DE102012004273B4 - Procedure for recovering lost and / or corrupted data - Google Patents

Abstract

A method of recovering lost and / or corrupted data that has been transmitted from a sending device to a receiving device, the method comprising the steps of: encoding this data by an encoder connected to the sending device, transmitting this data from the sending device to the receiving device over a transmission channel and decoding this data by a decoder connected to the receiving device, recovering lost and / or damaged data during decoding, using a generator matrix for coding and decoding, the data being encoded by concatenation an MDS (Maximum Distance Separable) code and a fountain code, the generator matrix of the MDS code has a Vandermonde or Cauchy structure, the inverse matrix of the Vandermonde or Cauchy matrix being calculated during the decoding, and during the decoding solving the system of equations defined by the matrix G ~ T, which is a concatenation of the matrix G ~ 'T of the MDS code and matrix G ~' 'T of the fountain code, comprises the following method steps: rewriting the matrix G ~ T as a concatenation of four matrices in the following form: where the rows of Vm '× m' are a set of m '<k rows of a Vandermonde matrix of order m' and A, B and C are two matrices without a Vandermonde structure, .. .

Description

The invention relates to a method for restoring lost and / or damaged data.

It is known that data is transmitted during transmission, e.g. B. can be protected via a noisy channel by various error correction schemes. For this purpose, m parity packets are generated by an encoder added to k information packets so that n = k + m codeword packets are transmitted over the channel. Using the transmitted parity information, lost and / or corrupted data can be recovered. An encoding scheme known in the art is fountain coding. Fountain coding can, for. At packet level and is a simple and efficient technique for ensuring reliable transmission in a communication system ( 1 ).

It is also known in the art to protect data by concatenation of an MDS code and a fountain code. This is z. As described in the publication - F. Lazaro Blasco, G. Liva, "On the Concatenation of Non-Binary Random Linear Fountain Codes with Maximum Distance Separable Codes," in proceedings of International Conference on Communications, 5-9 June 2011, Japan ,

For example, a class of simple and effective fountain codes is given by Linear Random Fountain Codes (LRFC) over Galois higher order fields described in G. Liva, E. Paolini, and M. Chiani, "Performance versus Overhead for Fountain Codes over Fq ", IEEE Communications Letters, vol. 14, no. 2, pages 178-180, February 2010.

Software implementations of packet level encoders and decoders are particularly attractive because they do not require much design / implementation effort and allow more flexibility compared to hardware implementations. Software modules can be easily implemented in a hardware architecture that was not specifically designed to support packet level coding. Since software packet level encoders and decoders are also designed to run on energy-constrained platforms (eg on-board units of spaceships, dropships, rovers, orbiters in near-earth and deep-space missions, satellites) efficient procedures are necessary.

The generator matrix of a concatenation of an MDS code and a Linear Random Fountain Code, as known in the art, is shown below, as a concatenation of two matrices G 'and G ".

G 'is a k × n MDS matrix and G "is a k × 1 generator matrix of a fountain code. Since the Linear Random Fountain Code is a non-random code, the number 1 of the columns of G "can in principle increase infinitely.

The coding is thus performed as c = u * G where u is in row vector containing the k input symbols and c is a row vector containing the n + 1 output symbols.

On the receiver side, a subset of m symbols is received. We denote by J = {j ₁ , j ₂ , ..., j _m } the set of indices on the symbols of c that have been received. The received vector y is therefore given by y = (y ₁ , y ₂ , ..., y _m ) = (c _j1 , c _j2 , ..., c _jm ) and it can be related to the source block u as y = u · G. Here G denotes the km matrix composed of the columns of G with the indices in J, ie

The restoration of u is reduced to solving the system of m = k + δ linear equations in k unknown G ~ ^T · u ^T = Y ^T with the matrix of the system in (2), which can be rewritten as described in ( 3), where m 'is the collected output symbols associated with the MDS code. Thus one can partition G ~ ^T as shown below.

