PRML code encoding and decoding methods dor high-density data storing device

0001The present invention relates to partial response maximum likelihood (PRML) code encoding and decoding methods for a high-density data storage device, and more particularly, to a PRML code encoding and decoding methods for a high-density data storage device. in which digital data is magnetically recorded on and reproduced from a disk memory device with high density without interference between signals.

0002Research has been conducted to more efficiently utilize mass quantities of information in a rapidly developing information society, resulting in great progresses in many fields. In a storing device field. the focus of research has been to transmit much information and to reduce the time required to process information to satisfy information needs in the information-competitive society. That is, efforts are being made toward rapid transmission of reliable information. while increasing the amount of data recorded (recording density) in a given storage device. For high-speed. large capacity data storage devices. material aspects are considered by improving the physical characteristics or increasing the accuracy of a storage medium. Further. signal processing aspects are considered by increasing the recording density of the storage device and facilitating detection of a reproduced signal through efficient encoding and decoding, or reducing data detection errors. relying on signal processing technology.

0003Studies have been made to increase the amount (recording density) of data recorded on a given storage device during recording and reproducing of data and to rapidly transmit reliable information. A storing device having data recorded with a high density enables efficient encoding, which in turn reduces redundancy and facilitates signal detection. However. the problem of intersymbol interference (ISI) becomes more prominent with higher-density recording. In general, data recorded on a storage device is encoded to run length limited (RLL) codes. The RLL code limits succession of recorded data symbols for timing control of data sampling clock signals and signal detection. That is. in the RLL code, the number of zeros between ones is limited to a minimum of d for easy signal detection and to a maximum of k for timing of data during playback of a signal. Recently used RLL code encoding methods include code rate ½ RLL(2, 7), 2/3 RLL(1, 7), 8/9 RLL(O, 3), and 8/9 RLL(O. 4/4) encoding methods. The first two encoding methods have dis of 1 and 2. respectively. Thus, they allow one zero and two zeros between ones, respectively, thereby reducing interference between signals. Despite the advantage of the decrease in signal interference, these methods require many bits to transmit given user data due to large redundancy caused by their low code rates. Hence, a low-code rate encoding method causes more intersymbol interferences than a high-code rate encoding method such as the 8/9 RLL(0, 3) and 8/9 RLL(0, 4/4) methods, thus nullifying the advantage of allowance of at least one zero between transitions. The high-code rate encoding method is more favorable for recording and reproducing data due to a smaller redundancy than the low-code rate encoding method. The high-code rate encoding method increases a channel input SNR, reduces interference between data due to a small redundancy, and enables high-density recording.

0004In general, a channel should be modeled after an actual one in recording and reproducing data in a storing device. To reflect the channel characteristic of the storing device, channel characteristics can be expressed as (1+Dⁿ)(n=1, 2, ...) or (1-D)(1+D)ⁿ (n=1, 2....). In the PRML method, mutually controlled intersymbol interference is set between current data and previous data by precoding an input signal and then data is detected in a Viterbi decoder by modifying a target response d_k to a_k+a_k-1 or a_k-a_k-2. The PRML method shows excellent detection performance with a recording density of the signal interference given under a channel characteristic of (n=1). As data is recorded at a higher density, the distance between transitions becomes smaller, thus worsening the intersymbol interference between data. To reduce the interference in high-density recording, the distance between transitions should be increased. An RLL(1, 7) encoding method, employing this concept, encodes data to have at least one zero between symbols. However, though the RLL(1, 7) encoding method limits transitions, it has a low code rate. Therefore, to transmit given data, more bits than the high-code rate encoding methods 8/9 RLL(0, 3) and 8/9 RLL(0, 4/4), are required leading to a small distance between recorded data. As a result, interference between data becomes serious. That is, a high-code rate encoding method is more useful in recording and reproducing data in channels of a storage device than a low-code rate encoding method. In addition, the high-code rate encoding method reduces interference between data and non-linearity, relative to the low-code rate encoding method, thus enabling high-density recording.

