KR19990066105A - Error Correction Method of 2D Reed-Solomon Code - Google Patents

Abstract

The present invention relates to an error correction method of a two-dimensional Reed-Solomon code, and to a different error correction method in longitudinal decoding. In the error correction method of the present invention, transverse-decoding of (n1, k1) codes in which n1-k1 parity symbols are added to data having the number of information symbols k1, and erasing a symbol having an error among (n1, k1) codes Setting the code (Erasure Code), comparing the number of erase codes with a predetermined reference number, and if the number of erase codes is larger than the reference number, performing error correction with (n2, k2) codes on the vertical axis. And decoding the erase code by an error-error correction method if the number of erase codes is equal to or less than the reference number.

Description

Error Correction Method of 2D Reed-Solomon Code

The present invention relates to a two-dimensional Reed-Solomon error correction code.

In general, when digital data is transmitted, recorded on a storage medium, or reproduced, digital data to be processed is often encoded with an error correction code to reduce errors. Reed-Solomon codes are commonly used for these error correction codes. In particular, Reed-Solomon error correction codes are widely used in digital satellite broadcasting and compact disks (CD).

1 shows a two-dimensional structure of this Reed-Solomon code. This two-dimensional Reed-Solomon code adds n1-k1 parity symbols 11 for error correction for k1 information symbols 10 in the horizontal axis direction, and n2-k2 number for k2 information symbols in the longitudinal axis direction. The parity symbol 12 has a form added. That is, the Reed-Solomon code is a code converted into data larger than the original data by adding the parity symbols 11 and 12 to the information symbol 10.

2 is a block diagram illustrating a general encoding and decoding process of a Reed-Solomon code having a two-dimensional structure. The encoding process through the block diagram shown in FIG. 2 will now be described. First, the input data Din is input to the vertical axis encoder and encoded in the vertical axis direction. The data encoded in the longitudinal axis direction is stored in the memory, input to the horizontal axis encoder in a predetermined processing unit, and encoded in the horizontal axis direction. Data encoded in both the longitudinal and transverse directions is recorded on a recording medium 20 such as a magnetic tape.

The decoding process through the block diagram shown in FIG. 2 will now be described. First, data recorded on the recording medium 20 such as a magnetic tape is input to the horizontal axis decoder. The horizontal axis decoder decodes the input data in the horizontal axis direction and stores the received data in the memory. Data stored in the memory is applied to the vertical axis decoder and decoded in the longitudinal axis direction and output.

That is, the encoding process of the two-dimensional Reed-Solomon code is performed in the order that the data is first encoded in the longitudinal axis in the longitudinal axis encoder and stored in the memory, and then the encoded code is encoded in the horizontal axis in the horizontal axis encoder and recorded on the recording medium. . The decoding process of the 2D Reed-Solomon code is performed in the reverse order of the encoding process. First, the horizontal axis decoder first decodes the data recorded on the recording medium in the horizontal axis direction, and then stores the data in the memory. Data is stored in the longitudinal axis direction and reduced to the original data.

In general, Reed-Solomon codes with k information symbols and n-k parity symbols added are named (n, k) lead-solomon codes. The nature of this (n, k) lead-solomon code is:

First, when the location of an error symbol is not known, the (n, k) lead-solomon code may correct '(n-k) / 2' errors. In addition, when the position of an error symbol is identified, error-correction correction may be performed on the symbol. At this time, when the number of error symbols whose position is unknown is x and the number of erasure symbols is y, the error symbols are correctly corrected if the condition of 2x + y≤n-k is satisfied. If it is assumed that the positions of the error symbols are all known, since x = 0, the error symbols are correctly corrected under the condition of y≤n-k.

Referring to the conventional decoding method using the properties of the Reed-Solomon code as follows. 3 is a flowchart illustrating a method of decoding a Reed-Solomon code. If error correction is performed on the code in the horizontal axis direction, the error symbols are correctly corrected when the number of error symbols is less than (n1-k1) / 2.