The decoding according to the prior art consists of solving the equation system G ~ ^T · u ^T = Y ^T , the matrix of which is shown in (3), with Gaussian elimination. Thus, u is a row vector containing the k input symbols, y is the received vector, and G ~ ^T is the matrix of the system. In particular, m 'equations belong to the MDS code and the remaining equations belong to the Fountain Code. Further background information concerning the present invention can be found in the following publications
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,
L. Richard Turner, "Inverse of Vandermonde matrix with applications," NASA Technical Note, August 1966,
Irving S. Reed, Gustave Solomon, "Polynomial codes over certain finite fields," Journal of the Society for Industrial and Applied Mathematics. [SIAM J.], 8, 1960, M. Karpinski, R. Karp, M. Luby, J. Bloemer, M. Kalfane and D. Zuckerman, "An XOR-based erasure-resilient coding scheme," ICSI, Tech. Rep., Aug. 1995, TR-95-048.

DE 10 2010 029 113 A1 describes a method for channel coding of digital data, the data being encoded by a concatenation of an MDS code and a fountain code.

US 2005/0138533 A1 describes a method for decoding a Reed-Solomon code, wherein the parity check matrix of the code has a Vandermond shape.

pamphlet US 2006/0212782 A1 describes the use of read-solomon codes defined by a generator matrix, which is a vandermonde matrix or cauchy matrix.

It is an object of the present invention to provide a method for recovering lost and / or corrupted data whereby data can be restored with reduced computational complexity.

This object is achieved by the features of claim 1.

According to the inventive method, data is transmitted from a transmitting device to a receiving device, data being encoded by an encoder connected to the transmitting device. Data is transmitted via a transmission channel, the z. B. is a broadcasting network in a satellite scenario. Any suitable transmission channel can be used. The transmitted data is decoded by a decoder connected to the receiving device, recovering lost and / or damaged data during decoding.

wobei L die Paketgröße in Bits ist, q die Ordnung des Galois Felds, über dem der Code erzeugt wurde.A generator matrix is used for encoding and decoding, data being encoded by a concatenation of an MDS code and a fountain code. According to the invention, the generator matrix of the MDS code has a Vandermonde or Cauchy structure. During decoding, an important step is calculating the inverse matrix of the Vandermonde or Cauchy matrix. In this step, the special structure of a Vandermonde or Cauchy matrix can be exploited to reduce the computational complexity for calculating the inverse matrix. The advantages of the method according to the invention compared to the prior art are shown below:
According to the prior art, the decoding complexity is cubic with the number of input symbols of the coding scheme k, ie ~ O (k ³ ). In fact, the number of operations needed during Gaussian elimination (GE) for a matrix of size (k + δ) x k, denoted N in the packet level (GE) / a and N (GE) / m and is shown below,

where L is the packet size in bits, q is the order of the Galois field over which the code was generated.

The proposed decoding algorithm uses the structure of the decoding matrix in (3) and the structure of a Vandermonde or Cauchy matrix. To illustrate the invention, first review the Reed-Solomon matrices and Vandermonde matrices and their inverses. Although no detailed explanation is given with respect to Cauchy matrices, it is also possible to invert a quadratic complexity Cauchy matrix as shown in M. Karpinski, R. Karp, M. Luby, J. Bloemer, M. Kalfane and D. Zuckerman, "An xor-based erasure-resilient coding scheme," ICSI, Tech. Rep., Aug. 1995, TR-95-048.

wobei q die Ordnung des Galois Felds ist, über dem der Code erzeugt wurde, und a das primitive Element des Felds ist.The following is an overview of a Reed-Solomon generator matrix and Vandermonde matrices:
The generator matrix of a Reed-Solomon code of length n = q-1 and the distance d = n-k + 1 is shown below and is referred to as G ',

where q is the order of the Galois field over which the code was generated, and a is the primitive element of the field.

