DE102013223801B4 - Procedure for recovering lost and / or corrupted data - Google Patents

Abstract

A method for recovering lost and / or corrupted data transmitted from a transmitting device to a receiving device via a lossy q-ary balanced channel, the method comprising the steps of encoding the data by an encoder connected to the transmitting device , Transmitting the data from the transmitting device to the receiving device via the transmission channel and decoding the data by a decoder connected to the receiving device, wherein recovery of lost and / or damaged data is performed by decoding the equation system of a parity check matrix H during decoding is performed, wherein in a first decoding step, a verification-based decoding method is performed and in the case of a decoder failure, d. H. if all data could not be recovered by the verification-based decoding method, a vector symbol decoding method is used for decoding.

Description

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

The transmitted data may be, for example, audio or video streams. These are transmitted by a transmitting device, which provides the data, for example, to a mobile receiving device. The mobile receiving device may be, for example, a mobile phone, a PDA or another mobile terminal. Alternatively, data may also be transmitted from a transmitting device to a stationary receiving device.

In order to ensure a desired transmission quality, it is necessary to check the correct transmission of the data or data packets to the receiving device. Various methods can recover lost and / or corrupted data that has not been correctly transmitted to the receiving device.

One known method of recovering lost and / or corrupted data is the low density parity check method or the low density parity check code. This is applied on a so-called erasure channel. Apart from an application by coding at the level of the physical layer, there are also applications in the area of a packet erasure channel (PEC).

However, the present invention relates to a scenario in which a bit stream is transmitted over a channel on which errors are generated in the bit transmission layer. This may, for example, be a binary, symmetric error channel with a defined error probability or with Gaussian white noise. In such a channel, swaps of bit values may occur, so that a bit is received as a zero even though it was sent as a 1. The peculiarity of such a channel is thus that the receiver does not know whether a bit received by it has the correct value or not. In contrast, in the so-called erasure channel, it can be assumed that all received bits have the correct value, since otherwise they would have been canceled out by the transmission channel. In such a scenario, encoding and decoding methods typically perform calculations on individual bits, so that the data structure is maintained. For example. There exist so-called "belief propagation techniques" which use the probability that each transmitted bit is zero or one. Calculations are performed to update these probabilities.

In contrast, in many practical applications, the basic unit of a data transfer need not be a single bit. Rather, bit blocks are transmitted, organized as packets. The term "packets" generally meant an accumulation of several bits. In individual cases, however, a package can also consist of only one bit. For example. Data packets can consist of thousands of bits, such as in Internet packages. On the other hand, a packet can simply be a 32-bit integer. In order to achieve high speeds, it is obvious to use coding methods which perform computational operations at the packet level rather than at the bit level. An example of this is the already mentioned packet-based LDPC codes.

Another motivation to consider packet-level methods is that they can be used in a concatenated code. For example. For example, an inner code that works at bit-level can be applied to individual packets. If this inner code fails, the bits in the packet may be erroneous in an unpredictable and seemingly arbitrary manner. An external code that is capable of handling packet-level errors could then be used to recover data that the inner code has failed. Concatenated codes allow the use of computation-intensive inner codes that are applied to packets of a fixed small size while at the same time being scalable to larger block lengths by using outer packet-level code. Alternatively, in a concatenated code, a less computation-intensive inner code may be used, then an outer packet-level code is used which handles the then more frequently occurring data not decoded by the inner code.

• [1] G. Liva, B. Matuz, E. Paolini, and M. Chiani, ”Pivoting algorithm for maximum likelihood decoding of LDPC codes over erasure channels”, IEEE Global Telecommun. Conf., Honolulu, HI, USA, Nov. 2009.
• [2] E. Paolini, G. Liva, B. Matuz, and M. Chiani, ”Maximum likelihood erasure decoding of LDPC codes: Pivoting algorithm and code design”, IEEE Trans. Commun. Vol. 60, no. 11, Nov. 2012.
• [3] J. J. Metzner and E. J. Kapturowski, ”A general decoding technique applicable to replicated file disagreement location and concatenated code decoding”, IEEE Trans. Inform. Theory, vol. 36, pp. 911–917, July 1990.
• [4] K. T. Oh and J. J. Metzner, ”Performance of a general decoding technique over the class of randomly chosen parity check codes”, IEEE Tans. Inform. Theory, vol. 40, no. 1, January 1994.
• [5] M. Luby and M. Mitzenmacher, ”Verification-based decoding for packet-based low-density parity-check codes,” IEEE Trans. Inf. Theory, vol. 51, no. 1, pp. 120–127, Jan. 2005.
• [6] F. Zhang and H. Pfister, ”List-message passing achieves capacity on the qary symmetric channel for large q,” in Proc. IEEE Global Telecommun. Conf., Washington, DC, USA, Nov. 2007, pp. 283–287.
• [7] B. Matuz, G. Liva, E. Paolini, and M. Chiani, ”Verification-based decoding with MAP erasure recovery,” in Proc. ITG Int. Conf. Syst., Commun. and Coding (SCC), Munich, Germany, Jan. 2013, pp. 1–6.

