Abstract:
A methodology for designing an implementing high rate RLL codes is optimized for application to 10-bit ECC symbols, and provides rate 20/21, rate 50/51, rate 90/91 and other modulation code rates for use in magnetic recording channels. A relatively small subcode encoding—one easy to implement—is applied to a portion of the input stream, and the resulting base codeword is partitioned into nibbles that, in turn, are interleaved among the unencoded ECC symbols. Code constraints on the subcode word nibbles depend upon the values of adjacent unencoded symbols. The resulting codes provide excellent density and error propagation performance.

Description:
RELATED APPLICATION  
       [0001]    This is a continuation of U.S. patent application Ser. No. 09/350,685 filed on Jul. 9, 1999, for “Method and Apparatus for “High Rate Runlength Limited Codes for 10-Bit ECC Symbols”, McEwen et al, now pending. 
     
    
     
       TECHNICAL FIELD  
         [0002]    The present invention relates to channel modulation codes and methods for implementation in magnetic recording systems such as disk drives. More specifically, the present invention relates to high rate run-length limited (RLL) modulation codes for use in a PRML channel.  
         BACKGROUND OF THE INVENTION  
         [0003]    Modulation codes are used in magnetic recording channels in order to limit recorded bit sequences to those that are most reliably detectable. In particular, run length limited (RLL) modulation codes have been used within partial response signaling, maximum likelihood (PRML) data recording and playback channels, decision feedback equalization (DFE) channels, and the like. Partial response systems of interest for magnetic data storage devices such as disk drives and magnetic tape include a PR4 (1-D 2 ) channel and EPR4 (1+D−D 2 −D 3 ) channel as well as other nonclassical polynomials. The present invention can be used in any PR channel.  
           [0004]    In general, magnetic recording systems employ Viterbi detectors to achieve maximum likelihood detection of user data as it is played back from the recording medium. A modulation code for a PRML data recording and playback channel is selected to balance code efficiency against timing/gain loop reliability and the Viterbi detector path memory, as well as error propagation during decoding.  
           [0005]    Run length limited modulation codes are often described using the format “(rate) RLL (d,G/I)”, where the “rate” is expressed as a ratio of the number of input bits to be encoded to the number of output bits in the resulting codeword. For example, a rate 8/9 modulation code converts an 8-bit input byte into a 9-but codeword. Rate 8/9 encoding is well known in the art, as described, for example, in U.S. Pat. No. 4,797,681 and U.S. Pat. No. 5,260,703. Rate 8/9 encoding for PRML data channel also is described in U.S. Pat. No. 5,196,849. As the code rate approaches unity, the code is deemed to be more efficient, in that relatively fewer code characters are required to encode user data values. Thus, rate 8/9 code is more efficient than a rate 2/3 code.  
           [0006]    Similarly, rate 16/17 code is more efficient than a rate 8/9 code. A rate 16/17 code (=0.941) achieves an approximately 6% increase in recording density over a standard rate 8/9 modulation code. One example of a rate 16/17 modulation code is described in commonly assigned U.S. Pat. No. 5,635,933 incorporated herein by this reference. Another rate 16/17 code is described in U.S. Pat. No. 5,784,010 assigned to IBM.  
           [0007]    Early PRML read channel used the well-known rate 8/9 RLL(0,4/4) channel code. In accordance with prior art, this channel code is combined with a 1/(1⊕D 2 ) modulo 2 precoder to obtain the {+1,−1} valued magnetic write-current pattern. On the decoder side, the signal is first equalized to the partial response target and then the +1/−1 write-current waveform is maximum-likelihood detected. The write current is then “unprecoded” (or postcoded) with a 1⊕D 2  modulo 2 function. This “undoes” the precoding to generate a {0,1 } valued sequence. The data is then RLL decoded for the user. Examples of RLL encoders and decoders are disclosed in the patent identified above.  
           [0008]    The rate 8/9 code can be extended to a rate 16/17 code by either bit-wise or byte-wise interleaving unencoded bytes with the encoded sequence. While the G and I constraints will become considerably larger (G=12, I=8 for byte-wise interleaved case), the roughly 6% in increases code rate is often considered worthwhile. Still, the need remains for improvements in recording channel encoding efficiency in order to improve storage capacities in recording systems and lower costs. The codes in this patent application are (0,k) codes. The k constraint is equivalent to the G constraint. The 0 means that consecutive ones are allowed, i.e. there is no restriction on the minimum run length of zeros.  