The transposed matrix of G 'is referred to as G' ^T and is defined as follows,

wobei x_i mit i = 1, ..., γ, γ unterschiedliche Elemente sind. Es ist zu beachten, dass das j-te Element in der i-ten Reihe von V^γ geschrieben werden kann als (x_i)^j-1, für j = 1, ..., γ. Eine wichtige Eigenschaft einer Vandermonde Matrix der Ordnung γ ist, dass sie eine Matrix mit vollem Rang ist, d. h. Rang (V^γ) = γ. A quadratic Vandermonde matrix of order y is denoted v ^γ and is defined as follows,

where x _i with i = 1, ..., γ, γ are different 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, for j = 1, ..., γ. An important property of a Vandermonde matrix of order γ is that it is a full rank matrix, ie rank (V ^γ ) = γ.

für i = 1, ..., m', und j = 1, ..., m'. Als ein Beispiel wird angenommen, dass J = {j₁ = 1, j₂ = 2, ..., j_m' = k}, d. h. m' = k und wir betrachten die Matrix, die sich zusammensetzt aus den ersten k Reihen von G'^T. Es gibt eine Vandermonde Matrix mit Rang k, wobei die k unterschiedlichen Elemente x_i = α^i-1, für i = 1, ..., k sind und die anderen Elemente (x_i)^j-1 = (α^i-1)^j-1, für j = 1, ..., k.With J = {j ₁ , j ₂ , ..., j _{m '} } we denote a random set of m'<= k indices of series of G ' ^T. Each of the [ n / m ' ] Matrices consisting of the m 'series, whose indices are defined in Y, is a Vandermonde matrix of order m', for each m '<= k. In particular

for i = 1, ..., m ', and j = 1, ..., m'. As an example, it is assumed that J = {j ₁ = 1, j ₂ = 2, ..., j _{m '} = k}, ie m' = k, and we consider the matrix composed of the first k rows G ^'T. There is a Vandermonde matrix of rank k, where the k different elements are x _i = α ^i-1 , for i = 1, ..., k and the other elements (x _i ) ^j-1 = (α ^i-1 ) ^j-1 , for j = 1, ..., k.

The special structure of a vandermonde matrix of order y can be exploited to give its inverse (V ^(γ) ) ^-1 with quadratic complexity in y, ie ~ O (γ ² ) instead of a cubic complexity in γ ~ O (γ ³ ) calculate, for. With a Gaussian elimination algorithm.

It is preferred that the computation of the inverse vowel probe or Cauchy matrix takes place with a reduced computational complexity compared to a matrix that does not have a vandermonde or cauchy structure, especially with quadratic complexity, compared to a cubic complexity for a matrix without the vandermonde or Cauchy Structure.

It is further preferred that k output symbols are generated by the MDS code and l output symbols are generated by the fountain code.

• a. l_1,1 = 1,
• b. l_i,j = 0 für j > i, d. h. untere Dreiecksmatrix
  
  für j <= i, außer für j = i = 1,

Further, it is preferable that the concatenation of the MDS code and the fountain code results in a guessless code. It is preferred that a Reed-Solomon code be used as an MDS code. The inverse matrix of a Vandermonde matrix can be obtained by a product of the two matrices U ^-1 and L ^-1 , where U is an upper triangular matrix and L is a lower triangular matrix. The calculation of the matrix L ^-1 comprises the following method steps:
Defining coefficients of the matrix L ^-1 , ie 1 with i = 1, ..., γ and j = 1, ..., γ by the following equations:

• a. _1,1 = 1,
• b. l _{i, j} = 0 for j> i, ie lower triangular matrix
  
  for j <= i, except for j = i = 1,

Starten von dem Element auf der Diagonalen l_i=j,i gegeben bei c. (außer l_l,l) als,

Verwenden rekursiver Gleichungen, um die Elemente von L^–1 zu berechnen: für jede Spalte j, j = 1, ..., γ,
Beginnen mit dem Berechnen des Elements auf der Diagonalen, d. h. i = j,
Berechnender anderen Elemente für jede Reihe i = j + 1, ..., γ.Fixing j, focusing on the coefficients of the jth column of L ^-1 , where the following recursive equation holds for i = j + 1, ..., γ,

Starting from the element on the diagonal l _{i = j, i} given at c. (except l _{l, l} ) as,

Using recursive equations to calculate the elements of L ^-1 : for each column, j, j = 1, ..., γ,
Starting with calculating the element on the diagonal, ie i = j,
Calculating other elements for each row i = j + 1, ..., γ.

