**CHANNEL CODE CONSTRUCTION FOR DECODER REUSE**

ZHAO, Zhipeng (Riesstr. 25, Munich, 80992, DE)

LEUNG, Wai, Kong, Raymond (Riesstr. 25, Munich, 80992, DE)

FOSSORIER, Marc (Riesstr. 25, Munich, 80992, DE)

LAND, Ingmar (Riesstr.25, Munich, 80992, DE)

ZHANG, Huijian (Riesstr. 25, Munich, 80992, DE)

*;*

**H03M13/03***;*

**H03M13/05**

**H03M13/19**US6131178A | 2000-10-10 |

CHRISTIAN HÄGER ET AL: "Approaching Miscorrection-free Performance of Product and Generalized Product Codes", ARXIV.ORG, CORNELL UNIVERSITY LIBRARY, 201 OLIN LIBRARY CORNELL UNIVERSITY ITHACA, NY 14853, 21 November 2017 (2017-11-21), XP080839029

MACWILLIAMS F J AND SLOANE N J A: "Theory of error-correcting codes passage", THEORY OF ERROR-CORRECTING CODES, X, XX, 1 January 1977 (1977-01-01), pages 23 - 29, XP002226718

MORELOS-ZARAGOZA R H: "The Art of Error Correcting Coding, PASSAGE", ART OF ERROR CORRECTING CODING, XX, XX, 1 January 2002 (2002-01-01), pages 101 - 120, XP002366026, DOI: 10.1002/0470847824.CH6

None

Claims 1. Device (100) for encoding and/or decoding data transmitted in a communication channel (101), the device (100) being configured to encode and/or decode the data based on an {N K '} code (102) generated from an {/V, K} code (103), wherein N and N’ are code lengths, K and K’ are code dimensions, N’ - N = q > 0, and K - K’ = p > 0, and wherein the {N’, K’} code (102) is defined by a check matrix (200), the check matrix (200) includes {N - K + p + q} rows (201) and { /V + q } columns (202), {27- K} elements (203) in each of {p + q} columns (204) of the check matrix (200) are all zeros, and {N - K} elements (205) in each of the remaining {N - p} columns (206) of the check matrix (200) are the elements of a check matrix defining the {TV, K} code (103). 2. Device (100) according to claim 1, wherein the upper {N - K} elements (203) of the left {p + q} columns (204) of the check matrix (200) are all zeros, and the upper {N - K} elements (204) of the remaining {N - p} columns (206) of the check matrix (200) are the elements of the check matrix defining the {TV, K} code (103). 3. Device (100) according to claim 2, wherein the lower {p + q} elements (300) of the left {p + q} columns (204) of the check matrix build an identity matrix. 4. Device (100) according to claim 2 or 3, wherein each of the lower {p + q} rows (301) is filled with alternating ones and zeros. 5. Device (100) according to one of the claims 1 to 4, wherein the {N K} and the {N’, K’} code (103, 102) are Hamming codes or Bose- Chaudhuri-Hocquenghem codes. 6. Device (100) according to one of the claims 1 to 4, wherein the {N K} code (103) is a Hamming code with N = 127 and K = 120, and the {N', K’} code (102) is a code with N’ = 128 and K’ = 119. 7. Device (100) according to one of the claims 1 to 6 including a code generator (400) according to one of the claims 8 to 15. 8. Code generator (400) for generating a {N’, K’} code (102) for encoding and/or decoding data transmitted in a communication channel (101) from a {/V, K} code (103), wherein N and N’ are code lengths, K and K’ are code dimensions, and the code generator (400) is configured to shorten the {N K} code (103) to obtain an intermediate code (401), and extend the intermediate code (401) to obtain the {N’, K’} code (102). 9. Code generator (400) according to claim 8, wherein N’ - N = q > 0, K - K’ = p > 0, and the code generator (400) is configured to shorten the {N K} code (103) by p positions to obtain an intermediate {N -p, K’} code (401), and extend the intermediate {N-p, K’} code (401) by p + q positions to obtain the {N’, K’} code (102). 10. Code generator (400) according to claim 9, configured to, for generating the {N’, K’} code (102) from the {N, K} code (103), modify a first check matrix that defines the {N, K} code (103) to obtain a second check matrix (200) that defines the {N’, K’} code (102), the modification of the first check matrix including removing p columns of the first check matrix to obtain a first intermediate matrix, adding {p + q}) left or right columns filled with all zeros to the first intermediate matrix to obtain a second intermediate matrix, and adding {p + q} rows to the second intermediate matrix to obtain the second check matrix (200). 11. Code generator (400) according to claim 10, configured to set {p + q} elements (300) of the left or right {p + q} columns (204) of the second check matrix (200) such that they build an identity matrix. 12. Code generator (400) according to claim 10 or 11, configured to set the elements of each of the {p + q} rows (301) of the second check matrix (200) such that it includes alternating ones and zeros. 13. Code generator (400) according to one of the claims 8 to 12, wherein the {TV, K} code (103) and the {N’, K’} code (102) are Hamming codes or Bose- Chaudhuri-Hocquenghem codes. 14. Code generator (400) according to one of the claims 8 to 13, wherein the {TV, K) code (103) is a Hamming code with N = 127 and K = 120, and the {N’, K’} code (102) is a code with I = 128 and K' = 119. 15. Code generator (400) according to one of the claim 8 to 14 included in a device (100) for encoding and/or decoding data based on the {N’, K’} code (102). 16. Method (600) for constructing a {TV’, K’} code (102) for encoding and/or decoding data transmitted in a communication channel (101) from a {TV, K} code (103), wherein N and N’ are code lengths, K and K’ are code dimensions, and the method (600) comprises shortening (601) the {TV, K} code (103) to obtain an intermediate code (401), and extending (602) the intermediate code (401) to obtain the {TV’, K’} code (102). |