           [0009]    Another limitation of prior art is that virtually all known channel coding schemes are based on 8-bit ECC symbols, as they are historically the de facto standard. We anticipate use of a 10-bit ECC symbol and thus new methods are required to achieve improved density and error propagation performance in the context of 10-bit ECC symbols.  
         SUMMARY OF THE INVENTION  
         [0010]    In view of the foregoing background, a general object of the present invention is to improve the effective a real density of data recorded on magnetic media.  
           [0011]    Another object is to improve recording efficiency by reducing the relative number of non-data bits or “overhead” in the data encoding process.  
           [0012]    An object of the invention is to provide very high rate modulation codes having reasonable zero run length limitations for use in magnetic recording and playback systems.  
           [0013]    A further object of the invention is to minimize implementation complexity in the context of high rate RLL codes, by providing a relatively small subcode.  
           [0014]    A further object of the invention is to enable effective RLL encoding of 10-bit symbols for magnetic recording.  
           [0015]    A more specific object of the invention is to provide a 50/51 modulation code which limits error propagation in the context of 10-bit ECC symbols.  
           [0016]    A further object of the present invention is to provide encoding schemes having improved ratios of data bits to code word length without degrading run length limiting in encoded data.  
           [0017]    Another object of the invention is to record data on a magnetic media so as to prevent long strings of no transition on the magnetic media thereby allowing for reliable timing and gain recovery.  
           [0018]    According to one aspect of the invention, methodologies and constraints are disclosed to enable the creation of a variety of high rate channel codes primarily for use in a PRML channel of a magnetic recording and playback system. The new method of designing and implementing a desired code generally includes the following steps:  
           [0019]    First, for a desired rate code, selecting a suitable base code (or “subcode”), having a rate n/(n+1) where n is a multiple of the ECC symbols size. Examples are rate 10/11, 20/21, 30/31 etc. for a 10-bit ECC symbols size.  
           [0020]    Second, encoding one or more of the ECC symbols using the selected base code. Specifically, the number of ECC symbols to be encoded is the number of symbols necessary to provide the number of input bits appropriate to the selected base code. For example, a rate 10/11 base code will require encoding one 10-bit ECC symbol, while a rate 30/31 base code will require encoding three ECC symbols (to enable 30 input bits).  
           [0021]    Third, partitioning the codeword produced by the base code into a plurality of m nibbles. In one version m is the number of unencoded ECC symbols. For example, if a rate 50/51 RLL code is desired, the base code rate 10/11 is selected, and one ECC symbol is encoded to form the 11-bit subcode word. That subcode word is partitioned into m=4 nibbles. Four ECC symbols remain unencoded. In one embodiment, three nibbles have three bits each, while a fourth nibble has two bits. However, other partitions are possible as described later.  
           [0022]    The fourth step, which is optional but preferred, entails modifying the subcode nibbles in response to the values of corresponding unencoded symbols that will be positioned adjacent to the x i  nibbles in the target codeword. Specifically, the invention forbids all zeros in a subcode nibble if the immediately preceding bit (i.e. the last bit of the preceding unencoded symbol) is zero. Conversely, we forbid all ones in a subcode nibble if the immediately preceding bit (i.e. the last bit of the preceding unencoded symbol) is a one. These constraints ensure that at least one magnetic flux transition per nibble.  
           [0023]    Finally, the resulting modified nibbles are interleaved among the unencoded ECC symbols. The order of the unencoded symbols and the order of the subcode nibbles interleaved among them is not limited to any specific predetermined sequence. The resulting codeword can begin with either an unencoded ECC symbol or a subcode nibble.  
           [0024]    In one embodiment of the present invention, before interleaving the nibbles, at least one unencoded symbol is partitioned into smaller portions, such that the nibbles are then interleaved among said smaller portions of the unencoded symbol. In another embodiment, the nibbles are interleaved such that there are two or more unencoded symbols between at least one pair of nibbles.  
           [0025]    The foregoing techniques can be applied to design rate 20/21, 30/31, 40/41, 50/51, 80/81, 90/91,110/111 and other similar rate codes for encoding 10-bit symbols. For each code, many different arrangements of the unencoded symbols and encoded nibbles can be used. FIG. 3 shows just one example for each code. The size and arrangements of the nibbles, however, has implications for the maximum length of uninterrupted strings of ones and zeros in the resulting codeword, as further explained later. In general, the nibbles will be similar to one another in length, if not uniform. This arises from, first, designing the base code so as to provide an adequate number of codewords. Second, the base code table is designed so as to be easy to implement. These design criteria will tend to result in codes that have good run length properties and result in nibbles that have about the same size. FIG. 3 shows some examples.  