• a. u_i,i = 1, wobei die Hauptdiagonale nur Elemente gleich 1 hat,
• b. u_i,j = 0 für j < i, d. h. obere Dreiecksmatrix
• c. u_i,j = u_i-1,j-1 – u_i,j–1x_j-1 mit u_0,j = 0, für j > i,

Verwenden der rekursiven Gleichung in c, um die Elemente von U^–1 zu berechnen durch Fixieren der Reihe i für jedes i = 1, ..., γ, und Berechnen der Elemente von U^–1 für jede Spalte j mit j = i + 1, ..., γ.The calculation of the matrix U ^-1 comprises the following method steps:
Defining the coefficients of the matrix U ^-1 , ie u _{i, j} with i = 1, ..., γ and j = 1, ..., γ 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 upper triangular matrix
• c. u _{i, j} = u _{i -1, j -1} - u _{i, j -1} x _{j -1} with u _{0, j} = 0, for j> i,

Using the recursive equation in c to calculate the elements of U ^-1 by fixing the row i for each i = 1, ..., γ, and calculating the elements of U ^-1 for each column j with j = i + 1, ..., γ.

In the following, the number of operations necessary to invert V ^{(δ) is} calculated according to the recursive formulas presented. In particular, the calculation takes into account Vandermonde matrices with elements in a Galois field of order q and a primitive element a, as shown above for Reed-Solomon codes. Thus, the elementary operations are additions and multiplications over GF (q).

für die Additionen und als

für die Multiplikationen und kann ausgedrückt werden, als

wobei die erste Serie für die Operationen steht, um die Elemente auf der Hauptdiagonalen von L^–1 zu berechnen, und die zweite Serie für die Operationen steht, um die anderen Terme zu berechnen. Somit ist die asymptotische Komplexität der Berechnung von L^–1 quadratisch in γ, d. h. ~O(γ²).The complexity of calculating L ^-1 is referred to as

for the additions and as

for the multiplications and can be expressed as

where the first series represents the operations to compute the elements on the main diagonal of L ^-1 and the second series represents the operations to compute the other terms. Thus, the asymptotic complexity of calculating L ^{-1 is} quadratic in γ, ie ~ O (γ ² ).

für die Additionen und als

für die Multiplikationen und kann ausgedrückt werden als

und basiert auf dem rekursiven Term. Somit ist die asymptotische Komplexität der Berechnung von U^–1 ebenfalls quadratisch in γ, d. h. ~O(γ²).The complexity of calculating U ^-1 is referred to as

for the additions and as

for the multiplications and can be expressed as

and is based on the recursive term. Thus, the asymptotic complexity of the computation of U ^{-1 is} also quadratic in γ, ie ~ O (γ ² ).

wobei die Reihen von V_m'×m' ein Set von m' < k Reihen einer Vandermonde Matrix der Ordnung m' und A, B and C zwei Matrizen ohne Vandermonde Struktur sind,
wobei das System, das auf der Empfängerseite gelöst werden muss, ist G ~^T·u^T = y^T wobei G ~^T, die m × k Matrix des Systems ist, u^T die k × L/log₂(q) Matrix ist, die sich zusammensetzt aus den k Informationspaketen von L Bits und y^T eine m × L/log₂(q) Matrix ist, die die bekannten Terme des Systems darstellt.According to the invention, it is provided that, during the decoding, the solving of the system of equations defined by a matrix G ~ ^T , which is a concatenation of the matrix G ~ ' ^{T of} the MDS code and the matrix G " ^{T of} the fountain code, is the following method steps includes:
Rewriting the matrix G ~ ^T as a concatenation of four matrices in the following form:

where the rows of V _{m '× m' are} a set of m '<k rows of a vandermonde matrix of order m' and A, B and C are two matrices without a vandermonde structure,
the system that needs to be solved on the receiver side is G ~ ^T · u ^T = y ^T where G ~ ^T , the m × k matrix of the system, u ^{T is} the k × L / log ₂ (q) matrix, which is composed of the k information packets of L bits and y ^{T is} an m × L / log ₂ ( q) is the matrix representing the known terms of the system.

