DE102013219863B4 - A method of transmitting a message from a sender to a receiver via a communication medium using packet-level encoding - Google Patents

Abstract

Method for transmitting a message from a sender to a receiver via a communication medium by means of packet-level coding, comprising the following steps: at the sender side splitting the message to be transmitted into k information packets of L bits (input characters) Encoding the k information packets into n codeword packets of L bits (output characters), where n is greater than k and where m = n-k parity packets are generated, and - protecting the n codeword packets by errors Detection and correction codes, transmitting the n codeword packets and m parity packets over the communication medium from the sender to the receiver, and at the receiver side for each of the transmitted codeword packets, performing an error correction by means of the Error correction codes and detection of residual errors by means of the error detection code, wherein a codeword packet on which an error is detected, considered lost and as Lösc hung, and for the purpose of recovering the message, decoding the received codeword packets by means of a concatenated parity check matrix comprising an information part and a parity check part, if a sufficient number of codeword packets have been received, characterized in that - the coding of the k information packets is carried out by means of Maximum Distance Separable (MDS) codes based on Vandermonde or Cauchy matrices which are checked with non-binary or binary low-density parity checks. (LDPC) codes are serially concatenated, which results in the concatenated total parity check matrix having: (I) as its information part (i) a densely packed first sub-matrix which the parity check matrix ...

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, as is basically known from [3].

Packet-level encoding is a simple and efficient technique for guaranteeing reliable transmission in a communication system [1]. The main idea is to further protect the information to be sent by encoding data packets on higher layers of the communication stack. The scenario to which this invention pertains, ie packet level encoding with serial concatenated MDS codes (Vandermonde / Cauchy matrix based codes) and LDPC codes, is disclosed in U.S.P. 1 shown. (The coding of the information packets by means of MDS codes with serially concatenated LDPC codes is described in [9].)

First, the message to be transmitted, z. A FILE, is divided into k info packets of L bits (input characters) and encoded into n code characters which are packets of L bits (output characters). Then, the n output characters are generated by the packet-level encoder.

Second, the n output characters on the physical layer (within the PHY layer frame) are protected by error correction codes (eg, Turbo, LDPC ...) and error detection codes (eg, cyclic redundancy checks (CRC)) and they are sent. Third, a physical layer error correction is made on each packet at the receiver side, and residual errors are detected by means of the error detection code. If errors are detected, the packet is deemed lost and marked as deletion. Thus, the layers located above the physical layer regard the communication medium as a packet deletion channel (PEC) at which the packets are either correctly received or lost. Finally, if a sufficient amount of packets has been received, the packet level decoder will recover the original message.

wobei

die Paritätsprüfmatrix des MDS-Codes der Größe m_MDS × n_MDS ist,

der Informationsteil der Paritätsprüfmatrix des LDPC-Codes der Größe m_LDPC × k_LDPC ist,

der Paritätsteil der Paritätsprüfmatrix des LDPC-Codes der Größe m_LDPC × m_LDPC ist, und 0 die Null-Matrix der Größe m_MDS × m_LDPC ist.The parity check matrix H of the concatenated code can be written as

in which

the parity check matrix of the MDS code of size m _MDS × n _MDS is,

the information part of the parity check matrix of the LDPC code of size m is _LDPC × k _LDPC ,

the parity part of the parity check matrix of the LDPC code of size m is _LDPC × m _LDPC , and 0 is the zero matrix of size m _MDS × m _LDPC .

vorgenommen, wobei

ein Vektor ist, der die gelöschten kodierten Symbole enthält, x_k ein Vektor ist, der die korrekt empfangenen kodierten Symbole enthält,

die Sub-Matrix ist, die aus den Spalten in der Paritätsprüfmatrix zusammengesetzt ist, welche den gelöschten Zeichen entsprechen, und H_k die Sub-Matrix ist, die aus den Spalten in der Paritätsprüfmatrix zusammengesetzt ist, welche den empfangenen Zeichen entsprechen.The decoding is done on the PEC by solving the linear equation system

made, where

is a vector containing the deleted coded symbols, x _{k is} a vector containing the correctly received coded symbols,

is the sub-matrix composed of the columns in the parity-check matrix corresponding to the deleted characters, and H _{k is} the sub-matrix composed of the columns in the parity-check matrix corresponding to the received characters.

die dünnbesetzte Matrix ist, die das Gleichungssystem beschreibt, die gelöst werden sollen, um die Löschungen rückzugewinnen.Note that the right side of the system is known in (1), and that

is the sparsely populated matrix that describes the equation system that should be solved to recover the deletions.

The prior art decoding consists of solving the equation system in (1) by means of hybrid iterative / maximum likelihood techniques based on smart Gaussian elimination (GE), in which the sparse nature of the system is used to advantage ; see, for. For example, [7], [8]. These decoding techniques can also be used to decode non-binary LDPC codes [2].

In particular, the complexity of smart GE techniques is cubic with respect to the number of pivots p of the code, ie, O (p ³ ).

Hybrid iterative / maximum likelihood decoding based on smart Gaussian elimination (GE)

A brief overview of the hybrid decoding method will be given. Furthermore, upper limits of the number of operations to be performed on the Galois field are required, which are required of each step of the method.