           [0026]    Thus in one preferred embodiment, in a rate 50/51 code, the base code has rate 10/11, the number of the unencoded ECC symbols is m=4, and the length 11 word produced by the base code encoder is divided into 4 nibbles of lengths 3, 3, 3, and 2. The resulting codeword consists of four unencoded ECC symbols interleaved with the nibbles mentioned above. The codeword arrangement is shown in FIG. 3C. This just a simple example; there are various ways of mapping the input bits to the codewords within the scope of the present invention. Further illustrations are given later.  
           [0027]    The described techniques provide simplicity of implementation along with enviable recording density and error propagation performance. The codes described also allow use of a simple precoder 1/1⊕D to limit the length of the Nyquist sequence ( . . . 1010101 . . .) where ⊕ denotes modulo-2 addition.  
           [0028]    The run length constraint k can be reduced by imposing additional constraints on the base codeword nibbles. For example, additional patterns (besides the all ones and zeros patterns) can be excluded. And, as noted above, the nibble lengths can be “smoothed” (i.e. variation minimized) to reduce k as well.  
           [0029]    The present invention is quite different from other modulation coding schemes. For example, in the rate 24/25 code described in U.S. Pat. No. 5,757,294, one input symbol or byte is encoded, and the resulting codeword is partitioned and interleaved among the unencoded bytes. There, the encoded byte (rate 8/9 base code) was produced by a fixed encoding, i.e. without regard to the unencoded bytes. By contrast, according to the present invention, the nibbles first produced by the base code encoding are then subjected to modifications (x→y i ), the fourth step above, depending upon the adjacent unencoded byte (the adjacent bit). Another example of the prior art is the &#39;933 patent, directed to a rate 16/17 encoding scheme that again depends solely on the 16-bit (2 byte) input word, without regard to neighboring (unencoded) data. By considering the states of adjacent unencoded bits, the present invention achieves high code rates and improved performance.  
           [0030]    Another aspect of the invention is high rate run length limited code designed in accordance with the foregoing principles.  
           [0031]    The foregoing and other objects, features and advantages of the invention will become more readily apparent from the following detailed description of a preferred embodiment of the invention which proceeds with reference to the accompanying drawings.  
       
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0032]    [0032]FIG. 1 is a simplified block diagram illustrating data flow in a magnetic data storage device, such as a hard disk drive, employing partial response signaling and maximum likelihood detection.  
         [0033]    [0033]FIG. 2 illustrates a preferred bit sequence for a rate 50/51 codeword according to the present invention.  
         [0034]    FIGS.  3 A-G are examples of codeword arrangements for various high rate RLL codes.  
         [0035]    [0035]FIG. 4 is a conceptual diagram illustrating a high rate RLL encoding path according to the present invention.  
         [0036]    [0036]FIG. 5A lists logic equations for implementing the final encoding step for the 50/51 RLL code.  
         [0037]    [0037]FIG. 5B lists logic equations for implementing the first decoding steps for the rate 50/51 code.  
         [0038]    [0038]FIG. 6 sets forth 10/11 subcode final encoding logic equations. 
     
    
     DETAILED DESCRIPTION OF THE INVENTION  
       [0039]    [0039]FIG. 1 sets for a simplified block diagram of a magnetic recording and playback system such as a hard disk drive. While a hard disk drive is one application for the present invention, those skilled in the art will appreciate that the principles of this invention may be usefully applied to other devices, such as magnetic tape recording, for example. User data blocks  12  are received from a source, such as a host computer (not shown). The blocks are passed through an error correction encoder  14  which generates and appends ECC remainder bytes to the blocks in accordance with a preestablished ECC polynomial and scheme. The error correction encoder  14  may be conventional, and it is not further described herein. Each data block (now having a predetermined ECC bytes appended) then passes through a modulation encoder  16 . The modulation encoder  16  is in accordance with principles of the present invention, and it encodes data to form codewords as described in greater detail hereinafter. Each codeword is then passed serially through, for example, a PR4 precoder  18  having a function 1/(1⊕D 2 ). The precoded codewords are then recorded as sequences of magnetic flux transitions within a data track defined on a storage surface of a magnetic recording disk  20 .  