0005To reduce the intersymbol interference, there are two schemes: increasing code rate; and setting the interval between transitions. However, both have a trade-off relationship. That is, it is impossible to set some interval between transitions without reducing the code rate, and the code rate cannot be increased with the interval being kept between transitions.

0006The 2/3 RLL(1, 7) encoding method relies on the latter method. On the other hand, the 8/9 RLL(0, 3) and 8/9 RLL(0, 4/4) encoding methods are suggested from a different viewpoint. These encoding methods have higher channel SNRs due to their high code rates than low-code rate encoding methods. Further, less bits are used than in the low-code rate decoding methods to record given data. Thus, intersymbol interference and thus non-linearity can be reduced. However, intersymbol interference becomes a serious problem as the recording density of a data storing device is increased.

0007To solve the above problem, it is an object of the present invention to provide PRML code encoding and decoding methods for a high-density data storage device, which has optimized input and output lookup tables and a simplified codeword generating equation for encoding and decoding a block code with a code rate of 8/9 and a maximum transition run of 3.

0008According to the present invention, there is provided encoding and decoding methods for a high-density data storage device. In the encoding and decoding methods, input 8-bit data is compressed and error-correction encoded to be immunized to noise and other signal-distortion causing factors. The error-corrected data is encoded to 9-bit codewords modulated to be suitable for the channel characteristics of the storing device. A signal is generated for the input 8-bit data encoded to modulation codes. The generated signal is pre-compensated and recorded. The generated signal is processed to be easily detected by reproducing the 9-bit codewords. The processed signal is detected to minimize an error rate. The detected signal is decoded according to channel characteristics, and recovered to the input 8-bit user data through error correction and compression decoding.

0009In the PRML encoding and decoding methods for a high-density data storing device, the 9-bit codeword encoding step is for receiving the 8-bit input data, which are a predetermined series of 8-bit binary data symbols with an integer time of k as an index, X_k of 1 or 0, and k of 1, 2, ..., 8 in a storage device or a communications channel, and encoding the 8-bit input data to predetermined 9-bit codewords with Y_k of 1 or 0 and k of 1, 2, ..., 9. The 9-bit codeword encoding step includes the substeps of (step 100 - step 600).

0010In step 100, a codeword with a maximum transition run (MTR) of 3 is generated. In step 200, a codeword with a maximum zero run length k of 7 is generated with respect to the codeword generated in step 100. In step 300, it is determined whether the fifth bit of the 9-bit codeword satisfying step 100 and step 200 is zero, and the procedure goes to step 400 if the determination result is zero, and jumps to step 500 if the determination result is not zero. In step 400, the first four bits and last four bits of the input 8-bit user data are mapped to the first four bits and last four bits of the codeword without variation. In step 500, the input 8-bit user data excluded in step 300 is mapped to the most approximate 9-bit codewords which have ones in the fifth bits thereof. In step 600, encoder and decoder mapping principles are applied between the input 8-bit user data and the 9-bit codewords according to the mapping results of step 400 and step 500, and simplified to Karnaugh maps.

0011In the input 8-bit data decoding step, the 9-bit codewords are decoded to the 8-bit binary data symbols by generating predetermined 8-bit binary data sequences with X_k of 1 or 0, and k of 1, 2, ..., 8 from the 9-bit codewords with Y_k of 1 or 0 and k of 1, 2, ..., 9. This step is performed by reversely performing the step of encoding the 8-bit user data to the 9-bit codewords.

0012A maximum transition run (MTR) method is suggested to record data in a data storage device with high density. This encoding method allows an MTR of 2 similar to the 2/3 RLL(1, 7) encoding method for recording data with at least one sample between transitions, and increases a code rate influencing the channel input SNR. Thus, the MTR encoding method is suitable for high-density recording for a data storing device. The present invention pertains to encoding modulation codes suitable for a partial response channel for maximum likelihood (ML) detection expressed as (1-D)(1+D)ⁿ (n = 1, 2, ...), and limits MTR to 3. Thus, an MTR of 3 is allowed and, on the other hand, the code rate influencing a channel input SNR is increased relative to a conventional MTR encoding method, so that data is efficiently recorded and reproduced.