The complexity of the prior art decoding algorithm is given in terms of the number of field operations required. The number of field additions is given as N _a , and the number of field multiplications is given as N _m .

Feld-Elementen enthält.It specifies: with L (bits) the package size of the schema, with d (max) / v the maximum variable node degree of the code, with d (max) / c the maximum check node degree of the code, with e _IT the number of erasures retrieved by iterative decoding, e 'the number of erasures remaining after IT decoding, and p the number of pivots of the code , It should be noted that if codes are created on a Galois field of order q, where q is a power of 2, each packet size of L bits is a number of bits

Contains field elements.

The following operations are performed:

1. IT decoding step

Equations with only a single unknown are solved by an IT decoder.

Upper limits for the number of operations involved:

2. Triangulation and Pivoting Step

in eine tiefere triangulare Form gebracht, indem nur Reihen und Spalten permutiert werden. Falls die Triangulation steckenbleibt, wird eine Pivoting-Strategie angewandt, was bedeutet, dass Spalten der nicht triangulierten Matrix ausgewählt und zur Seite der Matrix bewegt werden. Die kodierten Zeichen, die den bewegten Spalten entsprechen, werden als pivots bezeichnet. Innerhalb dieses Schritts sind keine Feld-Operationen involviert.If the iterative (IT) decoder fails, the remaining matrix becomes

into a deeper triangular shape by permuting only rows and columns. If the triangulation gets stuck, a pivoting strategy is applied, which means that columns of the non-triangulated matrix are selected and added to the Side of the matrix to be moved. The coded characters corresponding to the moving columns are called pivots. Within this step, no field operations are involved.

3. Cancel step

The matrix below the main diagonal of the sub-triangular matrix is transformed by row additions into a zero matrix. Upper limits for the number of operations involved:

4. Gaussian elimination (GE)

A rollover GE is applied to solve the size system (p + δ) × p, which is described only by the pivot variables. δ is the overhead of the system.

This involved (for k = p):

5. IT decoding step

If the GE was successful, an IT decode step is sufficient to recover the remaining e'-p unknowns, which are a linear combination of the pivots.

Upper limits for the number of operations involved:

dünnbesetzt ist. Somit wird die Gesamt-Komplexität von Schritt 4 dominiert, wobei das von den Pivot-Variablen beschriebene lineare System aufgrund der Operationen von Schritt 3 dichtbesetzt ist. Insbesondere weist Schritt 4 die Komplexität O(p³) auf, wobei p die Anzahl der Pivots ist. In der Tat ist der Algorithmus effizient, solange die Anzahl der Pivots klein gehalten wird. Hochentwickelte Pivoting-Strategien sind in [7] untersucht worden.It should be noted that row permutations and series additions are also made on the known term of the system given by the right-hand side of (1). It should also be noted that the matrix

is sparse. Thus, the overall complexity of step 4 is dominated, with the linear system described by the pivot variables being densely populated due to the operations of step 3. In particular, step 4 has the complexity O (p ³ ), where p is the number of pivots. In fact, the algorithm is efficient as long as the number of pivots is kept small. Sophisticated pivoting strategies have been studied in [7].

The decoding complexity is cubic with respect to the number of pivots, and the number of pivots increases with concatenated codes.

The median number of concatenated code pivots is higher in the context of the standalone LDPC code (see 4 and 5 ). Let θ + p be the number of concatenated code pivots referred to, wherein θ _max <θ (Δ) ≤ 0 and θ = θ _max (Δ = 0); where Δ is the overhead of the RX page, ie the number of received coded characters in addition to k. The asymptotic complexity of the prior art decoding is thus O ((θ + p) ³ ).

The task addressed by this invention is to reduce the complexity of decoding serial concatenated MDS codes (based on Vandermonde or Cauchy matrices) and LDPC codes when applied at the packet level.

• – an der Sender-Seite
• – Aufteilen der zu übertragenden Nachricht in k Informationspakete von jeweils L Bits (Eingabe-Zeichen),
• – Kodieren der k Informationspakete zu n Codewort-Paketen von jeweils L Bits (Ausgabe-Zeichen), wobei n größer als k ist und wobei m = n – k Paritäts-Pakete erzeugt werden, und
• – Schützen der n Codewort-Pakete durch Fehler-Detektions- und -Korrektur-Codes,
• – Übertragen der n Codewort-Pakete und m Paritäts-Pakete über das Kommunikationsmedium vom Sender zum Empfänger, und
• – an der Empfänger-Seite
• – für jedes der gesendeten Codewort-Pakete, Durchführen einer Fehlerkorrektur mittels des Fehler-Korrektur-Codes und Detektion von Restfehlern mittels des Fehler-Detektions-Codes, wobei ein Codewort-Paket, an dem ein Fehler detektiert wird, als verloren betrachtet und als Löschung markiert wird, und
• – zwecks Rückgewinnen der Nachricht erfolgendes Dekodieren der empfangenen Codewort-Pakete mittels einer konkatenierten Paritätsprüf-Matrix, die einen Informations-Teil und einen Paritätsprüf-Teil aufweist, und zwar sofern eine hinreichende Anzahl an Codewort-Paketen empfangen worden ist,
• – wobei das Kodieren der k Informationspakete mittels MDS-Codes durchgeführt wird, die auf Vandermonde- oder Cauchy-Matrices basieren, welche mit nicht-binären oder binären LDPC-Codes seriell konkateniert sind, was dazu führt, dass die konkatenierte Gesamt-Paritätsprüf-Matrix aufweist: (I) als ihren Informations-Teil (i) eine dichtbesetzte erste Teil-Matrix, welche die Paritätsprüf-Matrix des MDS-Codes ist und dementsprechend vom Vandermonde- oder Cauchy-Matrix-Typ ist, und (ii) eine dünnbesetzte zweite Teil-Matrix, die aus dem Informations-Teil der Paritätsprüf-Matrix des LDPC-Codes besteht, und (II) als ihren Paritätsprüf-Teil (i) eine dritte Teil-Matrix, die eine Null-Matrix ist, und (ii) eine dünnbesetzte vierte Teil-Matrix, die aus dem Paritätsprüf-Teil der Paritätsprüf-Matrix des LDPC-Codes besteht,
• – wobei das Dekodieren der empfangenen Codewort-Pakete unter Berücksichtigung derjenigen Spalten der konkatenierten Gesamt-Paritätsprüf-Matrix durchgeführt wird, die den gelöschten Codewort-Paketen zugeordnet sind und die folgende Gleichung lösen:
  