TECHNICAF FIEFD The present invention relates to an encoding and/or decoding device, which uses a modified channel code that bases on an original channel code. Thus, the invention also relates to a code generator for generating a modified code from an original code, by modifying the original code. In particular, according to the invention the modified code, which is defined by a check matrix, is obtained by modifying a check matrix of the original code. The present invention also relates to a channel code generating method.

BACKGROUND

Channel codes are essential in all digital communications systems. A typical system for forward error correction (FEC) coding, also called a coding scheme, is shown in FIG. 7. The system includes an encoder (at the transmitter side) and a decoder (at the receiver side), which are connected via a communication channel. The encoder adds redundancy to data to be transmitted in the communication channel, i.e. adds redundant data. The decoder exploits this redundancy to correct transmission errors, such that the receiver ideally obtains the transmitted data free of errors despite noise in the communication channel.

In particular, in FIG. 7, the data u to be transmitted (termed information word) is given to the encoder, which produces a codeword x containing redundancy. This codeword x is then transmitted over the noisy communication channel, which typically introduces errors. The output vector y is provided to the decoder, which produces estimates of the transmitted codeword and the transmitted data. A set C of possible codewords is called the code (or channel code), and the following description is particularly concerned with such a code.

For reasons of complexity at the encoder and decoder side, typically linear codes over finite fields are employed. The following explanation is thus provided for the finite field F _{2 } = {0, 1} of size 2 - for the sake of simplicity. However, the following explanation holds in a similar manner for other fields or rings. In particular, a code C of length N and dimension K (labelled in this document an‘{/V, K) code’), may be defined by a generator matrix G of size K N as: C = {x = u G u E F }

In that case, the encoder that maps the information word u of length K to the codeword x of length N is given by x = u - G where addition and multiplication are over the binary field {0, 1 }. Alternatively, the code C may be defined by the parity check matrix H (in this document short‘check matrix’) of size (N- K) x N as: By this definition a vector x is a codeword if and only if, x - H ^{T } = 0

For a given generator matrix, check matrices can be determined, and vice versa.