The calculation of the inverse of V using the structure of a Vandermonde matrix can be performed with the following upper limit on the number of operations called N (1) / a and N (1) / m (see the previous section).

d. h. M·G ~^T·u^T = M·γ^T,
sodass die Matrix des Systems wird zu

sodass der neue bekannte Term bezeichnet wird als y ≈^T und das Gleichungssystem nun geschrieben werden kann, als G ≈^T·u^T = y ≈^T.Furthermore, it is preferred that the decoding algorithm comprises the following additional method steps: multiplying the equations G ~ ^T · u ^T = γ ^T by. the matrix M given below

ie M · G · ^T · u ^T = M · γ ^T ,
so that the matrix of the system becomes too

so that the new known term is called y ≈ ^T and the system of equations can now be written as G ≈ ^T · u ^T = y ≈ ^T.

und

) plus die Anzahl an Operationen, die notwendig sind um V^–1 mit y^T zu multiplizieren, sodass man y ≈^T (bezeichnet als

) erhält. Durch Verwendung der zerlegten Struktur von V^–1 = U^–1L^–1 kann man zuerst die Anzahl der Operationen, die notwendig ist, um L^–1 mit A zu multiplizieren und L^–1 mit y^T erhalten, wobei zu berücksichtigen ist, dass die Multiplikation mit U^–1 die gleiche Anzahl an Operationen beinhaltet. Es ist insbesondere

The number of operations involved in this step is the number of operations necessary to multiply V ^-1 by A, so that A 'is obtained (referred to as

and

) plus the number of operations necessary to multiply V ^-1 by y ^T such that y ≈ ^T (denoted as

) receives. By using the decomposed structure of V ^-1 = U ^-1 L ^-1 one can first of ^all obtain the number of operations necessary to multiply L ^-1 by A and L ^-1 by y ^T , bearing in mind that multiplication by U ^-1 involves the same number of operations. It is special

The number of operations involving the above step (referred to as Nn (2) / a and N (2) / m ) is thus

The decoding algorithm according to the invention may further comprise the following additional method step:
Nulling the matrix B in G ≈ ^T by multiplication and row additions, where the matrix of the system resulting from this step is denoted as G = ^T and given below

The number of operations on G = ^T necessary during this step is denoted as Nn (B) / a and as N (B) / m . Nn (B) / a = m '(m - m') (k - m ') N (B) / m = m '(m - m') (1 + k - m ')

The number of operations necessary on the known term y ≈ ^T leading to the new known term y = ^T and thus to the system G = ^T * u ^T = y = ^T is referred to as

The number of operations involved in step (3) (denoted as N (3) / a and N (3) / m ) therefore

It is further preferred that the solving of the system of equations of the matrix C 'comprises the Gaussian elimination, thus restoring k - m' information symbols, namely packets of L bits.

The number of operations required during Gaussian elimination (GE) on a matrix of size (k + δ) x k in the packet level is denoted as N (GE) / a _and N (GE) / m .

The number of operations involved in this step is referred to as N (4) / a and N (4) / m and is obtained by evaluating the above equations about k = k - m '.

It is preferred that the remaining m 'input symbols be restored by back substitution.

The number of operations required by this step is designated as N (5) / a and N (5 m and is

Thus, the overall complexity of the invention is referred to as N (Invention) / a and N (Invention) / m and is N (invention) / a = N (1) / a + N (2) / a + N (3) / a + N (4) / a + N (5) / a N (invention) / m = N (1) / m + N (2) / m + N (3) / m + N (4) / m + N (5) / m.

The key point of the proposed decoding algorithm is the step bgzl of calculating the inverse of a quadratic complexity vowel probe or Cauchy matrix m ', ie, O ((m') ² ), where z. B. the proposed algorithm can be used.

The invention allows to save a significant number of operations for decoding codes based on concatenated MDS codes and fountain codes. The complexity reduction is achieved by exploiting the structure of the matrices of the MDS code.

In the following, preferred embodiments of the invention are explained in the context of the figures:

1 shows a packet level coding scenario in which the method according to the invention can be used,

2 - 4 show matrices that are used in different steps of the method according to the invention.