  wobei
  
  einen Vektor bezeichnet, der die gelöschten kodierten Codewort-Pakete (Ausgabe-Zeichen) enthält, x_k einen Vektor bezeichnet, der die korrekt empfangenen kodierten Codewort-Pakete enthält,
  
  die Sub-Matrix ist, die aus den Spalten in der Gesamt-Paritätsprüf-Matrix besteht, welche den gelöschten Codewort-Paketen (Ausgabe-Zeichen) entsprechen, H_k die Sub-Matrix der Gesamt-Paritätsprüf-Matrix bezeichnet, die aus den Spalten in der Gesamt-Paritätsprüf-Matrix besteht, welche den korrekt empfangenen Codewort-Paketen (Ausgabe-Zeichen) entsprechen, und das hochgestellte T die Transponierte der betreffenden Matrix bezeichnet, und
• – wobei
  
  eine dünnbesetzte Matrix ist, die das Gleichungssystem beschreibt, welche zum Rückgewinnen der gelöschten Codewort-Pakete (Ausgabe-Zeichen) zu lösen sind.

Thus, the invention provides a method for transmitting a message from a sender to a receiver via a communication medium using packet-level coding, comprising the steps of:

• - at the transmitter side
• Splitting the message to be transmitted into k information packets of L bits (input characters),
• Encoding the k information packets into n codeword packets of L bits (output characters), where n is greater than k and where m = n-k parity packets are generated, and
• Protect the n codeword packets by error detection and correction codes,
• Transmitting the n codeword packets and m parity packets over the communication medium from the sender to the receiver, and
• - at the receiver side
• For each of the transmitted codeword packets, performing error correction by means of the error correction code and detection of residual errors by means of the error detection code, wherein a codeword packet on which an error is detected is considered lost and as deletion is marked, and
• Decoding the received codeword packets by means of a concatenated parity check matrix comprising an information part and a parity check part, provided that a sufficient number of codeword packets have been received, for the purpose of recovering the message;
• Wherein the coding of the k information packets is performed by means of MDS codes based on Vandermonde or Cauchy matrices serially concatenated with non-binary or binary LDPC codes, resulting in the concatenated total parity check matrix comprising: (I) as its information part (i), a sparse first sub-matrix which is the parity-check matrix of the MDS code and is accordingly of the Vandermonde or Cauchy matrix type, and (ii) a sparse second one Sub-matrix consisting of the information part of the parity check matrix of the LDPC code, and (II) as its parity check part (i) a third sub-matrix which is a null matrix, and (ii) a sparse fourth sub-matrix consisting of the parity check part of the parity check matrix of the LDPC code,
• Wherein the decoding of the received codeword packets is performed taking into account those columns of the concatenated total parity check matrix associated with the deleted codeword packets and solving the following equation:
  
  in which
  
  denotes a vector containing the deleted coded codeword packets (output characters), x _k denotes a vector containing the correctly received coded codeword packets,
  
  is the sub-matrix consisting of the columns in the total parity check matrix corresponding to the deleted codeword packets (output characters), H _k denotes the sub-matrix of the total parity check matrix consisting of the columns in the total parity check matrix, which correspond to the correctly received codeword packets (output character), and the superscript T denotes the transpose of the respective matrix, and
• - in which
  
  is a sparse matrix describing the system of equations to be solved for recovering the deleted codeword packets (output characters).

• – die k Eingangs-Zeichen werden zuerst mittels eines (n_MDS, k)-MDS-Codes (basierend auf einer Vandermonde- oder Cauchy-Matrix) zu einer Anzahl n_MDS von Zeichen kodiert,
• – die n_MDS MDS-kodierten Zeichen werden dann mittels eines (n, k_LDPC = n_MDS) LDPC-Codes zu n Ausgangs-Zeichen kodiert.