In a communication system, the information word u is encoded into the codeword x, and this codeword x is then transmitted over the noisy communication channel, yielding the observation vector y of length N. Based on the observation vector y, the decoder determines the most-likely codeword (codeword estimate) x and the corresponding information word (information word estimate) u. This is called decoding.

For example, the maximum- likelihood (ML) decoder minimises the probability of a wrong decision, however, often at high decoding complexity. Other decoding methods, like Chase decoding or syndrome decoding, typically approximate this decision under lower decoding complexity.

An important property of a channel code is its minimum distance d, which is the minimum Hamming distance, i.e. the number of different positions between any two codewords. Due to the linearity of the code, this is also equal to the minimum Hamming weight, i.e. the number of non-zero positions of any code word. A second important property of a channel code is the number of such minimum-distance codewords, also called the multiplicity. The minimum distance together with its multiplicity determine the error rate of a code under ML decoding and many other decoding methods at low noise levels.

Two conventional methods to modify length N or dimension f of a given code are so- called‘shortening’ and‘extending’. The effect of these operations on the check matrix of a code is depicted in FIG. 8. Assumed is specifically an {/V, if} code with check matrix H. Shortening the code by p positions leads to an {N-p, K -p} code, i.e. both length N and dimension if are reduced by p. The check matrix H’ of the shortened code can be obtained from the check matrix H by removing p columns such that the rank of the matrix does not change. Extending by q positions leads to an {N + q, if} code, i.e. the length N increases by q and the dimension if stays the same. The check matrix H” of the extended code can be obtained from the check matrix H by first adding q zero columns and then adding q rows in a way such that the rank of the matrix increases by q.

For many good algebraic codes, like Hamming codes or Bose-Chaudhuri-Hocquenghem (BCH) codes, efficient decoding algorithms are available, i.e. algorithms that achieve low error rates at low decoding complexity. Such codes, however, are only available for specific values of length N and dimension if. If an application requires other lengths or dimensions, e.g. requires an {N’, if’} code while so far only {N, if} codes are provided, new decoding algorithms need to be developed that are matched to the specific {N’, if’} code. These algorithms may be less efficient than those for the {/V, if} codes, even if 1ST and f’ are close to N and if.

A conventional way of constructing a new code {N’, if’} from and original {/V, if} code - with the constraints defined above - is illustrated in FIG. 9 in terms of the check matrices of the codes. The original code is first extended to the desired length 1ST and then expurgated to the desired dimension if’ thus adding further check constraints. The process shown in FIG. 9 in terms of the check matrices is as follows: starting from the check matrix H of the original code, conventionally, first q zero columns are appended to if, and then q + p rows are added to obtain the check matrix if of the new code. Since the q + p new check constraints are adjoined in an unstructured way, a decoder of the original {TV, K} code cannot be used efficiently to decode the new { /V K’} code.

SUMMARY

In view of the above-mentioned problems and disadvantages, the present invention aims to improve the conventional code modification schemes. In particular, the present invention has the objective to provide a code generator for generating a modified code from an original code such that the modified code can be efficiently reused by a decoder (or encoder) of the original code. The present invention aims also for an encoding and/or decoding device that operates efficiently based on the modified code. In particular, the invention desires modifying the original code by increasing its code length N and at the same time decreasing its code dimension K such that a decoder for the new {N’, K’} code can efficiently reuse the decoder of the original {N K} code.

Such a desired decoder is illustrated in FIG. 10. The decoder is originally a decoder of the original {N K} code, i.e. it decodes the observation vector y with the original code to obtain the codeword estimate x and the information word estimate u. However, the decoder can also use the new {N', K’} code for performing this decoding.

The objective of the present invention is achieved by the solution provided in the enclosed independent claims. Advantageous implementations of the present invention are further defined in the dependent claims.