Information on the methods known from the prior art can be found in the following publications:

• [1] G. Liva, B. Matuz, E. Paolini, and M. Chiani, "Pivoting algorithm for maximum likelihood decoding of LDPC codes over erasure channels", IEEE Global Telecommun. Conf., Honolulu, HI, USA, Nov. 2009.
• [2] E. Paolini, G. Liva, B. Matuz, and M. Chiani, "Maximum likelihood erosure decoding of LDPC codes: Pivoting algorithm and code design", IEEE Trans. Commun. Vol. 60, no. 11, Nov. 2012.
• [3] JJ Metzner and EJ Kapturowski, "A general decoding technique applicable to replicated file disagreement location and concatenated code decoding", IEEE Trans. Inform. Theory, vol. 36, pp. 911-917, July 1990.
• [4] KT Oh and JJ Metzner, "Performance of a general decoding technique over the randomly selected parity check codes", IEEE Tans. Inform. Theory, vol. 40, no. 1, January 1994.
• [5] M. Luby and M. Mitzenmacher, "Verification-based decoding for packet-based low-density parity-check codes," IEEE Trans. Inf. Theory, vol. 51, no. 1, pp. 120-127, Jan. 2005.
• [6] F. Zhang and H. Pfister, "List-message passing achievements capacity on the qary symmetric channel for large q," in Proc. IEEE Global Telecommun. Conf., Washington, DC, USA, Nov. 2007, pp. 283-287.
• [7] B. Matuz, G. Liva, E. Paolini, and M. Chiani, "Verification-based decoding with MAP erasure recovery," in Proc. ITG Int. Conf. Syst., Commun. and Coding (SCC), Munich, Germany, Jan. 2013, pp. 1-6.

DE 10 2011 102 503 B3 describes a method corresponding to publication [7].

From the prior art, the algorithm of Luby and Mitzenmacher is known (see reference [5]), which is called Verification-based decoding (VBD). An iterative algorithm is described here which thus does not have a particularly high computational complexity. This algorithm is used to handle packet errors. The idea is to mark variable nodes (VNs), each corresponding to a packet, and check nodes (CNs) as verified (i.e., as correct) if the packets (locally) satisfy the corresponding parity equation. In the event that the parity equation is not met, various strategies exist to correct the damaged packets, depending on the specific configuration of the algorithm. Depending on the parity checks, for example, candidate values for the packets can be propagated along the graph. In the most complex embodiment, verification-based decoding corresponds to a message-list-passing algorithm, as described, for example, in reference [6]. Due to the fast-growing list sizes, such an algorithm is not easy to implement.

Another algorithm known from the prior art is the so-called Vector-Symbol Decoding (VSD), which is described in reference [3]. The main idea is to perform manipulations on the syndrome vector to determine the positions of the errors in the transmitted codeword. Subsequently, the erroneous packages can be corrected.

It is an object of the present invention to provide a method of transmitting data over a bit-error channel having improved performance.

The object is achieved according to the invention by the features of claim 1.

The inventive method is used to restore lost and / or damaged data, which are transmitted from a transmitting device via a faulty channel that generates bit errors, in particular a q-ary balanced channel to a receiving device. The method comprises the following steps:
Encoding the data by an encoder connected to the transmitting device,
Transmitting the data from the transmitting device to the receiving device via the transmission channel and
Decoding the data by a decoder connected to the receiving device, wherein recovery of lost and / or damaged data is performed by decoding the equation system of a parity check matrix H during decoding.

According to the invention, a verification-based decoding method is carried out in a first decoding step. In case of a decoding failure, i. H. if all data could not be recovered by the verification-based decoding method, a vector symbol decoding method is used for decoding.