However, the decoding method of the two-dimensional Reed-Solomon code shown in FIG. 3 has the following problems. First, if there is one more symbol having an error than '(n1-k1) / 2', the decoder mis-corrects the wrong code or detects an error. Mis-correction means a case where error detection is not performed when more than '(n1-k1) / 2' errors occur in the received horizontal axis code and corrected by another codeword.

In the conventional two-dimensional decoding method of the Reed-Solomon code, the code in which the horizontal axis decoder error occurs is set as an erasure code, and the erasure code is used to perform erasure correction by using the erasure code. At this time, however, if the erase code becomes larger than n2-k2 or if a miscorrection occurs, the decoding error occurs when the subordinate decoder performs erase correction. For example, when one or more miscorrections occur in the horizontal axis decoding, and n2-k2 erase codes are set, normal erasure decoding is not performed when the erase correction is performed. That is, a decoding error occurs.

The present invention has been made to solve such a problem, and an object thereof is to minimize decoding errors occurring during vertical axis decoding.

1 illustrates the structure of a two-dimensional Reed-Solomon code.

2 is a block diagram showing the operation and structure of a typical decoder.

3 is a flowchart illustrating a conventional decoding process.

4 is a flowchart illustrating a decoding process of the present invention.

Fig. 5 is a graph showing the decoding error rate after longitudinal axis decoding, which changes according to the correction number of erase codes.

Explanation of symbols for main parts of the drawings

10: information symbol 11: horizontal axis parity symbol

12: vertical parity symbol 20: recording medium

In the present invention, instead of the conventional method of performing erasure correction when performing longitudinal decoding, when the erase codes are smaller than or equal to a predetermined number, error erasure correction is performed, and a larger number of erase codes have been generated. In this case, the erase code is ignored and the error is corrected. That is, the present invention is characterized in that the decoding is different depending on the number of erase codes. The core of the present invention is a method of setting the number of erase codes at different decoding points.

In order to set the number of erase codes at the time of different decoding, which is the core of the present invention, the error rate after the longitudinal axis decoding of the 2D Reed-Solomon code is calculated. 4 is a flowchart illustrating a decoding process of the present invention.

First, in the transverse decoding process, the (n1, k1) lead-solomon code is correctly decoded when the number of error symbols is less than or equal to '(n1-k1) / 2'. If the error symbols are randomly distributed in the input data and the probability that the error symbols occur is P _S , the probability P _C for correctly transversely decoding the data is as follows.

T1 = (n1-k1) / 2.

If the number of errors is greater than '(n1-k1) / 2', they may be miscorrected, resulting in more errors. At this time, the probability P _ud to be miscorrected is as follows.

However, Equation 2 is the following condition,

ego,

Must satisfy Q is the size of the code symbol. In addition, the fault is detected erasure code follows the probability P _er be (Erasure) occurs.

P _er = 1-P _C -P _ud

When an arbitrary symbol is selected after horizontal axis decoding, if the symbol is an error, it may be caused by miscorrection or an erase code may be generated. First, when an arbitrary symbol is selected after performing transverse axis decoding, the probability P _{ud, s} of error symbol occurrence due to miscorrection is as follows.

Then, the error is detected when a t1 + 1 of error occurs when the probability of occurrence as the cancellation code _{(Erasure) P er (t1 +} 1), and then the expression is satisfied.

_{P er (t1 + 1) =} P ( symbol error rate is caused t1 + 1 pieces) - P (symbol error probability is the cause ohjeongjeong code generated when t1 + 1 pieces)

In this case, the number of error symbols after decoding is (t1 + 1). Therefore, when an arbitrary symbol is selected from the code in which the erase code is detected, the probability P _{er, s} that the code is an error symbol is as follows.

Here, _Per (i) at this time is as follows.

In the conventional decoding method, only the erasure code is corrected when decoding in the vertical axis direction. In this case, the maximum number of erase codes that can be corrected is at most n2-k2. This is equal to the number of parity symbols in the (n2, k2) code. If the number of erase codes exceeds n2-k2, the decoding method of the conventional decoding method has occurred in the vertical decoder as described above.