In particular the present invention proposes modifying an original code in a specific manner - namely by a combined shortening and extending operation - to obtain a new code (modified code). The obtained new code can be efficiently used by an existing decoder of the original code.

A first aspect of the present invention provides a device for encoding and/or decoding data transmitted in a communication channel, the device being configured to encode and/or decode the data based on an {N', K '} code generated from an {N K} code, wherein N and N’ are code lengths, K and K’ are code dimensions, N’ - N = q > 0, and K - K’ = p > 0, and wherein the {N', K’} code is defined by a check matrix, the check matrix includes {N - K + p + q} rows and {N + q } columns, {N - K) elements in each of {p + q } columns of the check matrix are all zeros, and {N - K} elements in each of the remaining {N - p} columns of the check matrix are the elements of a check matrix defining the {TV, K} code.

Advantageously, the device of the first aspect may use the new {N’, K’} code for encoding the data, allowing a decoder of the original {N K} code to efficiently reuse the modified code. Further, the device of the first aspect may efficiently reuse the new {N’, K’} code for decoding the data, if it is a decoder of the original code. The device of the first aspect may even perform efficient encoding and/or efficient decoding on the data based on both the new {N’, K’} code and the original {N K} code.

In this document, N, N’, K, K’, p , and q are natural numbers greater than zero.

In an implementation form of the first aspect, the upper {A - if} elements of the left {p + q} columns of the check matrix are all zeros, and the upper {N - K} elements of the remaining {N - p} columns of the check matrix are the elements of the check matrix defining the {A K} code.

This check matrix is an example that yields particularly good results in terms of minimum distance and reduced multiplicity of the new code. Row and/or column permutations may be applied to the check matrix of the new (modified) code, and the resulting check matrix would lead to the same results.

In a further implementation form of the first aspect, the lower {p + q} elements of the left {p + q} columns of the check matrix build an identity matrix.

This allows for a particular simple implementation of the check matrix, and reduced computational complexity.

In a further implementation form of the first aspect, each of the lower {p + q} rows is filled with alternating ones and zeros. Preferably, any two adjacent rows of these lower {p + q) rows start with a one (first row) and a zero (second row), respectively. Thus, the adjacent rows are able to check odd and even indices of the code, respectively.

In a further implementation form of the first aspect, the { N, K} and the { N K’} code are Hamming codes or Bose-Chaudhuri-Hocquenghem codes.

These codes are particularly good algebraic codes, and moreover perform well with the code construction scheme of the present invention.

In a further implementation form of the first aspect, the { N, K} code is a Hamming code with N = 127 and K = 120, and the {N’, K’} code is a code with I = 128 and K' = 119.

For this specific example the minimum distance of the new code is the same as for a new code constructed in a conventional manner, but its multiplicity is largely reduced. This significantly improves error rates under both ML decoding and sub-optimal decoding methods.

In a further implementation form of the first aspect, the device includes a code generator according to a second aspect of the present invention or any of its implementation forms.

The device may particularly be an encoder (in a transmitter or transceiver) and/or a decoder (in a receiver or transceiver) of data, specifically in a mobile communication system or an optical (fibre) communication system. The device is thus advantageously able to modify an original {N, K} code into a new {N’, K’} code, which can be efficiently reused. This allows for new applications that operate on other codes.

The second aspect of the present invention provides a code generator for generating a {N’, K’} code for encoding and/or decoding data transmitted in a communication channel from a {N, K} code, wherein N and N’ are code lengths, K and K’ are code dimensions, and the code generator is configured to shorten the {N, K} code to obtain an intermediate code, and extend the intermediate code to obtain the {N’, K’} code. By applying a combined shortening and extending operation on the original {TV, K) code the code generator of the first aspect is able to generate the new {TV’, K’} code such that it can be efficiently reused by a decoder (and/or encoder) of the original code.

In an implementation form of the second aspect, N’ - N = q > 0, K - K’ = p > 0, and the code generator is configured to shorten the {TV, K} code by p positions to obtain an intermediate {N - p, K’} code, and extend the intermediate {N - p, K’} code by p + q positions to obtain the {TV’, K’} code.