         [0040]    During playback, flux transitions induced in a read head element are subjected to analog/digital filter-equalization processes, quantized as digital samples, and applied to a detector  22  implementing a Viterbi algorithm. The playback codeword is then demodulated in a modulation decoder  24  also in accordance with principles of the present invention explained later. Following decoding by the modulation decoder  24 , each playback data block is passed through an error correction decoder  26  which checks the playback ECC bytes to locate and correct any correctable error bursts. Error corrected user data  12  is then returned to a requester, such as the host computer (not shown). If the error correction decoder determines that a data block includes uncorrectable errors, an error flag is returned to the requester, and a second attempt is made to read the data block from the disk  20 .  
         [0041]    A primary goal of the present invention is to devise an encoding or modulation scheme that has the advantages of constrained error propagation while increasing recording density. Importantly, many prior art systems use 8-bit ECC symbols in the recording channel, and the various RLL encoding schemes summarized above assume that symbol size. To improve performance in the future, however, the trend is toward employing 10-bit ECC symbols, and this presents an opportunity to explore new encoding schemes. The present invention is directed to leveraging 10-bit ECC symbols in a magnetic recording channel. Specific examples of embodiments of the invention include codes with rates 20/21, 30/31, 40/41, 50/51, 80/81, 90/91, 110/11 summarized in Table 1 below.  
         [0042]    The codes of the present invention are characterized by m unencoded ECC symbols together with a base code with rate n/n+1, where n is a multiple of the ECC symbols size (e.g. 10 bits). The length n+1 codeword produced by the base code encoder is divided into m nibbles, each of which contains at least one transition.  
         [0043]    For example, in a rate 50/51 code, the base code has rate 10/11, the number of unencoded ECC symbols is m=4 (total 40 bits), and the length 11 word produced by the base code encoder is divided into 4 nibbles of length 3, 3, 3 and 2. Thus the new codeword consists of the unencoded ECC symbols interleaved with the base code nibbles, for a total of 51 bits. One example of the codeword sequence is shown in FIG. 2. However, the base code nibbles can be interleaved arbitrarily among the unencoded ECC symbols in any order, and the codeword can start with either a base code nibble or an ECC symbol. This feature is further explained later.  
         [0044]    In one embodiment of the present invention, before interleaving the nibbles, at least one unencoded symbol is partitioned into smaller portions, such that the nibbles are then interleaved among said smaller portions of the unencoded symbol. For example, a method of rate 20/21 encoding a 20-bit input sequence includes the steps of: receiving a series of two 10-bit input symbols, selecting one of the series of input symbols for rate 10/11 encoding, leaving the one non-selected 10-bit input symbols unencoded, rate 10/11 encoding the selected one of the input symbols so as to form an 11-bit base code word consisting of a series of four nibbles, each nibble containing at least one transition, partitioning the 10-bit unencoded symbol into four portions, and interleaving the four nibbles between the four portions of the unencoded 10-bit input symbol, thereby forming a 21-bit codeword. The interleaving step includes: inserting a first one of the series of nibbles following the first portion of the unencoded 10-bit input symbol, inserting a second one of the series of nibbles following the second portion of the unencoded 10-bit input symbol, inserting a third one of the series of nibbles following the third portion of the unencoded 10-bit input symbol, and inserting the fourth one of the series of nibbles following the fourth portion of the unencoded 10-bit input symbol, wherein the first, second, third and fourth nibbles are arbitrarily selected among the four nibbles of the 11-bit subcode word. In one example, the 10-bit unencoded symbol can be partitioned into four 3-bit, 2-bit, 2-bit and 3-bit portions.  
         [0045]    In another embodiment, the nibbles are interleaved such that there are two or more unencoded symbols between at least one pair of nibbles. For example, in a rate 90/91 code, the base code has rate 10/11, the number of unencoded ECC symbols is m=8 (total 80 bits), and a length 11 word produced by the base code encoder is divided into 4 nibbles of length 3, 3, 3 and 2. Thus the new codeword consists of the unencoded ECC symbols interleaved with the base code nibbles, such that there are 2 unencoded ECC symbols between encoded nibbles. The total codeword length is 91 bits. One example of the codeword sequence is shown in FIG. 3G. However, the base code nibbles can be interleaved arbitrarily among the unencoded ECC symbols in any order, and the codeword can start with either a base code nibble or an ECC symbol. This feature is further explained later.  