The scenario of the present invention, ie Packet Level Coding with concatenated MDS codes (Vandermonde / Cauchy matrix based codes) and Fountain Codes, is in 1 shown. First, the message to be transmitted, z. A FILE divided into k information packets of L bits (input symbols) and encoded in n + 1 code symbols which are packets of L bits (output symbols). Thus, the n + 1 output symbols are generated by the packet level encoder. In particular, n is the number of output symbols generated according to an MDS code, and l is the number of output symbols generated according to a fountain coding scheme. Second, the n + 1 output symbols in the physical memory are protected (within the PHY layer frame) by error correction codes (eg Turbo, LDPC ...), error detection codes (eg Cyclic Redundancy Check (CRC)) and they are transmitted , Third, physical layer error correction and remaining errors are applied to each packet on the receiver side are detected by the error detection code. If errors are detected, the packet is considered lost and marked as extinction. Thus, the layers above the physical layer look at the communication medium as a Packet Erasure Channel (PEC), where packets are either received correctly or lost. Finally, the Packet Level Decoder recovers the original message when a sufficient number of packets have been received.

In the following a concrete example for the calculation of the inverse of a Vandermonde matrix is described:
Consider the following Vandermonde matrix V ⁽³⁾ over GF (4) generated with the primitive polynomial p (x) = x ² + x + 1. The inverse of V ⁽³⁾ , V ^-1 = U ^-1 L ^-1 must be found

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

First, L ^-1 must be found, according to the presented procedure, this yields:

Now let U ^-1 be calculated. According to the presented method this results in:

This results

It is now easy to see that V ^-1 · V = I, where I is a 3 × 3 identity matrix. The inverse of V was found.

The form into which the matrix G ~ ^{T of} the system can be rewritten is on the left side of the 2 shown as a concatenation of four matrices (as already described in the context of equation (3)). The known term of the system of equations to be solved on the receiver side is on the right side of FIG 2 shown.

three shows the matrix of the system on the left and the known term of the system on the right after multiplying y ^T by the matrix M.

4 shows the same two matrices after the next step, namely after nulling out the matrix B.

A detailed description of the invention has already been given. The derived equations for the number of operations required for the proposed decoding algorithm have been implemented in SW. A comparison of the prior art with the invention was made. Now consider the following example for checking the complexity.

Consider a concatenated RS (255, 191) generated in GF (256) with a Linear Random Fountain Code over GF (256), so that the order of the Galois field in which q is worked is 256. Thus, k = 191, n = 255, q = 256. It is assumed that L = 40 bits, which is a block size of 3GPP / UMTS turbo codes. It is assumed that m '= 187, and thus m = 191. The complexity of the proposed prior art decoding method, e.g. B. compared to a Gaussian elimination algorithm.

The results were calculated according to the presented equations and are given in Table I.

Tabelle I. Komplexitätsvergleich für das präsentierte Beispiel im Hinblick auf die Gesamtanzahl der Operationen N = N_a + N_m. RS(255, 191) + LRFC über GF(256), q = 256, L = 40 bits, m' = 187. Stand der Technik vs. Erfindung.

Table I. Complexity comparison for the presented example with respect to the total number of operations N = N _a + N _m . RS (255, 191) + LRFC over GF (256), q = 256, L = 40 bits, m '= 187. Invention.

It should be noted that for this example, the invention reduces the number of operations required in the prior art by 84%.

The proposed decoding algorithm can be used in all types of commercial wireless and wired transmission systems, such as satellite communication, internet communication, etc.

As has been shown, the proposed decoding method allows a substantial reduction in the complexity of the prior art decoding algorithm, so that concatenated schemes based on MDS codes (e.g., Reed-Solomon codes) and fountain codes (e.g., LRFC, LT Codes, Raptor codes) can also be executed on platforms with low computing power.