0013An encoding method with a code-rate of 8/9 and an MTR of 3 shows a high code rate, and reduces viterbi detector paths like a 2/3 RLL(1, 7) with an MTR of 2 encoding method, thus decreasing the delay and complexity of a detector. In addition, in the present invention, the constraint k is 7 for timing and gain control to thereby keep the self-clocking characteristics of a signal and facilitate signal detection. The code-rate of 8/9 and an MTR of 3 encoding method is suitable for a high-density data storage device since it is byte-oriented and has minimal redundancy.

0014The present invention deals with modulation codes suitable for a partial response channel for ML detection. These modulation codes offer frequent non-zero samples and thus improves the timing of the partial response channel and the performance of a gain control circuit. Further, they limit the complexity of a ML detector by reducing paths during a process in a data estimator.

0015A modulation code has two parameters in the present invention: MTR and k. MTR represents a maximum transition run in a channel output code bit sequence and reduces intersymbol interference, whereas k represents a maximum zero run length in a coded sequence. and is used for accuracy of timing and gain control, and self-clocking. In the present invention. the MTR is 3. and k is 7.

0016The code constraints for encoding and decoding data are applicable to a partial response signal transmission system for ML detection. In the present invention. the constraints are a code rate of 8/9, an MTR of 3, and k of 7, and optimized lookup tables for encoding and decoding block codes under such code constraints are provided.

0017An example of the present invention will now be described in detail with reference to the accompanying drawings. in which: <ul id="ul0001" list-style="none"><li>FIG. 1 is a block diagram of a high-density data storage device for encoding and decoding PRML codes according to the present invention; and,</li><li>FIG. 2, ..., is a flowchart illustrating a method for generating PRML codes with a code rate of 8/9, an MTR of 3, and k of 7.</li></ul>

0018FIG. 1 is a block diagram of a high-density data storage device for PRML code encoding and decoding as a general process to which the present invention is applied.

0019As shown, the storage device includes a compression encoder 10 for compressing user data, an error-correction encoder 20 for performing an error-correction on the compressed data, an RLL modulation encoder 30 for encoding the error-corrected data to RLL modulation codes. a signal generator 40, a writing equalizer 50, a channel head/disk, a reading equalizer 60, a detector 70 for detecting a signal, an RLL modulation decoder 80 for decoding data to RLL modulation codes, an error-correction decoder 90 for performing an error-correction decoding, and a compression decoder 100.

0020The general process. to which the present invention is applied in the high-density data storage device, will be described.

0021In order to record and reproduce data with high density without interference between signals, a data encoder, a data decoder, an equalizer, and a detector are used. User data is compressed. error-corrected, and encoded to modulation codes suitable for a channel.

0022After the encoded signal is recorded and reproduced, a signal is detected in the detector through the equalizer, and decoded. to thereby recover the user data.

0023The operational principle of the present invention will be described as follows.

0024Referring to FIG. 1, a signal recorded in a storage device is compressed in the compression encoder 10 to protect previous data and encoded in the error-correction encoder 20, thus being protected from noise and other signal-distortion causing factors.

0025Then, the error-corrected signal is encoded to modulation codes suitable for channel characteristics of the storing device in the RLL modulation encoder 30. A signal is generated in the signal generator 40, and pre-compensated in the writing equalizer 50. Then, the pre-compensated signal is recorded and reproduced. The reproduced signal is processed in the reading equalizer 60 to facilitate signal detection, and a signal is detected in the detector 70 to minimize the error rate. The data is recovered by reversely performing the data recording process in the storage device.

0026FIG. 2 is a flowchart illustrating a method for generating a PRML code under a code rate of 8/9, an MTR of 3, and k of 7.