It is thus proposed to concatenate on the decoder side a verification-based decoding algorithm with a vector symbol decoding algorithm. Any type of verification-based decoding algorithm can be used. If the verification-based decoding algorithm does not enable successful decoding, the unrecovered data is passed to a subsequent vector symbol decoding algorithm.

It is preferred that this, after detecting the error positions in the codeword, include the use of a smart Gaussian elimination method. This particular embodiment of a VSD algorithm is described in more detail below.

On the one hand, the combination of a VBD method with a VSD method provides an increase in performance because the VSD algorithm can correct for some errors that the VBD algorithm could not correct. On the other hand, the foregoing VBD algorithm can resolve linearly dependent error patterns so that successful decoding by the VSD algorithm is possible. Furthermore, the complexity can be reduced compared to pure VSD decoding, since numerous errors are already corrected during the first decoding step by the VBD algorithm.

• a) nämlich einer Verifikation, indem, sofern alle mit einem Check Node verbundenen Variable Nodes die Paritätsgleichung erfüllen, sie als verifiziert ( V ) markiert werden und der Check Node als bestätigt ( C ) markiert wird,
• b) nämlich einer Korrektur, indem, sofern alle mit einem Check Node verbundenen Variable Nodes bis auf einen Variable Node verifiziert sind, der nicht-verifizierte Variable Node gemäß der Paritätsgleichung auf die elementweise Modulo-2-Summe der verifizierten Variable Node-Werte gesetzt wird, sodass sein Status als verifiziert markiert wird und der Status des Check Nodes als bestätigt markiert wird.

It is preferred that a VBD method is performed by initializing all variable nodes as unverified in the representation of a bipartite graph ( V ) and all check nodes are unconfirmed ( C ), whereby all check nodes are processed simultaneously by performing the following steps in parallel:

• a) Verification in that, provided that all variable nodes associated with a check node satisfy the parity equation, they are verified ( V ) and the check node is confirmed ( C ) is marked,
• b) namely a correction in that, provided that all variable nodes connected to a check node are verified except for one variable Node, the non-verified variable Node is set according to the parity equation to the elementary modulo-2 sum of the verified variable node values so that its status is marked as verified and the status of the check node is marked as confirmed.

• a) Berechnen einer Syndrom-Matrix S durch Multiplikation der Parity-Check-Matrix mit dem empfangenen Codewort,
• b) Anwenden des Gauß'schen Eliminationsverfahrens auf die Syndrom-Matrix, um eine neue Syndrom-Matrix S' zu erhalten, die zum Teil die Form einer Identitätsmatrix aufweist,
• c) Ausnullen des restlichen Teils der neuen Syndrom-Matrix S', der nicht die Form einer Einheitsmatrix aufweist, durch lineares Kombinieren mit den Zeilen der neuen Syndrom-Matrix S', die die Form einer Identitätsmatrix aufweisen, wobei die gleichen Linearkombinationen wie in der neuen Syndrom-Matrix S' in den entsprechenden Zeilen der Parity-Check-Matrix vorgenommen werden, sodass ein korrespondierendes Set an Nullkombinationen aus der Parity-Check-Matrix entsteht.
• d) Ermitteln, welche Spalten dieser Nullkombinationen aus der Parity-Check-Matrix ausschließlich Nullen aufweisen, sodass hierdurch die Positionen der Fehler im empfangenen Codewort bekannt sind,

If the decoding process could not be successfully completed by the VBD method, a VSD method is performed by

• a) calculating a syndrome matrix S by multiplying the parity check matrix by the received codeword,
• b) applying the Gaussian elimination method to the syndrome matrix to obtain a new syndrome matrix S 'which is partly in the form of an identity matrix,
• c) nulling out the remainder of the new syndrome matrix S ', which does not have the form of a unit matrix, by linearly combining with the lines of the new syndrome matrix S' having the form of an identity matrix, the same linear combinations as in FIG new syndrome matrix S 'be made in the corresponding lines of the parity check matrix, so that a corresponding set of zero combinations from the parity check matrix is created.
• d) determining which columns of these zero combinations from the parity check matrix have all zeros, so that the positions of the errors in the received codeword are known,

The erroneous data in the received codeword whose position has been detected is corrected using the correctly received data.

It is preferred that the VSD method be used to correct for linearly independent errors after performing the VBD method.

It is preferred that in the VSD method, the erroneous data whose positions were detected be corrected by the use of a smart Gaussian elimination method. This will be described in more detail below.