Claims (8)

A method for recovering lost and / or corrupted data transmitted from a sending device to a receiving device, the method comprising the steps of: encoding that data by an encoder connected to the sending device, transmitting that data from the sending device to the receiving device via a transmission channel and decoding this data by a decoder connected to the receiving device, recovering lost and / or corrupted data during decoding using a generator matrix for encoding and decoding, wherein the data is coded by a concatenation of an MDS (Maximum Distance Separable) code and a Fountain Code, the generator matrix of the MDS code has a Vandermonde or Cauchy structure, wherein during decoding the inverse matrix of the Vandermonde or Cauchy matrix is calculated, during decoding means solving the system of equations defined by the matrix G ~ ^T , which is a concatenation of the matrix G ~ ' ^{T of} the MDS code and matrix G ~'' ^{T of} the fountain code, comprising the following steps: rewriting the matrix G ~ ^T as a concatenation of four matrices in the following form:

where the rows of V _{m 'x m' are} a set of m '<k rows of a vandermonde matrix of order m' and A, B and C are two matrices without a vandermonde structure, the system which must be solved on the receiver side is: G ~ ^T · u ^T = y ^T where G ~ ^T , the m × k matrix of the system, is u ^T , the k × L / log ₂ (q) matrix, which is composed of the k information packets of L bits and y ^{T is} an m × L / log ₂ (q) is matrix representing the known terms of the system. A method according to claim 1, characterized in that the calculation of the inverse vowel probe or Cauchy matrix takes place with a reduced computation complexity compared to a matrix without a vandermonde or cauchy structure, namely with quadratic complexity compared to a cubic complexity for a matrix without the vandermonde or Cauchy structure. A method according to claim 1 or 2, characterized in that n output symbols are generated by the MDS code and l output symbols are generated by the fountain code. Method according to one of Claims 1 to 3, characterized in that the concatenation of the MDS code and the fountain code leads to a non-random code. Method according to one of claims 1 to 4, characterized in that a Reed-Solomon code is used as an MDS code. Method according to one of claims 1 to 5, characterized in that the inverse matrix of a Vandermonde or a Cauchy matrix is obtained by the product of two matrices U ^-1 and L ^-1 , where U is an upper triangular matrix and L is a lower triangular matrix, the calculation of the matrix L ^-1 comprises the following steps: defining coefficients of the matrix L ^-1 , ie l _{i, j} , where i = 1, ..., γ and j = 1, ..., γ by the following Equations: a. l _{l, l} = 1, b. l _{i, j} = 0 for j> i, ie lower triangular matrix

for j <= i, except for j = i = 1, fix j, focusing on the coefficients of the jth column of L ^-1 , where the following recursive equation holds for i = j + 1, ..., γ .

Starting from the element on the diagonal l _{i = j, j} given at c. (except l _{l, l} ) as,

Using recursive equations to calculate the elements of L ^-1 : For each column, j, j = 1, ..., γ, begin by calculating the element on the diagonal, ie i = j, computing the other elements for each Row i = j + 1, ..., γ and the calculation of the matrix U ^-1 comprises the following method steps: Defining the coefficients of the matrix U ^-1 , ie u _{i, j} with i = 1, ..., γ and j = 1, ..., γ 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 upper triangular matrix C. u _{i, j} = u _{i-1, j-1} -u _{i, j-1} x _j-1 with u _0,1 = 0, for j > i, using the recursive equation in c to compute the elements of U ^-1 by fixing the row i for each i = 1, ..., γ, and calculating the elements of U ^-1 for each column j with j = i + 1, ..., γ. Method according to one of claims 1-6,characterized, the decoding algorithm further comprises the following method steps: Multiplying the equations G ~^T· u^T =^γT through the matrix M given below

d. H. M · G ~^T· u^T = M · γ^T. so that the matrix of the system becomes too

so that the new known term is called y ≈^T and the system of equations can now be written as G ≈^T· u^T = y ≈^T , Zoom out of the matrix B into G ≈^T by multiplication and series addition, the matrix of the system resulting from this step being referred to as G =^T and given hereafter,

Method according to claim 7, characterized in that the solving of the system of equations of the matrix C 'comprises Gaussian elimination such that k - m' information symbols, namely packets of L bits, are reconstituted, the remaining m 'input symbols being restored by back substitution.

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