0027A block code with a code rate of 8/9, an MTR of 3, and k of 7 provides 257 9-bit codewords which correspond to 8-bit data bytes. Thus, every 8-bit data combination can be encoded and decoded to a corresponding 9-bit codeword.

0028In the present invention, read backward symmetry is kept by introducing a particular method for converting an 8-bit data to a 9-bit codeword, and the 9-bit codeword has a similar structure to that of the 8-bit binary data through byte partitioning. Byte partitioning facilitates one-to-one correspondence of the 8-bit binary data to the 9-bit codeword so that the complexity of an encoder and a decoder is kept to a minimum.

0029Assuming that Y is a 9-bit codeword with a code rate of 8/9, an MTR of 3, and k of 7, Y can be expressed as<maths id="math0001" num="(1)"><math display="block"><mrow><mtext>Y = {Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9}</mtext></mrow></math><img file="EP0899885A2_D0001.tif" /></maths>

0030In step 100, a group of 9-bit codewords with a code rate of 8/9 and an MTR of 3 can be generated by 9-bit sequences having a successive transition run of 2 or above at their left ends, a successive transition run of 3 or above at their right ends, or a successive transition run of 4 or above, from the total encoded sequences.

0031The constraint (MTR=3) is given as a Boolean equation as follows:<maths id="math0002" num="(2)"><math display="block"><mrow><mtext>(</mtext><mover accent="true"><mrow><msub><mrow><mtext>Y</mtext></mrow><mrow><mtext>1</mtext></mrow></msub><msub><mrow><mtext>Y</mtext></mrow><mrow><mtext>2</mtext></mrow></msub></mrow><mo>¯</mo></mover><mtext>) (</mtext><mover accent="true"><mrow><msub><mrow><mtext>Y</mtext></mrow><mrow><mtext>1</mtext></mrow></msub><msub><mrow><mtext>Y</mtext></mrow><mrow><mtext>2</mtext></mrow></msub><msub><mrow><mtext>Y</mtext></mrow><mrow><mtext>3</mtext></mrow></msub><msub><mrow><mtext>Y</mtext></mrow><mrow><mtext>4</mtext></mrow></msub></mrow><mo>¯</mo></mover><mtext>) (</mtext><mover accent="true"><mrow><msub><mrow><mtext>Y</mtext></mrow><mrow><mtext>2</mtext></mrow></msub><msub><mrow><mtext>Y</mtext></mrow><mrow><mtext>3</mtext></mrow></msub><msub><mrow><mtext>Y</mtext></mrow><mrow><mtext>4</mtext></mrow></msub><msub><mrow><mtext>Y</mtext></mrow><mrow><mtext>5</mtext></mrow></msub></mrow><mo>¯</mo></mover><mtext>)</mtext><mspace linebreak="newline" /><mtext> (</mtext><mover accent="true"><mrow><msub><mrow><mtext>Y</mtext></mrow><mrow><mtext>3</mtext></mrow></msub><msub><mrow><mtext>Y</mtext></mrow><mrow><mtext>4</mtext></mrow></msub><msub><mrow><mtext>Y</mtext></mrow><mrow><mtext>5</mtext></mrow></msub><msub><mrow><mtext>Y</mtext></mrow><mrow><mtext>6</mtext></mrow></msub></mrow><mo>¯</mo></mover><mtext>) (</mtext><mover accent="true"><mrow><msub><mrow><mtext>Y</mtext></mrow><mrow><mtext>4</mtext></mrow></msub><msub><mrow><mtext>Y</mtext></mrow><mrow><mtext>5</mtext></mrow></msub><msub><mrow><mtext>Y</mtext></mrow><mrow><mtext>6</mtext></mrow></msub><msub><mrow><mtext>Y</mtext></mrow><mrow><mtext>7</mtext></mrow></msub></mrow><mo>¯</mo></mover><mtext>) (</mtext><mover accent="true"><mrow><msub><mrow><mtext>Y</mtext></mrow><mrow><mtext>5</mtext></mrow></msub><msub><mrow><mtext>Y</mtext></mrow><mrow><mtext>6</mtext></mrow></msub><msub><mrow><mtext>Y</mtext></mrow><mrow><mtext>7</mtext></mrow></msub><msub><mrow><mtext>Y</mtext></mrow><mrow><mtext>8</mtext></mrow></msub></mrow><mo>¯</mo></mover><mtext>)</mtext><mspace linebreak="newline" /><mtext> (</mtext><mover accent="true"><mrow><msub><mrow><mtext>Y</mtext></mrow><mrow><mtext>6</mtext></mrow></msub><msub><mrow><mtext>Y</mtext></mrow><mrow><mtext>7</mtext></mrow></msub><msub><mrow><mtext>Y</mtext></mrow><mrow><mtext>8</mtext></mrow></msub><msub><mrow><mtext>Y</mtext></mrow><mrow><mtext>9</mtext></mrow></msub></mrow><mo>¯</mo></mover><mtext>) (</mtext><mover accent="true"><mrow><msub><mrow><mtext>Y</mtext></mrow><mrow><mtext>7</mtext></mrow></msub><msub><mrow><mtext>Y</mtext></mrow><mrow><mtext>8</mtext></mrow></msub><msub><mrow><mtext>Y</mtext></mrow><mrow><mtext>9</mtext></mrow></msub></mrow><mo>¯</mo></mover><mtext>) = 1</mtext></mrow></math><img file="EP0899885A2_D0002.tif" /></maths>