This new code yields particularly good results in terms of minimum distance and reduced multiplicity.

In a further implementation form of the second aspect, the code generator is configured to, for generating the {TV’, K’} code from the {TV K} code, modify a first check matrix that defines the {TV K} code to obtain a second check matrix that defines the {TV’, K’} code, the modification of the first check matrix including removing p columns of the first check matrix to obtain a first intermediate matrix, adding {p + q} left or right columns filled with all zeros to the first intermediate matrix to obtain a second intermediate matrix, and adding {p + q} rows to the second intermediate matrix to obtain the second check matrix.

The code generator can thus obtain an efficient new code by operating on the check matrix of the original code in a relatively simple manner. As an example, the p left columns of the first check matrix may be removed to obtain the first intermediate matrix. However, any p columns may be removed as long as the rank constraint is fulfilled. Further, {p + q} bottom rows may be added to the second intermediate matrix to obtain the second check matrix. However, the additional rows may be added above, below, or in between the other rows.

In a further implementation form of the second aspect, the code generator is configured to set {p + q} elements of the left or right {p + q} columns of the second check matrix such that they build an identity matrix.

This allows for a particular simple implementation of the check matrix, and for reduced computational complexity. For example, the {p + q} elements may be the lower {p + q} elements of the left or right {p + q} columns of the second check matrix. In a further implementation form of the second aspect, the code generator is configured to set the elements of each of the {p + q} rows of the second check matrix such that it includes alternating ones and zeros.

Preferably, any two adjacent rows of these lower {p + q} rows start with a one (first row) and a zero (second row), respectively. Thus, the adjacent rows are able to check odd and even indices of the code, respectively.

In a further implementation form of the second aspect, the {TV, K} code and the { /V K’} code are Hamming codes or Bose-Chaudhuri-Hocquenghem codes.

These codes are particularly good algebraic codes, and moreover perform well with the code construction scheme of the present invention.

In a further implementation form of the second aspect, the {TV, K} code is a Hamming code with N = 127 and K = 120, and the {TV’, K’} code is a code with N’ = 128 and tC = 119.

For this specific example the minimum distance of the new code is the same as for a new code constructed in a conventional manner, but its multiplicity is largely reduced. This significantly improves error rates under both ML decoding and sub-optimal decoding methods.

In a further implementation form of the second aspect, the code generator is included in a device for encoding and/or decoding data based on the {TV’, K’} code.

The device may be an encoder and/or decode, for instance, included in a transmitter, receiver, or transceiver for mobile communications.

A third aspect of the present invention provides a method for constructing a {N’, K’} code for encoding and/or decoding data transmitted in a communication channel from a {TV, K} code, wherein N and N’ are code lengths, K and K’ are code dimensions, and the method comprises shortening the {TV, K} code to obtain an intermediate code, and extending the intermediate code to obtain the {TV’, K’} code. In an implementation form of the third aspect, N’ - N = q > 0, K - K’ = p > 0, and the method comprises shortening the {N K} code by p positions to obtain an intermediate {N -p, K’} code, and extending the intermediate {N-p, K’} code by p + q positions to obtain the {N’, K’} code.

In a further implementation form of the third aspect, the method comprises, for generating the { N K’} code from the {TV, K} code, modifying a first check matrix that defines the {TV, K} code to obtain a second check matrix that defines the {TV’, K’} code, the modification of the first check matrix including removing p columns of the first check matrix to obtain a first intermediate matrix, adding {p + q} left or right columns filled with all zeros to the first intermediate matrix to obtain a second intermediate matrix, and adding {p + q} rows to the second intermediate matrix to obtain the second check matrix.

In a further implementation form of the third aspect, the method comprises setting {p + q} elements of the left or right {p + q} columns of the second check matrix such that they build an identity matrix.