In order to reduce the error rate after decoding, an error erasing method is performed until there are K erase codes, and when more than K + 1 occurs, error correction is performed to reduce the error rate after decoding. K at this time is determined as a value that can minimize the error rate after decoding. Then, in order to determine K, first, the probability P (j Erasure) of occurrence of j erase codes in the vertical axis decoder is calculated as follows.

Then, when j erase codes are generated and erased, the probability of an error is as follows.

In this case, t is the number of error symbols generated by miscorrection. Therefore, when the number of erase codes is less than or equal to K, the error rate after decoding of the error erasure definition is as follows.

According to the present invention, when the number of erase codes is larger than K, error correction is performed by disregarding the erase information generated in the horizontal axis decoder in the vertical axis decoder. In this case, i + t> (n2-k2) / 2 when i error symbols occur among the codes set to the erase code and t error symbols occur among the codes not set to the erase code, Occurs. First, the probability of generating i error symbols among the j erase codes is as follows.

In addition, when j erase codes are generated, a probability of generating an error symbol due to miscorrection among remaining symbols except for the erase codes is as follows.

Therefore, when error correction is performed when j erase codes are generated, the probability of decoding error is as follows.

At this time, t2 = (n2-k2) / 2,

to be.

Therefore, when the number of erase codes is larger than K, the probability of generating a decoding error is as follows.

Therefore, when decoding according to the present invention, the probability of generating an error code in the longitudinal decoder is as follows.

P _e (K) = P _e1 (K) + P _e2 (K)

When j erase codes occur, the vertical axis decoding error rates in case of error erasing correction and error correction are as shown in Equations 6 and 10, respectively. Therefore, it is preferable to perform decoding in a manner in which the error rate is minimized among the two methods. The occurrence rate of the error symbol of the input data is not a factor that can be controlled in the error correction process of decoding. However, since the set number of erase codes can be sufficiently determined in the error correction process, the decoder can be designed such that an error correction method is implemented differently according to the set number of erase codes. Therefore, the decoder of the present invention can be designed with a K value that minimizes the error rate that occurs in the longitudinal decoder.

A suitable embodiment for determining the K value of the present invention will be described with reference to the following table.

(Example)

Table 1 to Table 4 set k1 = 100, k2 = 100, n2 = 110 and n1 to 106, 108, 110, 112 in the two-dimensional (n, k) lead-solomon code as shown in FIG. According to the following four cases, the optimal K value according to the error symbol occurrence rate of the input data is shown.

(n1, k1) BER K0 K2 K4 K6 K8 K10 (106,100) 0.02 1.42E-04 1.42E-04 1.42E-04 1.42E-04 2.38E-04 4.23E-04 (106,100) 0.01 1.01E-09 1.01E-09 1.05E-09 1.19E-08 9.89E-07 1.12E-06 (106,100) 0.005 5.96E-16 5.96E-16 5.47E-16 5.53E-17 3.99E-16 4.06E-16 (106,100) 0.002 6.55E-25 6.55E-25 5.87E-25 2.01E-29 7.57E-30 7.57E-30 (106,100) 0.001 6.15E-32 6.15E-32 5.52E-32 3.52E-38 2.95E-38 2.95E-38

(n1, k1) BER K0 K2 K4 K6 K8 K10 (108,100) 0.02 2.24E-06 2.24E-06 2.24E-06 2.26E-06 7.96E-05 1.34E-04 (108,100) 0.01 3.05E-13 3.05E-13 3.03E-13 2.59E-14 1.59E-13 1.66E-13 (108,100) 0.005 3.25E-21 3.25E-21 3.21E-21 8.05E-25 1.71E-26 1.71E-26 (108,100) 0.002 1.63E-32 1.63E-32 1.61E-32 7.16E-40 3.28E-43 3.28E-43 (108,100) 0.001 2.48E-41 2.48E-41 2.45E-41 1.35E-51 8.36E-53 8.36E-53