According to the invention, a serial concatenation of MDS codes with LDPC codes (the coding is in 2 shown), ie:

• The k input characters are first encoded into a number n _MDS of characters by means of an (n _MDS , k) MDS code (based on a Vandermonde or Cauchy matrix),
• - The n _MDS MDS coded characters are then coded by a (n, k _LDPC = n _MDS ) LDPC code to n output characters.

This concatenation, as proposed by the invention, improves the performance of the LDPC code under shared ML decoding, in particular in the error ground area (see 3 ). In particular, concatenations with nonbinary LDPC codes are taken into account, as it has been proven that these improve the performance of their binary counterparts [2], [3].

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

Because software packet-level encoders and decoders are also designed to operate on energy-constrained platforms (such as on-board units of spaceships, landing gear, rovers, orbiting near-Earth or space missions, satellites). , efficient schemes are required.

In particular, this invention proposes an efficient packet-level coder method for serial concatenated MDS codes (based on Vandermonde or Cauchy matrices) or non-binary or binary LDPC codes.

The invention makes it possible to reduce the complexity of decoding according to the prior art.

Proposed decoding scheme

The decoding step proposed according to the invention utilizes the sparse nature of the system's decoding matrix in (1) and the structure of a vandermonde or Cauchy matrix.

To present the invention, it is first necessary to review MDS matrices and Vandermonde matrices and their inverse. It is not intended to give an overview of Cauchy matrices, but it should be noted that it is also possible to invert a Cauchy matrix in quadratic complexity [6].

Overview of the MDS Parity Check Matrix and the Vandermonde Matrices

wobei β_i worin i = 1, ..., n_MDS die n_MDS Nicht-Null-Elemente des Galois-Felds der Ordnung q sind, auf dem der Code aufgebaut ist.The parity check matrix of an MDS code constructed on a Galois field of order q with the information size k and the block length n _MDS is as follows:

where β _i, where i = 1, ..., n _{MDS are} the n _MDS non-zero elements of the order q Galois field on which the code is constructed.

It should be noted that m _MDS = n _MDS - k.

wobei x_i, worin i = 1, ..., γ für γ distinkte Elemente stehen. Anzumerken ist, dass das j-te Element in der i-ten Reihe von V^(γ) geschrieben werden kann als (x_i)^j–1, mit j = 1, ..., γ. Eine wichtige Eigenschaft einer Vandermonde-Matrix der Ordnung γ besteht darin, dass sie eine vollrangige Matrix ist, d. h. mit einem Rang (V^(γ)) = γ.A quadratic Vandermonde matrix of order γ is designated V ^(γ) and is defined as follows:

where x _i , where i = 1,..., γ stand for γ-distinct elements. It should be noted that the j-th element in the i-th row of V ^(γ) can be written as (x _i) ^j-1, j = 1, ..., γ. An important property of a Vandermonde matrix of order γ is that it is a full-ranked matrix, ie with a rank (V ^(γ) ) = γ.

bezeichnet. Jede der

Matrices, die aus den m' Spalten (abgekürzt auf ihre m' Elemente) of

bestehen, deren Indices in J definiert sind, ist die Transponierte einer Vandermonde-Matrix der Ordnung m' für jedes m' <= m_MDS. With J = {j ₁ , j ₂ , ..., j _{m '} } let any set of m'<= m _{MDS be} indices of columns of

designated. Each of the

Matrices coming from the m 'columns (abbreviated to their m' elements) of

whose indices are defined in J, is the transpose of a vandermonde matrix of order m 'for each m'<= m _MDS .

d. h. m' = m_MDS, und der Fokus soll auf die Matrix gerichtet werden, die aus den ersten m_MDS Spalten von

besteht. Es liegt die Transponierte einer Vandermonde-Matrix des Rangs m_MDS vor, wobei es sich bei den m_MDS distinkten Elementen um x_i = β_i handelt, mit i = 1, ..., m_MDS.As an example, suppose that

ie m '= m _MDS , and the focus should be directed to the matrix consisting of the first m _MDS columns of

consists. The transpose of a vandermonde matrix of the rank m _{MDS is} present, where the m _MDS- distinct elements are x _i = β _i , where i = 1, ..., m _MDS .

The inverse of a vandermonde matrix

The special structure of a Vandermonde matrix of order γ can be used to calculate its inverse (V ^(γ) ) ^-1 in a quadratic complexity in γ [4], ie ~ O (γ ² ), rather than in a cubic complexity in γ ~ O (γ ³ ) to calculate, z. With a GE algorithm.

The inverse matrix of a Vandermonde matrix can be obtained as the product of two matrices called V ^-1 = U ^-1 L ^-1 , ie V ^-1 = U ^-1 L ^-1 , where U is an upper triangular matrix, L is a lower triangular matrix, and the superscript of V is omitted for better clarity of writing.