In a further implementation form of the third aspect, the method comprises setting the elements of each of the {p + q} rows of the second check matrix such that it includes alternating ones and zeros.

In a further implementation form of the third aspect, the {TV, K} code and the {TV’, K’} code are Hamming codes or Bose-Chaudhuri-Hocquenghem codes.

In a further implementation form of the third aspect, the {TV, K} code is a Hamming code with N = 127 and K = 120, and the {N’, K’} code is a code with I = 128 and K' = 119.

In a further implementation form of the third aspect, the method is carried out by a device for encoding and/or decoding data based on the {TV’, K’} code.

The method of the third aspect and its implementation forms provide the same advantages and effects as described above for the code generator of the second aspect and its respective implementation forms. It has to be noted that all devices, elements, units and means described in the present application could be implemented in the software or hardware elements or any kind of combination thereof. All steps which are performed by the various entities described in the present application as well as the functionalities described to be performed by the various entities are intended to mean that the respective entity is adapted to or configured to perform the respective steps and functionalities. Even if, in the following description of specific embodiments, a specific functionality or step to be performed by external entities is not reflected in the description of a specific detailed element of that entity which performs that specific step or functionality, it should be clear for a skilled person that these methods and functionalities can be implemented in respective software or hardware elements, or any kind of combination thereof.

BRIEF DESCRIPTION OF DRAWINGS The above described aspects and implementation forms of the present invention will be explained in the following description of specific embodiments in relation to the enclosed drawings, in which

FIG. 1 shows a device for encoding and/or decoding data transmitted in a communication channel according to an embodiment of the invention.

FIG. 2 shows a check matrix for a modified {N’, K’} code according to the invention. FIG. 3 shows a check matrix for a modified {N’, K’} code according to the invention.

FIG. 4 shows a code generator for generating a modified {N', K’} code from an original {/V, K} code according to an embodiment of the present invention.

FIG. 5 shows a construction of a modified {N’, K’} code from an original { /V, K} code an used in a device for encoding (encoder) or device for decoding (decoder) according to an embodiment of the present invention. FIG. 6 shows a method according to an embodiment of the invention.

FIG. 7 shows schematically a transmission system for FEC coding. FIG. 8 shows check matrices for an original code (middle), a shortened code (left) and an extended code (right).

FIG. 9 shows a check matrix for {N’, K’} code modified in a conventional manner from a {N, K} code.

FIG. 10 shows schematically a decoder for a new {N’, K’} code reusing a decoder of an original {TV, K} code.

DETAILED DESCRIPTION OF EMBODIMENTS

FIG. 1 shows a device 100 for encoding and/or decoding data according to an embodiment of the present invention. That is, FIG. 1 shows an encoder and/or decoder, which may be employed in a transmitter, receiver or transceiver, respectively. The data may be communications data and is transmitted in a (wireless or wired e.g. by an optical fibre) communication channel 101. The device 100 may encode the data, and then send the encoded data over the communication channel 101 to another device for decoding the data. The device 100 may also receive the data over the communication channel 101 from another device that encoded the data, and may then decode the received data. In particular, the device 100 of FIG. 1 is configured to encode and/or decode the data based on a new {TV’, K’} code 102 generated from an original {TV K} code 103, wherein N and N’ are code lengths, K and K’ are code dimensions, N’ - N = q > 0, and K - K’= p > 0. That is, the code length was increased, while the code dimension was decreased. Notably, the device 100 may also be configured to encode and/or decode data based also on the {N K} code 103.

The {TV’, K’} code 102 is defined by a specific check matrix 200 (FIG. 2 shows an example explained further below in detail). The specific check matrix 200 generally includes {TV— K + p + q} rows 201 and {TV + q} columns 202. Thereby, {TV- K} elements 203 in each of {p + q ) columns 204 of the check matrix 200 are all zeros, and { N - K } elements 205 in each of the remaining {N -p} columns 206 of the check matrix 200 are the elements of a check matrix defining the { N, K} code 103.