0032In step 200, to generate 9-bit codewords with k of 7, codewords are removed from the whole sequences. which have a zero run length of 5 from their left ends, a zero run length of 4 from their right ends. or a zero run length of 7. The constraint (k=3) is expressed as<maths id="math0003" num="(3)"><math display="block"><mrow><mtext>(Y1+Y2+Y3+Y4+Y5)(Y6+Y7+Y8+Y9) = 1</mtext></mrow></math><img file="EP0899885A2_D0003.tif" /></maths>

0033There are 293 9-bit binary codewords satisfying equation (2), and there are 257 9-bit codewords satisfying equations (2) and (3).

0034Therefore, the efficiency is very high since only one extra codeword exists. This extra codeword can be used to remove an unintended codeword pattern or detect errors.

0035In step 300, it is determined whether the middle bits of the 9-bit codewords satisfying equations (2) and (3) are zeroes, and a method for mapping the input 8-bit data to 9-bit codewords is varied.

0036In step 400. which is a step performed when the middle bits of the 9-bit codewords are zeroes, partitioning is used to give regularity between the 9-bit codewords and the input 8-bit user data sequence.

0037The first four bits and last four bits of the 8-bit data are mapped to the first four bits and last four bits of the 9-bit codewords without variations. In this partitioning, the middle bit. that is, the fifth bit of a 9-bit codeword is always zero.

0038That is, the first four bits and last four bits of the 9-bit codewords having middle bits as zeroes among the 9-bit codewords satisfying equations (2) and (3) are mapped from the 8-bit binary data having the same first four bits and last four bits. There are 143 pairs of 8-bit data bits and 9-bit codewords separated by partition using the symmetry between the 8-bit user data and the 9-bit encoded data.

0039In step 500. the other 8-bit sequences excluded In step 400 are mapped to 9-bit codewords.

0040That is, when the 8-bit user data excluded In step 400 is mapped to 9-bit codewords, the 9-bit codewords are most approximate to the 8-bit user data among 28-143 9-bit codewords having the middle bits as zeroes, in order to reduce the complexity of an encoder and a decoder.

0041In step 600. encoder and decoder mapping principles between the input 8-bit user data and the 9-bit codewords are applied according to the mapping results In step 400 and step 500. and simplified to Karnaugh maps.