(n1, k1) BER K0 K2 K4 K6 K8 K10 (110,100) 0.02 1.24E-08 1.24E-08 1.24E-08 8.45E-09 2.42E-08 2.79E-08 (110,100) 0.01 3.13E-17 3.13E-17 3.13E-17 1.12E-19 1.17E-21 1.12E-21 (110,100) 0.005 5.96E-27 5.96E-27 5.94E-27 1.33E-32 1.02E-37 9.75E-38 (110,100) 0.002 1.32E-40 1.32E-40 1.32E-40 8.47E-51 3.48E-57 3.48E-57 (110,100) 0.001 3.24E-51 3.24E-51 3.24E-51 6.05E-65 3.43E-68 3.43E-68

(n1, k1) BER K0 K2 K4 K6 K8 K10 (112,100) 0.02 2.84E-11 2.84E-11 2.84E-11 4.79E-12 2.81E-13 1.26E-13 (112,100) 0.01 1.34E-21 1.34E-21 1.34E-21 1.34E-25 3.52E-30 1.37E-30 (112,100) 0.005 4.45E-33 4.45E-33 4.45E-33 6.81E-41 2.94E-49 1.26E-49 (112,100) 0.002 4.32E-49 4.32E-49 4.32E-49 3.08E-62 1.00E-71 1.00E-71 (112,100) 0.001 1.68E-61 1.68E-61 1.68E-61 8.81E-79 2.50E-84 2.50E-84

In Tables 1 to 4, (n2, k2) is (110, 100), and BER is an error rate of input data symbols. The number after K is a setting value of K, which indicates that error erasure is corrected up to the number of erasure codes corresponding to the number after K, and that error correction is performed when more erase codes occur.

In Tables 1 to 4, each code error rate when the incidence of error symbols is 10 ⁻² , that is, when the BER value is 0.01, is shown in Table 5 below.

BER (n1, k1) K0 K2 K4 K6 K8 K10 0.01 (106,100) 1.01E09 1.01E-09 1.01E-09 9.48E-10 1.48E-08 1.12E-06 0.01 (108,100) 3.05E-13 3.05E-13 3.05E-13 1.14E-13 4.85E-15 1.66E-13 0.01 (110,100) 3.13E-17 3.13E-17 3.13E-17 2.62E-18 3.18E-21 1.12E-21 0.01 (112,100) 1.34E-21 1.34E-21 1.34E-21 1.90E-23 6.24E-28 1.37E-30

According to Table 5, when the error rate of the input data is assumed to be 0.01, the K value at which the decoding error rate is minimum when n1 is 106 is 6, and the K value at which the decoding error rate is minimum when n1 is 108 is 8, n1. When the value is 110 or 112, it can be seen that the K value at which the decoding error rate is minimum is 10. Fig. 5 shows the decoding error rate at the time of vertical axis decoding which changes according to the correction number of erase codes.

However, as shown in Tables 1 to 4, even if the two-dimensional Reed-Solomon code of the same structure, it can be seen that the optimum K value is different if the error rate of the input data is different. Therefore, the present invention improves the error correction effect by differently decoding the K value according to the error rate of the input data and the structure of the Reed-Solomon code during longitudinal axis decoding. According to the present invention, the K-value may be set and decoded according to the structure of the Reed-Solomon code.

In the error correction method of the present invention, since the decoding method is different depending on the number of erase codes when the 2D Reed-Solomon code is decoded, the probability of error occurrence after decoding is minimized as compared to the conventional error correction method by simple decoding. There is. Therefore, the error correction method of the present invention increases the transmission reliability of digital data after longitudinal axis decoding, as compared with the conventional error correction method.

Claims (3)

Performing horizontal axis decoding with (n1, k1) codes in which n1-k1 parity symbols are added to data having the number of information symbols k1; Setting an error symbol among the (n1, k1) codes as an erasure code; Comparing the number of erase codes with a predetermined reference number; Error correcting and decoding the (n2, k2) code of a vertical axis if the number of erase codes is larger than the reference number; And, And if the number of erase codes is equal to or less than the reference number, decoding the erase codes by an error erasure correction method. The error correction method of claim 1, wherein the reference number is minimized when a predetermined input error rate is given, and the error occurrence rate after vertical axis decoding is minimized. The error correction method of claim 2, wherein the reference number varies according to the input error rate.