FIG. 2 illustrates an exemplary structure of such a specific check matrix 200, which may define the {N’, K’} code 102 employed by the device 100 of FIG. 1. The exemplary check matrix 200 of FIG. 2 includes {N - K + p + q} rows 201 and {N + q } columns 202. Here particularly the upper {N- K} elements 203 of the left {p + q} columns 204 of the check matrix 200 are all zeros. Further, particularly the upper {N - K} elements 204 of the remaining {N - p} columns 206 (i.e. right columns) of the exemplary check matrix 200 are the elements of the check matrix defining the {TV, K} code 103.

For encoding of the obtained new code 102, an explicit generator matrix may be obtained or other methods may be applied. Notably, equivalent codes 102 may be obtained by changing the order of the columns 202 of the check matrix 200 shown in FIG. 2, without any impact on the performance parameters of a decoder of the original code 103 using the new code 102. The same holds for changing the order of the rows 201 of the check matrix 200 or a combination of columns 202 and rows 201. The structure of the check matrix 200 as shown in FIG. 2 may further be obtained in several ways, wherein a preferred way is presented further below.

FIG. 3 shows another exemplary structure of a check matrix 200, which builds on the check matrix 200 shown in FIG. 2. In this check matrix 200 of FIG. 3, preferably the lower {p + q} elements 300 of the left {p + q} columns 204 of the check matrix 200 build an identity matrix I. Further, preferably each of the lower {p + q} rows 301 is filled with alternating ones and zeros. Thereby, the lower rows 301 start preferably with ones and zeros, respectively, in an alternating manned, in order to check odd and even indices of the new code 102, respectively.

FIG. 4 shows a code generator 400 according to an embodiment of the invention. The code generator 400 may be included in the device 100 for encoding and/or decoding data based on the {N’, K’} code 102 as shown in FIG. 1. That is, the code generator 400 may generate the code 102 from the { /V, K} code 103 for direct use in the device 100. The device 100 may thereby be configured to encode and/or decode the data based both on the new {N’, K’} code 102 or the original { /V, K) code 103. Thereby, again N and N’ are code lengths, and K and K’ are code dimensions.

The code generator 400 is particularly configured to shorten the original {TV, K} code 103 to obtain an intermediate code 401, and then extend the intermediate code 401 to obtain the new {N’, K’} code 102. For instance, the code generator 400 may be configured to shorten the {TV, K} code 103 by p positions to obtain the intermediate {N - p, K’} code 401, and to extend the intermediate {N - p, K’} code 401 by p + q positions to obtain the {TV’, K’} new code 102, wherein N’ - N= q > 0, K- K’ = p > 0, i.e. the code dimension is decreased while the code length is increased to obtain the new code 102.

FIG. 5 illustrates the generation of the new {TV’, K’} code 102 from the original {TV, K} code 103 as performed by the code generator 400 of FIG. 4 in more detail. Assumed is here a given {TV, K} code 103 with a check matrix H of size (N - K) x N. Further assumed is here a desired {TV’, K’} code 102 of larger length N’ and smaller dimension K’ and with a check matrix H’ of size (TV - 1C) x TV . The task of the code generator 400 is to construct H’ from TG, in order to allow efficient reuse of the new code 102 by a decoder of the original code 103.

The code construction, as preferably carried out by the code generator 400, includes the following two steps to obtain the new {TV’, K’} code 102. After these steps the new {N’, K’} code 102 may be provided to and/or used in an encoder and/or a decoder (e.g. in a device 100 as shown in FIG. 1):

Step 1: Shorten the {N, K} code 103 by p = K - K’ positions to obtain an {N - p, K - p} code 401 (the intermediate code), i.e. to obtain a {N - p, K’} code with the desired code dimension K’ = K -p.

Step 2: Extend the obtained {N-p, K’} intermediate code 401 by p + q positions to obtain a new {N-p + (p + q), K’} code 102 - i.e. an {N + q, K’} code - with the desired code length N’ = N+ q.