0042Table 1 shows 9-bit codewords with a code rate of 8/9, an MTR of 3, and k of 7, expressed as sexidecimal forms. <tables id="tabl0001" num="0001"><img file="EP0899885A2_D0004.tif" /></tables><tables id="tabl0002" num="0002"><img file="EP0899885A2_D0005.tif" /></tables>

0043Table 2 illustrates the encoding mapping principle between the 8-bit user data and the 9-bit codewords with a code rate of 8/9, an MTR of 3, and k of 7, in order to simplify an encoder circuit. <tables id="tabl0003" num="0003"><table frame="none"><title>(TABLE 2A)</title><tgroup cols="1" colsep="1" rowsep="0"><colspec colnum="1" colname="col1" colwidth="157.50mm" /><tbody valign="top"><row><entry namest="col1" nameend="col1" align="left">G = (A₇A₆'=A₇'(A₆+A₅+A₄))(A₃+A₂+A₁+A₀)(A₂'+A₁'+A₀')</entry></row><row><entry namest="col1" nameend="col1" align="left">H = A₇A₆A₇'(A₂+A₁+A₀)(A₂'+A₁'+A₀')</entry></row><row><entry namest="col1" nameend="col1" align="left">J = A₇A₆A₃'(A₂'+A₁'+A₀')</entry></row><row><entry namest="col1" nameend="col1" align="left">K = (A₆+A₅+A₄)(A₆'+A₅'+A₄')(A₂A₁A₀+A₃'A₂'A₁'A₀')</entry></row><row><entry namest="col1" nameend="col1" align="left">L = A₇'A₆'A₅'A₄'A₃'(A₂+A₁+A₀)(A₂'+A₁'+A₀')</entry></row><row><entry namest="col1" nameend="col1" align="left">LL = A₇'A₆'A₅'A₄'A₃'(A₂+A₁+A₀)(A₂'+A₁'+A₀')</entry></row><row><entry namest="col1" nameend="col1" align="left">M = A₇'A₆'A₅'A₄'(A₂A₁A₀+A₂'A₁'A₀')</entry></row><row><entry namest="col1" nameend="col1" align="left">N = (A₇'A₆A₅A₄+A₇A₆'A₅'A₄')(A₂A₁A₀+A₃'A₂'A₁'A₀')</entry></row><row><entry namest="col1" nameend="col1" align="left">O = A₇A₆A₅A₄(A₂A₁A₀+A₃'A₂'A₁'A₀')</entry></row><row><entry namest="col1" nameend="col1" /></row><row><entry namest="col1" nameend="col1" align="left">C₈ = GA₇+H+KA₃+M</entry></row><row><entry namest="col1" nameend="col1" align="left">C₇ = GA₆+JA₂+ KA₇A₃'+ L(A₂A₁ +A₂A₀) + LL(A₂+A₁A₀) + NA₂</entry></row><row><entry namest="col1" nameend="col1" align="left">C₆ = GA₅+HA₅+JA₁+K((A₃⊕A₂)+A₇A₃)+N(A₃A₂+A₃'A₂')+OA₂</entry></row><row><entry namest="col1" nameend="col1" align="left">C₅ = GA₄+HA₄+JA₀+L+LL+M+NA₂'</entry></row><row><entry namest="col1" nameend="col1" align="left">C₄ = H+J+K+L+LL+M+N+O</entry></row><row><entry namest="col1" nameend="col1" align="left">C₃ = GA₃+K+L(A₁A₀+A₁'A₀')+LL+M</entry></row><row><entry namest="col1" nameend="col1" align="left">C₂ = GA₂+HA₂+JA₅A₄+K(A₆A₅+A₆A₄)+L(A₁⊕A₀)+N+O</entry></row><row><entry namest="col1" nameend="col1" align="left">C₁ = GA₁+HA₁+J(A₅⊕A₄)+K(A₅A₄+A₅'A₄')+LA₁+LLA₁+MA₃+NA₇+OA₃</entry></row><row rowsep="0"><entry namest="col1" nameend="col1" align="left">C₀ = GA₀+HA₀+JA₄'+KA₄'+LA₀+LLA₀+MA₂+NA₆+OA₃'</entry></row></tbody></tgroup></table></tables>