In the following decoding steps, the errors in the received codeword whose positions are known in the codeword are considered as cancellations. Thus, rather than being a q-ary symmetric channel, the lossy transmission channel may be considered an erasure channel. For an erasure channel, it is assumed that a received data unit (eg, a bit value) represents a correct value, otherwise an erasure would occur.

Since the channel is now considered an erasure channel, techniques that are commonly used in data transmission over an erasure channel can be used for further decoding. For solving the equation system of the parity check matrix, the parity check matrix H is converted into a triangular form by means of column and / or line interchanges. Columns of a sub-matrix B of the matrix H that obstruct the triangularization process are shifted into a submatrix P of the matrix H, so that the triangularization process can continue until the matrix H except the matrix P is completely in one Triangular shape was brought. Subsequently, the Gaussian elimination method is applied to a part P1 of the submatrix P.

The aim of the method according to the invention is to keep the area of the parity check matrix H to which the Gaussian elimination method is applied as small as possible, since this method requires a high computing power owing to the necessary row additions for large arrays. The complexity of the maximum likelihood method is O (n ³ ), where n is the block length, that is, the number of columns of the parity check matrix H. If the Gaussian elimination method is applied to only a small part of the parity check matrix H, the required computing power can be considerably reduced. For this purpose, the submatrix A is gradually increased.

Thus, the sub-matrix P can be kept as small as possible, so that the computational effort for the Gaussian elimination method applied to the submatrix P can be minimized.

Before the Gaussian elimination method is applied to the submatrix P, the submatrix A is placed in the diagonal form and the submatrix D is nullified. The Gaussian elimination method therefore only has to be applied to the lower part of the submatrix P, namely to P1. P1 is densely occupied. If the Gaussian elimination of P1 has been successfully performed, the remaining unknowns can be determined iteratively or by back substitution. The iterative method is preferred here. This is applied to the original matrix, in which the now discovered unknowns are used.

The exact procedure of the Vector Symbol Decoding Algorithm is described in more detail below:
It is assumed that H is a (n-k) × n parity check matrix of a parity check block code. H defines the (n, k) code of the code symbol x of length l bits as the set of codewords that satisfy the parity equations:

When a sequence of n symbols has been received, a syndrome matrix is calculated:

Here, e = y + x is the error vector in the position j.

Each row of the matrix H defines a parity check equation on the vector symbol which must be satisfied by a codeword. If it is possible to find a line of H satisfying this condition, then any position in which this n-bit line vector contains a 1 is error-free, assuming that the I-bit error vectors are linearly independent , Such a vector in the line space of H is called a null combination.

There is an error matrix in which some lines have only zeros and other lines that are not just zeros represent the error pattern.

To obtain the syndrome matrix, this error matrix must be multiplied by the parity check matrix of the code. It is known that elementary column operations in a matrix do not change which combinations of rows of S together yield zero. This is important because column operations are performed on the matrix S, which operate according to a Gaussian elimination, so that a new matrix S 'is obtained. If the t errors are linearly independent, then the matrix S 'is composed of t rows of an identity matrix and other seemingly random lines. It is possible that in the Matrix S 'one or more row vectors that are zero can be found. This means that the corresponding rows of the matrix H represent a zero combination (see publication [4]). After these operations, the combinations of rows of S 'which together make zero are known (by nulling out the part of the matrix that does not have the form of an identity matrix). Likewise, the corresponding lines of the parity check matrix are known. In fact, there will be no such zero indicators in the matrix. If the modulo-2 sums of the corresponding rows of H are vector-wise formed, nkt null combinations are found. After this step, the so-called "union" of these zero combinations must be formed. In this case, it is determined which columns of these zero combinations from the parity check matrix have all zeros. The result is a logical vector that can be used to find the positions of the t error vectors. This union of zero combinations will, under the conditions described, have zeros in all t positions where the errors are located. On the other hand, it will have a 1 at the positions without error.

The method steps described below are only used in the vector symbol decoding algorithm, which is known from the prior art, and are replaced, as described below, in the context of the method according to the invention by an alternative method:
When the error positions are found, a t × t sub-matrix is generated whose columns are the t columns of H that correspond to the vector positions of the identity matrix in S '. Now, this sub-matrix is inverted (Gaussian elimination can be used here) and the exact error values can be found by multiplying the inverse matrix by the corresponding t rows of S.

In the method according to the invention, this second part of the Vector Symbol Decoding Algorithm becomes the inversion of a sub-matrix of H (by Gauss-Jordan elimination with a complexity of O (n ³ )) and the multiplication between two matrices (which has a complexity of O ( n ³ ), since the matrices are densely populated) has been replaced with a maximum a posteriori (MAP) erasure decoding algorithm using smart Gaussian elimination.