• 1. Berechnen von L^–1
• 2. Berechnen von U^–1

The algorithm for calculating the inverse of V ^(γ) goes through the following two steps:

• 1. Calculate L ^-1
• 2. Calculate U ^-1

Step 1 - Calculate L ^-1

• a) l_i,i = 1
• b) l_i,j = 0 für j > i , d. h. eine untere triangulare Matrix

The coefficients of the matrix L ^-1 , ie l _{i, j} where i = 1, ..., γ and j = 1, ..., γ, are given by the following equations:

• a) l _{i, i} = 1
• b) l _{i, j} = 0 for j> i, ie a lower triangular matrix

Let j be set, focusing on the coefficients of the jth column of L ^-1 .

und zwar ausgehend von dem Element l_i=j,j auf der Diagonalen gemäß obiger Definition c (unter Ausschluss von l_i,i) als

It should be noted that for i = j + 1, ..., γ the following recursive equation can be written:

starting from the element l _{i = j, j} on the diagonal according to the above definition c (excluding l _{i, i} ) as

Thus one can use recursive equations to compute the elements of L ^-1 : for each column j, j = 1, ..., γ, one begins by calculating the element on the diagonal, ie i = j, and then computes the other elements for each row i, where i = j + 1, ..., γ.

Step 2 - Calculation of U ^-1

• a) u_i,i = 1, wobei die Haupt-Diagonale nur Elemente hat, die gleich 1 sind
• b) u_i,j = 0 für j < i , d. h. die obere triangulare Matrix
• c) u_i,j = u_i-1,j-1 – u_i,j-1x_j-1 wobei u_0,j = 0, für j > i

The coefficients of the matrix U ^-1 , ie u _{i, j} where i = 1, ..., γ and j = 1, ..., γ, are given by the following equations:

• a) u _{i, i} = 1, where the main diagonal has only elements equal to 1
• b) u _{i, j} = 0 for j <i, ie the upper triangular matrix
• c) u _{i, j} = u _{i-1, j-1} -u _{i, j-1} x _j-1 where u _{0, j} = 0, for j> i

Thus, one can use the recursive equation in equation c) to compute the elements of U ^-1 : first, the series i is set for each i = 1, ..., γ, and then the elements of U ^-1 is calculated for each column j, where j = i + 1, ..., γ.

Now, let us calculate the number of operations required for inverting V ^(γ) according to the recursive formulas cited. In particular, Vandermonde matrices should be considered with elements in a Galois field of order q.

Thus, the elementary operations are additions and multiplications made to GF (q).

für die Additionen und als

für die Multiplikationen aufgeführt, was geschrieben werden kann als

wobei die erste Gleichung die Operationen zum Berechnen der Elemente auf der Hauptdiagonalen von L^–1 betrifft und die zweite Gleichung die Operationen zum Berechnen der anderen Terme betrifft. Somit ist die asymptotische Komplexität von Schritt 1 hinsichtlich γ quadratisch, d. h. ~ O(γ²).The complexity of step 1 is called

for the additions and as

listed for multiplications, which can be written as

wherein the first equation relates to the operations for computing the elements on the main diagonal of L ^-1 and the second equation concerns the operations for computing the other terms. Thus, the asymptotic complexity of step 1 with respect to γ is quadratic, ie ~ O (γ ² ).

für die Additionen und als

für die Multiplikationen aufgeführt, was geschrieben werden kann als

und sie basieren auf dem rekursiven Term.The complexity of step 2 is called

for the additions and as

listed for multiplications, which can be written as

and they are based on the recursive term.

Thus, the asymptotic complexity of step 2 in γ is also quadratic, ie ~ O (γ ² ).

Example. Consider the following Vandermonde matrix V ⁽³⁾ at GF (4), which is generated with primitive polynomials p (x) = x ² + x + 1. a is the primitive element of the field. The inverse of V ⁽³⁾ , V ^-1 = U ^-1 L ^{-1 is to be determined} .

Thus, x ₁ = α ⁰ = 1, x ₂ = α ¹ = α, x ₃ = α ² .

First, L ^{-1 should be} found. According to the method presented here:

Now let U ^-1 be calculated. According to the method presented here:

Thus one obtains:

It is easy to find that V ^-1 · V = I, where I is a 3 × 3 identity matrix. Thus, the inverse of V has been determined.

It should be noted that for a general full matrix M, the following property applies: (M ^T ) ^-1 = (M ¹ ) ^T

Thus, if the inverse of the transpose of a matrix M is to be computed, then it is possible to compute the inverse of M and then capture its transversal.

Description of the invention

The invention aims to solve the equation system in (1) by taking advantage of the structure of the system matrix and the properties of a vandermonde (or Cauchy) matrix.

Let e _{i be} the number of deletes concerning the information part of the concatenated code parity check matrix, and e _p the number of deletes concerning the parity part of the parity check matrix. Thus, the total number of deletions is e = e _i + e _p .

It should be noted that the maximum number of deletions that can be recovered is e _max = m _MDS + m _LDPC .

The invention provides an improved shared decoding method consisting of 4 decoding algorithms, one for each type of erase pattern, as indicated in Table I.

Tabelle I. Löschmuster-Typen und zugeordnete Algorithmen.

Table I. Deletion pattern types and associated algorithms.

As already explained, the parity check matrix of the concatenated code can be written as

bezeichnet) die Matrix bezeichnet, die aus den Spalten von H zusammengesetzt ist, welche den e gelöschten Positionen des empfangenen Codeworts entsprechen.It was with H (previously through

denotes) the matrix composed of the columns of H corresponding to the e-deleted positions of the received codeword.