0044Table 2B illustrates the decoding mapping principle between the 8-bit user data and the 9-bit codewords with a code rate of 8/9, an MTR of 3, and k of 7, in order to simplify a decoder circuit. <tables id="tabl0004" num="0004"><table frame="none"><title>(TABLE 2B)</title><tgroup cols="1" colsep="1" rowsep="0"><colspec colnum="1" colname="col1" colwidth="157.50mm" /><tbody valign="top"><row><entry namest="col1" nameend="col1" align="left">G = C₄'</entry></row><row><entry namest="col1" nameend="col1" align="left">H = C₈C₇'C₄C₃'</entry></row><row><entry namest="col1" nameend="col1" align="left">J = C₈'C₄C₃'(C₂'+C₁')(C₂'+C₀')</entry></row><row><entry namest="col1" nameend="col1" align="left">K = C₅'C₄C₃(C₈'+C₇')(C₂'+C₁')</entry></row><row><entry namest="col1" nameend="col1" align="left">L = C₈'C₆'C₅C₄(C₇'+C₃')(C₃+C₂)</entry></row><row><entry namest="col1" nameend="col1" align="left">LL = C₈'C₆'C₅C₄C₃C₂'(C₇+(C₁⊕C₀))</entry></row><row><entry namest="col1" nameend="col1" align="left">M = C₈C₇'C₆'C₅C₄C₃C₂'</entry></row><row><entry namest="col1" nameend="col1" align="left">N = C₈'C₄C₃'C₂(C₇+C₆)(C₆+C₅')(C₇+C₅)(C₁ +C₀)</entry></row><row><entry namest="col1" nameend="col1" align="left">O = C₈'C₇'C₅'C₄'C₃'C₂(C₆+C₀)(C₁+C₀)</entry></row><row><entry namest="col1" nameend="col1" /></row><row><entry namest="col1" nameend="col1" align="left">A₇ = GC₈+H+J+K(C₇+C₈C₆)+NC₁+O</entry></row><row><entry namest="col1" nameend="col1" align="left">A₆ = GC₇+H+J+K(C₂+C₁C₀)+NC₀+O</entry></row><row><entry namest="col1" nameend="col1" align="left">A5 = GC₆+HC₆+J(C₁C₀+C₁'C₀')+K(C₁⊕C₀)+NC₀+O</entry></row><row><entry namest="col1" nameend="col1" align="left">A₄ = GC₅+HC₅+JC₀'+KC₀'+NC₀+O</entry></row><row><entry namest="col1" nameend="col1" align="left">A₃ = GC₃+J+KC₈+LL+MC₁+NC₇C₆+OC₁</entry></row><row><entry namest="col1" nameend="col1" align="left">A₂ = CG₂+HC₂+JC₇+K(C₈+C₆)+L(C₇+C₂'C₁'C₀')+LL(C₇⊕(C₁C₀)) +MC₀+NC₇+OC₆</entry></row><row><entry namest="col1" nameend="col1" align="left">A₁ = GC₁+HC₁+JC₆+K(C₈+C₆)+LC₁+LLC₁+MC₀+NC₇+OC₆</entry></row><row rowsep="0"><entry namest="col1" nameend="col1" align="left">A₀ = GC₀+HC₀+JC₅+K(C₈+C₆)+LC₀+LLC₀+MC₀+NC₇+OC₆</entry></row></tbody></tgroup></table></tables>

0045In the present invention data is efficiently recorded and reproduced by increasing a code rate which affects a channel input SNR, optimized input and output lookup tables and a simplified codeword generating equation for encoding and decoding a block code with a code rate of 8/9 and an MTR of 3 are provided. and a higher recording density can be enabled than an encoding method having an MTR of 2.