MAP erasure decoding already knows the true values of the erased symbols instead of just the error pattern as known from the VSD algorithm of the prior art. Therefore, it is not necessary to recover the true values of the erroneous symbols by elementally forming the modulo-2 sum between the received symbols and the error patterns.

It should be noted that the o. G. Method step b) of the VSD algorithm can also have an intelligent Gaussian elimination algorithm. However, this would be of little use, since the syndrome is usually sparse.

In the following, preferred embodiments of the invention will be explained with reference to figures.

Show it:

1 the function of a VBD algorithm,

2 a schematic representation of a decoder according to the invention,

3 a graphical representation of the performance of the method according to the invention.

The following is a simple example of the VSD algorithm. The calculation of the syndrome is done as a product of the parity check matrix and the error vector.

By performing column operations on the syndrome (which corresponds to Gaussian elimination) one arrives at the modified syndrome S ':

The lower four rows of the modified syndrome S 'can be nullified by linearly combining the first three rows. Thus, it can be concluded that the corresponding parity checks (which also consist of linear combinations of the first three parity check equations) do not contain any erroneous symbols. Otherwise, the syndrome would not be zero. Then the union of zero combinations is generated (see reference [5]), d. H. it is determined which columns of this zero combination from the parity check matrix have all zeros, so that thereby the positions of the errors in the received codeword are known. Now, the erroneous symbols can be corrected, for example, by erasure decoding techniques or by inversion of the syndrome equations (see reference [3]).

The following is an example of a VBD algorithm. It looks at the transmission of packets over a channel that generates errors. A (6.2) code is used that has the following parity check matrix:

It is assumed that the codeword x = [0 0 0 0 0 0] is transmitted. This codeword has only zeros. Every x _i must be understood as a GF (q) field element. Furthermore, it is assumed that the channel generates an error pattern e = [0 0 0 0 ab], where a, b are nonzero elements of GF (q). At the beginning of the VBD algorithm, all nodes (variable nodes and check nodes) are unverified / unacknowledged. The states of the variable nodes and check nodes as well as the current values associated with the variable nodes are in 1 for the time of the first iteration of the VBD algorithm. Immediately thereafter the decoder stops because there is no check-node whose parity is satisfied. The verification step described above under process step a) can thus not be carried out. Also if there is no check node associated with a single unverified variable node (the above-described correction step b) can not be performed either). Therefore, in the illustrated example, decoding is unsuccessful since only three out of six variable nodes could be verified.

In the context of the method according to the invention, it will now be shown below how successful decoding can be carried out in the example just considered. For this purpose, a concatenation of a VBD method with a VSD method takes place on the decoder side. More specifically, the encoder receives a set of k-packets and generates a set of n-packets at its output. From this, the packet vector x is formed. The communication channel, which may be, for example, a q-ary symmetric channel, generates errors so that decoder-side y = x + e is received, in case of an error e ≠ 0. The decoder wants to recover the transmitted packets x. In the first step, VBD is performed. Due to the suboptimal nature of this decoding algorithm, not all errors are recovered. Linear dependent errors within a codeword may be resolved unless they occur within the same parity check equation. In such a case, the decoder might incorrectly verify corrupted variable nodes. Document [5] has shown that the probability of a wrong verification at a check node decreases with the packet size, since 1 / (q - 1), where q = 2 ^l . Therefore, this probability already drops very low for small packet sizes (ie eg l = 50 bits) and is virtually non-existent. In the event the VBD algorithm fails (see the above example), the original message can still be recovered using an optimal (but not practical) Maxium likelihood decoder. According to the invention, however, the use of a VSD algorithm is provided in this step. In the event that no linearly dependent errors are left after the VBD algorithm, the VSD algorithm can determine the positions of the errors in the transmitted codeword and then correct the erroneous symbols using the correctly received data.

The parity check matrix H already shown above is again considered:

It is assumed that the codeword which consists only of zeros is transmitted with I = 2 bits and that the error pattern has the following form:

After the variable nodes 1, 3, and 4 have been verified (ie, identified as correct) by the VBD method, an adjusted parity check matrix H 'is generated in which all rows of H that are not the unknown ones have been removed Variable Nodes 2, 5 and 6 included. In this case, line 1 was removed. Therefore, the syndrome is calculated as S '= H' × y so that:

It is easy to see that the last line of the syndrome can be nullified by adding the first two lines. If the same additions are performed on the right hand side (ie if the first two lines of the parity check matrix) are added, the result is the row [0 1 0 1 0 0], which shows that the variable nodes 2 and 4 are correct. It is known from the preceding VBD algorithm that the variable nodes 1, 3 and 4 are correct. Therefore, it is now known that the variable nodes 5 and 6 contain the error.