The complexity of the proposed algorithms is given in terms of the number of required field operations. The number of field additions is indicated as N _a , and the number of field multiplications is indicated as N _m .

Feld-Elemente enthält.L (bits) specifies the package size of the schema, with d (max-LDPC) / v the maximum variable node degree of the LDPC code, and with d (max-LDPC) / c the maximum check node degree of the LDPC code. It should be noted that when creating nodes on a Galois field of order q, where q is a power of 2, each packet size is L bits

Contains field elements.

Deletion pattern of type 1 - Algorithm 1

Several algorithms can be used to decode quadratic complexity MDS codes.

A possible decoding algorithm goes through the following steps:

Step 1 - Calculate the inverse of the vandermonde matrix of degree e _i

This step involves the following number of operations:

Step 2 - Multiplication with the inverse of V

The known term of the system is multiplied by the calculated inverse of V.

This step involves the following number of operations:

The total number of operations for Algorithm 1 is thus:

Type 2 deletion pattern - Algorithm 2

For this deletion pattern, the matrix of the equations system Hals can be written a concatenation of 3 matrices. It should be noted that V is the transpose of a vertex probe matrix of size m _MDS × m _MDS that can be inverted with quadratic complexity.

The proposed decoding algorithm goes through the following steps:

Step 1. Calculation of the inverse of the Vandermonde matrix V of degree m _MDS .

This step involves the following number of operations:

Step 2. Multiplication by the inverse of V.

The known term of the system and the matrix A must be multiplied by the calculated inverse of V.

This step involves the following number of operations on the matrix A:

This step involves the following number of operations on the known term of the system:

The complexity of this step is thus:

The matrix of the equation system after this step becomes

Step 3 - Degrade the part of the matrix under the diagonal matrix I.

The complexity of this step, that of the matrix H of the system can be limited to the top by

The complexity of this step, which is done on the known term of the system, can be limited upwards by

An upper limit to the overall complexity of this step is

The matrix of the system after step 3 becomes

It should be noted that the matrix B is densely populated after step 3.

Step 4 - Gaussian elimination on B.

e _i - m _MDS deletions are recovered with a GE step. The overhead of the equation system to be solved in this step is δ = m _LDPC - (e _{i -m MDS} ).

Thus, for the number of operations to be performed both on the matrix of the system and on the known term:

Step 5 - Back Substitution

The remaining m _MDS deleted characters are recovered in a back-substitution step.

This involves the following number of operations:

The overall complexity of Algorithm 2 is limited upwards by:

Deletion pattern of type 3 - Algorithm 3.

For this erase pattern, the matrix of the system to be solved can be written as

It should be noted that V is the transpose of a vandermonde matrix of degree e _i .

Step 1 - Invert the vandermonde matrix V of degree e _i

The following number of operations is required.

Step 2 - Multiplication with the inverse of V

In this step, e _i deleted symbols are recovered.

This step does not involve any operations on the matrix of the system.

The known term of the system is multiplied by the calculated inverse of V.

The operations to be performed on the known term are

The complexity of this step is thus

The matrix of the system after this step is

mit den e_i rückgewonnenen Unbekannten.Step 3. Updating the known term with respect to the system of equations given by the lower part of the matrix, ie

with the e _i reclaimed unknowns.

An upper bound on the number of operations involved in this step is

und den aktualisierten bekannten Term gegeben ist.Step 4 - Smart GE on the sparse system of equations passing through the matrix

and the updated known term is given.

beschriebenen Code kubisch.Since the unknowns involved in this step are functions of the (already recovered) e _i unknowns and the system is sparse, a smart GE algorithm can be applied to recover the remaining e _p unknowns. The asymptotic complexity of this step is in terms of the number of pivots for the parity check matrix

described code cubic.

It should be noted that one needs zero pivots and only a single decode step to recover the e _p unknowns.

The overall complexity of Algorithm 3 is limited upwards by:

Deletion pattern of type 4 - Algorithm 4

For this erase pattern, the matrix of the system to be solved is as follows:

It should be noted that V is the transpose of a vandermonde matrix of degree m _MDS .

Step 1 - Inverted vertex probe matrix V of degree m _MDS

The operations involved in this step are:

Step 2 - Multiplication with the inverse of V.

The known term of the system and the matrix A must be multiplied by the calculated inverse of V.

This step involves the following number of operations on the matrix A:

Step involves the following number of operations on the known term of the system:

The total number of operations in step 2 is

The matrix of the system after this step is

Step 3 - Degrade the part of the matrix below the diagonal matrix I.

The complexity of this step, which is done on the matrix of the system, can be limited upwards by

The complexity of this step, which is done on the known term of the system, can be limited upwards by

The overall complexity of this step is

The matrix of the system after this step is

It should be noted that the matrix B is densely populated after step 3.

Step 4 - Smart GE on the sparse system of equations given by the lower part of the system, ie (0 B H ^{(LDPC parity)} ) e _p + (e _{i -m MDS} ) Erasures are recovered in this step.

The number of pivots in this step may be limited upwards by e _{i -m MDS} + p, where p is the number of pivots of the original LDPC code.