For decoding, the structure and relation to the original {N, K} code 103 can efficiently be exploited. The intermediate code 401 is a shortened code of the original {N, K} code 103. Decoding methods of the original code 103, e.g. Chase decoding or syndrome-based decoding, can easily deal with the shortening. The modified code 102 is an extended code of the intermediate code 401. Decoding is typically based on metric calculations, like the ML metric mentioned above, and the metric of an extended codeword can easily be computed from the metric of an intermediate codeword. Thus, an existing decoding algorithm for the original code 103 can efficiently be exploited for decoding of the new modified code 102.

FIG. 6 shows a method 600 according to an embodiment of the present invention. The method 600 is for constructing a { /V K’} code 102 for encoding and/or decoding data transmitted in a communication channel 101 from a { /V, K) code 103. The method 600 may be carried out by a device, like the code generator 400 shown in FIG. 3 or the device 100 shown in FIG. 1.

The code construction method 600 of the present invention is now explained in more detail based on an example. In particular, the aim is to specifically construct a { 128, 119} code 102 from a { 127, 120} Hamming code 103. This means that the code length N is to be increased by q = 1 and the code dimension K is to be decreased by p = 1. Notably, for Hamming codes, efficient syndrome decoding algorithms are available.

The check matrix H of the original Hamming code 103 consists of all binary non-zero column vectors of length 7. The check matrix of the cyclic Hamming code 103 can be constructed in the following way. Let a denote a root of the primitive polynomial g(a) = a ^{7 } + a ^{3 } + 1, and let (/) denote the binary representation of a! in the form of a column vector. The check matrix can then be defined as

H = [(0) (1) ... (126)]

Following the above-described general code construction, the original Hamming code 103 is now first shortened by p = 1 positions, and for example the first position is chosen. The check matrix of the resulting intermediate code 401 is

H _{t } = [(1) ... (126)] Then, this intermediate code 401 is extended by p + q = 2 positions, for instance by appending two all-zero columns (at the left of the matrix) and by further adding two check row vectors (i.e. effectively two additional check constraints for the new code 102) at the bottom, so that the check matrix 200 of the new code 102 becomes

Without loss of generality, the lower left part of this check matrix 200 may always be chosen to be the identity matrix (compare also the check matrix 200 in FIG. 3). Because of the two all-zero columns, computational complexity may be reduced. Further, the row vectors hi and fe are preferably chosen to be vectors with zeros and ones alternating (compare gain the check matrix 200 in FIG. 3), wherein hi preferably starts with 1 and h? preferably starts with 0. In this way, the penultimate row may advantageously be designed to check the odd indices, and the last row to check the even indices of the new code 102.

The proposed code construction according to the present invention is advantageous for the decoding of the new {N K’} code 102 (especially the { 128, 119} code). Namely, the decoding algorithm for the original {TV, K} code 103 (especially the { 127, 120} code) can efficiently be reused, as already detailed above.

The code construction according to the invention further provides a better distance profile. As an example, the minimum distance and its multiplicity was numerically evaluated for the example with the { 128, 119} code 102 and an alternative code construction, obtained according to the check matrix shown in FIG. 9. The minimum distance is the same, namely 4, in both constructions. However, the multiplicity is largely reduced with the code construction according to the invention. This is also observed clearly in error rate simulations under both ML decoding and sub-optimal decoding methods.

The present invention has been described in conjunction with various embodiments as examples as well as implementations. However, other variations can be understood and effected by those persons skilled in the art and practicing the claimed invention, from the studies of the drawings, this disclosure and the independent claims. In the claims as well as in the description the word“comprising” does not exclude other elements or steps and the indefinite article“a” or“an” does not exclude a plurality. A single element or other unit may fulfill the functions of several entities or items recited in the claims. The mere fact that certain measures are recited in the mutual different dependent claims does not indicate that a combination of these measures cannot be used in an advantageous implementation.

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