To recover the values of variable nodes 5 and 6, the simplest approach is to assume that variable nodes 5 and 6 are dropouts, so it is possible to use smart gaussian elimination for error recovery , If the deletions are designated as ŷ ₅ , ŷ ₆ , the following must be solved:

This leads to: ŷ ₅ = [0 0], ŷ ₆ = [0 0].

The above equation can be efficiently solved using the smart Gaussian elimination method. Furthermore, the VBD method requires fewer equations to be considered in the VSD process. This leads to a reduced complexity. It should be noted that two linearly dependent error patterns, e.g. For example, e ₅ = [1 0], e ₆ = [1 0] would cause the third line of the syndrome to have only zeros and thus [1 1 1 1 1 1] would be a valid null combination (all variable Nodes are apparently correct).

The following is based on 3 an example of the performance of the method according to the invention is shown. A transmission over a q-ary symmetric channel is considered. It is assumed that a low-density parity check code with n = 200 and k = 100 and a packet size l of 64 bits. Considered is the performance of the code in terms of codeword error rate (CER) using VBD, VSD and the method of the invention (considered here as VBDSR, namely Verification-based decoding with syndrome reprocessing). The VBD method and the VBD step in VBDSR are based on the approach described in reference [7], ie a modified version of the VBD algorithm described above is used.

It can be seen that the VBD algorithm provides the worst performance, while a strong improvement is seen when VSD is used instead. Not only can the complexity be reduced over VSD by the method of the invention, but also a recognizable gain in the channel erosure rate can be achieved.

Claims (3)

A method of recovering lost and / or corrupted data transmitted to a receiving device by a transmitting device via a faulty channel that generates bit errors, the method comprising the steps of Encoding the data by an encoder connected to the transmitting device, transmitting the data from the transmitting device to the receiving device via the transmission channel and decoding the data by a decoder connected to the receiving device, wherein recovery of lost and / or damaged during decoding Data is performed by solving the equation system of a parity check matrix H, wherein in a first decoding step, a verification-based decoding method is performed and in case of a decoding failure, that is, if not all data by the verification-based decoding method a vector symbol decoding method is used for decoding, in which the positions of errors in a received codeword are determined during decoding, the erroneous data in the received codeword whose positions were determined using the correctly received corrected data. A method according to claim 1, characterized in that a verification-based decoding method is performed by starting in the representation of a bipartite graph at the beginning of all variable nodes as unverifiziert ( V ) and all check nodes are unconfirmed ( C ), whereby all check nodes are processed simultaneously by performing the following steps in parallel: a) namely a verification in that, if all variable nodes connected to a check node satisfy the parity equation, they are verified ( V ) and the check node is confirmed ( C ), b) a correction in that, provided that all variable nodes connected to a check node are verified with the exception of one variable Node, the non-verified variable Node according to the parity equation applies to the elementary modulo-2 sum of the verified variable Node Values are set so that their status is marked as verified and the status of the check node is marked as confirmed, whereby, if the decoding process could not be successfully completed by the verification-based decoding method, a Vector-Symbol-Decoding- A method is carried out in which the positions of errors in a received codeword are determined during decoding as follows: a) calculating a syndrome matrix S by multiplying the parity check matrix with the received codeword, b) applying the Gaussian elimination method on the syndrome matrix to obtain a new syndrome matrix S ', which partly takes the form of an identity matrix c) nulling out the remainder of the new syndrome matrix S 'which does not have the form of a unit matrix by linearly combining with the lines of the new syndrome matrix S' having the form of an identity matrix, the same linear combinations as in the new syndrome matrix S 'in the corresponding lines of the parity check matrix, so that a corresponding set of null combinations arises from the parity check matrix. d) Determine which columns of these null combinations from the parity check matrix have all zeros, thereby the positions of the errors in the received codeword are known, the erroneous data in the received codeword whose position was determined using the correctly received data be restored. A method according to claim 1 or 2, characterized in that the vector symbol decoding method is used to correct existing linearly independent errors after performing the verification-based decoding method.

Images