It should be noted that one for IRA / Geira codes only e _i - has m _MDS Pivots, as you always the matrix H ^{(LDPC parity)} can triangulate with row and column permutations.

Step 5 - Back Substitution.

The remaining m _MDS deletions are recovered in a back-substitution step.

The following number of operations are involved:

The overall complexity of Algorithm 4 can be limited upwards by:

The invention allows to save a considerable number of operations for decoding codes based on a serial concatenation of MDS codes and PDPC codes. The complexity reduction is achieved by taking advantage of both the structure of the matrices of the MDS code and the sparse nature of the LDPC code.

A detailed description of the invention has already been given above. The derived equations for the number of operations (or upper bounds) required by the proposed decoding algorithm have been implemented in software (SW). A comparison was made between the prior art and the invention. The following example will be considered, and the complexity will be examined.

example

Consider serial concatenations with the (256, 128) LDPC code at the GF (256) designated in [2].

The maximum VN degree is d (max LDPC) / v = 100, and the maximum CN grade is d (max-LDPC) / c = 10.

Let L = 40 bits be assumed, which is a block size of 3GPP / UMTS turbo codes.

• 1. einem in dem GF(256) aufgebauten (128, 126)-MDS-Code, der somit 2 MDS-Gleichungen in der Paritätsprüf-Matrix des LDPC-Codes enthält
• 2. einem in dem GF(256) aufgebauten (128, 124)-MDS-Code, der somit 4
• MDS-Gleichungen in der Paritätsprüf-Matrix des LDPC-Codes enthält
• 3. einem in dem GF(256) aufgebauten (128, 120)-MDS-Code, der somit 8 MDS-Gleichungen in der Paritätsprüf-Matrix des LDPC-Codes enthält
• 4. einem in dem GF(256) aufgebauten (128, 88)-MDS-Code, der somit 40 MDS-Gleichungen in der Paritätsprüf-Matrix des LDPC-Codes enthält
• 5. einem in dem GF(256) aufgebauten (128, 68)-MDS-Code, der somit 60 MDS-Gleichungen in der Paritätsprüf-Matrix des LDPC-Codes enthält Die Leistung der ersten drei konkatenierten Codes in Form der Codewort-Fehlerrate (CER) gegenüber der Lösch-Wahrscheinlichkeit ε an dem Kanal ist in 3 gezeigt. Zu beachten ist, dass die Leistung verbessert wird, wenn mehr Gleichungen aus dem MDS-Code an die Paritätsprüf-Matrix des LDPC-Codes angefügt werden.

A concatenation should be made on this, with regard to:

• 1. a (128, 126) MDS code constructed in GF (256), thus containing 2 MDS equations in the parity check matrix of the LDPC code
• 2. a (128, 124) -MDS code constructed in the GF (256), which is thus 4
• Contains MDS equations in the parity check matrix of the LDPC code
• 3. a (128, 120) MDS code constructed in the GF (256), thus containing 8 MDS equations in the parity check matrix of the LDPC code
• 4. a (128, 88) MDS code built in the GF (256), thus containing 40 MDS equations in the parity check matrix of the LDPC code
• 5. a (128, 68) -MDS code constructed in the GF (256), thus containing 60 MDS equations in the parity check matrix of the LDPC code The power of the first three concatenated codes in the form of the codeword error rate ( CER) against the deletion probability ε at the channel is in 3 shown. Note that performance improves as more equations from the MDS code are added to the parity check matrix of the LDPC code.

The average number of pivots against the overhead Δ on the RX side for the designated codes when using the prior art decoding is shown in FIG 4 and 5 shown. (The results are averaged for 1000 trial runs where the IT decoder is unsuccessful and further steps are required.) The overhead is the number of characters received in addition to k. It should be noted that the number of pivots and thus the complexity increases along with the number of MDS equations attached to the parity check matrix of the LDPC code.

Let us now explain the serial concatenation with a (128, 68) -MDS code. The average number of pivots for this code for prior art decoding is shown in FIG 5 shown.

Now, the number of required operations for decoding according to the prior art and for the proposed algorithm will be calculated. In particular, it should be said that e _p = 64 and e _i = 60. The total number of symbols received is thus equal to 256 - 124 = 132, and an overhead Δ = 132 - 68 = 64 is used. The average number of pivots in the prior art decoding of θ + p at this overhead is ~ 6 ( 5 ), and on average 6 deletions are recovered in the prior art with IT decoding.

The results were calculated according to the presented equations and are listed in Table II.

Tabelle II. Komplexitätsvergleich für das präsentierte Beispiel anhand der Gesamt-Anzahl von Operationen N = N_a + N_m. (128, 68) MDS + (256, 128) LDPC an GF(256), q = 256, L = 40 Bits. Aufgeführt ist der Stand der Technik im Vergleich zur Erfindung.

Table II. Complexity comparison for the presented example, based on the total number of operations N = N _a + N _m . (128, 68) MDS + (256, 128) LDPC at GF (256), q = 256, L = 40 bits. Listed is the prior art compared to the invention.

It should be noted that for this example, and based on the derived upper limits, the invention reduces the number of operations required by the prior art by 79%.

