing the performance of their binary counterpart.

As will be shown below, nonbinary LDPC codes from a particular class of partially structured entities achieve very good performance in terms of codeword error rates and low error floor. Furthermore, these codes improve the performance of their binary counterparts. In the context of the following embodiment, a transmission scheme based on non-binary LDPC codes taken from a partially structured entity will be described.

4 shows the nonbinary, partially structured entity proposed by the invention. VNs are indicated by circles and CNs by squares. Each edge is assigned a non-zero symbol of a Galois field of order q.

Especially

• - m / 2 variable degree 2 nodes exist (shown at the top of 4 )
• - all grade 2 VNs are separated, i. H. they have no common neighbors (the neighbors of a VN are the CNs connected to it by an edge),
• So that each edge permutation is allowed for an edge going off of residual VNs (ie, outside the group of m / 2 separate VNs of degree 2), which typically have a degree different from 2 and which are separate can or not,
• An edge connecting a VN to a CN is to be understood as meaning that in the corresponding position of the parity check matrix there is no non-zero symbol of the Galois field on which the code is formed.

The invention proposes a class of nonbinary, partially structured LDPC codes that results in high performance and low error floors. Therefore, the proposed solution appears very attractive in transmission systems where such features are required. In particular, the amount of grade 2 VNs and their connectivity are carefully examined.

6 Figure 12 shows the simulated power, expressed in terms of the codeword error rate (CER) versus packet deletion probability, for (256, 128) LDPC codes on Galois fields of order 4 and 16, respectively. The codes are in accordance with the present invention designated. Both codes have a code rate equal to 0.5 and satisfy the degree distribution polynomials according to FIG 5 ,

The obtained power curves were tested against the singleton boundary, a lower bound on the block error probability of any (n, k) linear block code. It should be noted that the binary code is outperformed by non-binary codes. Further, even with a short codeword length n = 256, the performance of the non-binary codes closely approximates the singleton boundary, and for a code on the GF (16), no error floor for CER is visible down to at least 10 ^-8 ,

The proposed class of non-binary LDPC codes can be used in any commercial wireless and wireline transmission system. As pointed out here, the proposed transmission scheme already offers low-order Galois fields, i. h with low complexity, high performance, so that it can also be applied to low-performance platforms.

List of abbreviations used

• PEC

  Packet Erase Channel (Packet Erasure Channel)

  LDPC

  Parity check (code) with low density (Low-density parity-check (code))

  GF

  Galois field

  ML

  highest probability (maximum-likelihood)

  CRC

  cyclic redundancy check (Cyclic Redundancy Check)

  CERIUM

  Codeword Error Rate

  CN

  Check node (check node)

  VN

  variable node (variable node)

literature

• [1] J. Byers, M. Luby, M. Mitzenmacher, and A. Rege, "A digital fountain approach to reliable distribution of bulk data", SIGCOMM Comput. Commun. Rev., vol. 28, no. 4, pages 56-67, Oct. 1998
• [2] G. Garrammone, B. Matuz, "Short Erasure Correcting LDPC IRA Codes over GF (q)", GLOBECOM 2010, pages 1-5
• [3] G. Garrammone, E. Paolini, B. Matuz, G. Liva, M. Chiani, "Non-Binary Low-Density Parity-Check Codes for the q-ary Erasure Channel", IEEE International Conference on Communications (ICC ), Jun. 2013, Budapest, Hungary
• [4] "Channel Coding", 3rd edition, Prof. Dr.-Ing. Martin Bossert, Oldenbourg Wissenschaftsverlag 2013, pages 126-129
• [5] "Low Density Parity Check Codes" - An Introduction, Tilo Strutz, 2010-2013, February 26, 2013
• [6] T. Richardson, A. Shokrollahi, R. Urbanke, "Finite-length Analysis of Various Low-Density Parity-check Codes for the Binary Erasure Channel", IEEE International Symposium on Information Theory (ISIT), 2002, Lausanne, Switzerland
• [7] Klinc, D., Ha, J .; McLaughlin, SW: Optimized Puncturing and Shortening Distributions for Nonbinary LDPC Codes over the Binary Erasure Channel. In: 46 ^th Annual Allerton Conference on Communication, Control and Computing, 2008, pp 1053-1058

Claims (3)

A method for transmitting a message from a transmitter to a receiver via a communication medium by means of packet-level coding, comprising: on the transmitter side: dividing the message to be transmitted into k information packets of L bits each, encoding the k information packets in n codeword packets of L bits each, where n is greater than k and - where m = n - k parity packets are generated, and - protecting the n codeword packets by means of a non-binary LDPC error detection and check code, - where the LDPC code is defined by A number n of variable nodes, this number being identical to the number n of codeword packets, each variable node having one degree and the grades of the variable nodes being the same or different, a number m of check nodes, and Edges connecting the variable nodes to the check nodes, wherein the number of edges that depart from each of the variable nodes are the same or different, and the number of edges that depart from each of the check nodes are the same or different - transferring the n codeword packets from the transmitter to the receiver via the communication medium, and - at the receiver side - for each of the transmitted codeword packets, performing error correction by means of the non-binary LDPC error correction code and detecting residual errors by means of the error detection code, wherein a code word for which an error is detected is considered lost and marked as deletion, and if so e has been received sufficient amount of codeword packets, decoding the received codeword packets to recover the message, by means of a parity check matrix, characterized in that - the protection of the n codeword packets is performed by means of a non-binary LDPC code is defined as follows: a number b of the variable nodes has a degree 2, which means that two edges emanate from each of these b variable nodes, b (i) being equal to m / 2 or at most m / 2 or (ii) is the respective preceding integer smaller than m / 2, and all of the b variable nodes are separated, which means that, with respect to all the b variable nodes, the edges originating from each of the b variable nodes are connected to respective different check nodes and decoding the received codeword packets by considering those columns of the parity check matrix associated with the deleted codeword packets ordered by solving the following equation:

in which

denotes a vector containing the deleted codeword packets, x _k denotes a vector containing the correctly received codeword packets,

is the sub-matrix composed of the columns in the parity-check matrix corresponding to the deleted codeword packets, H _k denotes the sub-matrix of the parity-check matrix composed of the columns of the parity-check matrix corresponding to the correctly received codeword packets and the superscript T denotes the transpose of the respective matrix. A method according to claim 1, characterized in that in addition to said b variable nodes having a degree 2, all edge permutation is allowed for all other variable nodes. Method according to claim 1 or 2, characterized in that, apart from those b variable nodes having a degree 2, all other variable nodes are separate or not separated and / or have degrees other than 2.

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