The proposed algorithm can be used for all types of commercial wireless and wired transmission systems, such as: As has been stated, the proposed decoding method allows a considerable reduction of the prior art decoding algorithm such that serially concatenated schemes based on MDS codes (e.g., Reed Solomon codes) and LDPC codes can also be applied to platforms with low computational capacity.

Abbreviations

• CERIUM

  Codeword Error Rate

  CRC

  cyclic redundancy check (Cyclic Redundancy Check)

  CN

  Check node

  Geira

  Generalized irregular-repeat accumulate

  GE

  Gaussian elimination

  GF

  Galois field

  IT

  iterative

  IRA

  Irregular-repeat accumulate

  LDPC

  Low density parity check (low density parity check)

  MDS

  maximum separation distance (maximum distance separable)

  PHY

  Physically

  P

  EC packet clear channel (Packet Erasure Channel)

  VN

  variable node

literature

• [1] J. Byers, M. Luby, M. Mitzenmacher, A. Rege, "A digital fountain approach to reliable distribution of bulk data", SIGCOMM Comput. Commun. Rev., vol. 28, No. 4, pp. 56-67, Oct. 1998.
• [2] Giuliano Garrammone, Balazs Matuz, "Short Erasure Correcting LDPC IRA Codes over GF (q)", GLOBECOM 2010, pp. 1-5.
• [3] G. Garrammone, B. Matuz, G. Liva, "An Overview of Efficient Coding Techniques for Packet Loss Recovery in CCSDS Telemetry Links," 5th European Space Agency Workshop on Tracking, Telemetry and Command Systems for Space Applications, TTC 2010 , Noordwijk (Netherlands), 21.-23. September 2010.
• [4] L. Richard Turner, "Inverse of Vandermonde matrix with applications", NASA Technical Note, August 1966.
• [5] Irving S. Reed, Gustave Solomon, "Polynomial codes over certain finite fields," Journal of the Society for Industrial and Applied Mathematics. [SIAM J.], 8, 1960.
• [6] J. Blömer, M. Kalfane, R. Karp, M. Karpinski, M. Luby, D. Zuckerman, "An xor-based erasure-resilient coding scheme", ICSI, Tech. Rep., Aug. 1995, TR-95-048.
• [7] G. Liva, B. Matuz, E. Paolini, M. Chiani, "Pivoting Algorithms for Maximum Likelihood Decoding of LDPC Codes over Erasure Channels", IEEE Globecom 2009 (Communication Theory Symposium), Honolulu, November 2009.
• [8] D. Burshtein, G. Miller, "An efficient maximum likelihood decoding of LDPC codes over the binary erasure channel", IEEE Trans. Inf. Theory, vol. 50, No. 11, pp. 2837-2844, Nov. 2004.
• [9] LIU, Qi, LU Yang, WANG Wensheng, CUI Huijuan, TANG Kun, "Robust Video Transmission Scheme Using Dynamic Rate Selection LDPC and RS codes. In: IMACS Multiconference on Computational Engineering in Systems Applications, Vol. 2, 2006, pp. 1673-1679. - ISBN 7-302-13922-9.

Claims (1)

Method for transmitting a message from a sender to a receiver via a communication medium by means of packet-level coding, comprising the following steps: at the sender side splitting the message to be transmitted into k information packets of L bits (input characters) Encoding the k information packets into n codeword packets of L bits (output characters), where n is greater than k and where m = n-k parity packets are generated, and - protecting the n codeword packets by errors Detection and correction codes, transmitting the n codeword packets and m parity packets over the communication medium from the sender to the receiver, and at the receiver side for each of the transmitted codeword packets, performing an error correction by means of the Error correction codes and detection of residual errors by means of the error detection code, wherein a codeword packet on which an error is detected, considered lost and as Lösc hung, and for the purpose of recovering the message, decoding the received codeword packets by means of a concatenated parity check matrix comprising an information part and a parity check part, if a sufficient number of codeword packets have been received, characterized in that The coding of the k information packets is carried out by means of Maximum Distance Separable (MDS) codes based on Vandermonde or Cauchy matrices which are used with non-binary or binary Low Density Parity Check (LDPC). Codes are serially concatenated, which results in the concatenated total parity check matrix comprising: (I) as its information part (i) a densely packed first sub-matrix which is the parity check matrix of the MDS code and, accordingly, from Vandermonde or Cauchy matrix type, and (ii) a sparsely populated second sub-matrix consisting of the information portion of the parity check matrix of the LDPC code, and (II) as its parity check portion (i) third sub-matrix being a null matrix, and (ii) a sparse fourth sub-matrix consisting of the parity check part of the parity check matrix of the LDPC code, and decoding the received codeword packets considering those columns of concatenation total parity check matrix associated with the deleted codeword packets, and performed by solving the following equation:

in which

denotes a vector containing the deleted coded codeword packets (output characters), x _k denotes a vector containing the correctly received coded codeword packets,

is the sub-matrix consisting of the columns in the total parity check matrix corresponding to the deleted codeword packets (output characters), H _k denotes the sub-matrix of the total parity check matrix consisting of the columns in the total parity check matrix, which correspond to the correctly received codeword packets (output character), and the superscript T denotes the transpose of the respective matrix, - where

is a sparse matrix describing the system of equations to be solved for recovering the deleted codeword packets (output characters).

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