         [0046]    Base Code Constraints  
         [0047]    The coding forbids either the all-ones nibbles, or the all-zeros nibble, depending on the value of the bit immediately preceding the nibble in question (i.e. the last bit of the unencoded symbol preceding the nibble). The bit immediately preceding is zero, the all-zeros nibble is forbidden; and if the preceding bit is a one, the all-ones nibble is forbidden. This ensures at least one transition per base code nibble. Implementation of this constraint is further described below.  
         [0048]    For a rate n/(n+1) code, there must be at least 2 n  codewords available. With the constraint described above, there are 2 L −1 possibilities for each length L nibble, and the number of possible base code codewords is equal to the product of the number of possibilities for each nibble. Thus, with the rate 10/11 base code, the number of possible codewords is (2 2 −1)*(2 3 −1) 3 =3*7 3 =1029 which is greater than 2 10 =1024. Thus the proposed 10/11 coding, as constrained, still provides an adequate number of possible codewords.  
         [0049]    Now to generalize, let k 1  be the maximum allowed number of consecutive zeros or ones; if all nibbles have the same length L, then k 1 =E*n+2L−1, where E is the length of the ECC symbol (E being 10 for purposes of illustration) and n is the number of uninterrupted, i.e., contiguous, unencoded symbols. For example, if all nibbles have length L=3 and if n=2, then k 1 =2E+2L−1=25. The nibbles need not all be of the same length, however. If the nibble lengths vary, then k 1  is determined by the two consecutive nibbles whose lengths have the largest sum. Thus, although the nibbles can be arbitrarily interleaved among unencoded ECC symbols, as noted above, the selected order of unequal length nibbles can affect the resulting zero/one run length. Concatenation of codewords (not subcode) must also be taken into consideration in determining the maximum number of zeros or ones; i.e. the last nibble and the first nibble should be considered in determining the two consecutive nibbles whose lengths have the largest sum. For the 90/91 code, k 1 =2*10+(2*3)−1=25. Note that these codes are RLL (0,k) codes with k=k 1 −1.  
         [0050]    Nyquist Sequence Consideration  
         [0051]    The code constraint described above does not per se limit the length of the Nyquist sequence . . . 010101 . . . . When a 1/(1⊕D) precoder is used, the maximum length of the Nyquist sequence is limited. Operation of the precoder is defined as follows: if the input at time j is a j , then the output at time j is b j =b j-1 ⊕a j  where ⊕ denotes modulo-2 addition. With this precoder, k 1  (and thus k) increases by 1 and the maximum length of a Nyquist sequence is k 1 . Table 1 shows a summary of some of the code constraints that can be obtained using the approach described above. Note that these are merely examples and other can be created from this description.  
                                     TABLE 1                           Examples of High Rate RLL Codes                Code   k (no   Base   BaseCode Capacity =           Rate   precoder)   Code   log 2 (#codewords)                       20/21    7   10/11   10.007           30/31   19   10/11   10.931           40/41   16   10/11   10.621           50/51   14   10/11   10.007           80/81   16   20/21   20.020           90/91   24   10/11   10.007           110/111   16   30/31   30.156                      
 
         [0052]    For each of the code constraints listed in Table 1, one possible codeword arrangement is shown in FIG. 3, wherein FIG. 3A illustrates a rate 30/31 codeword arrangement; FIG. 3B illustrates a 40/41 codeword arrangement; FIG. 3C illustrates a rate 50/51 codeword arrangement; FIG. 3D illustrates a rate 80/81 codeword arrangement; FIG. 3E illustrates a rate 110/111 codeword arrangement; FIG. 3F illustrates an example rate 20/21 codeword arrangement (coded nibbles are shaded, uncoded portions are unshaded); and FIG. 3G illustrates a rate 90/91 codeword arrangement (coded nibbles are shaded, uncoded portions are unshaded). As noted, the unencoded ECC symbols can be arranged in any arbitrary order, as can the subcode nibbles, subject to the discussion above. In other words, there any many ways of mapping the input buts to the codewords.  
         [0053]    If capacity is sufficient, the k constraint might be reduced by imposing additional constraints on the nibbles. This can be done by forbidding additional patterns (other than the all-zero and all-one patterns) for one or more of the nibbles. This can also be done by imposing a dependence on the encoding of consecutive nibbles. Since there is excess capacity for the constraints given above, there are many ways of choosing codewords that will be used in the code. In other words, there are many ways of mapping the input bits to the codewords.  
         [0054]    Rate 50/51 Code  
         [0055]    To more fully illustrate the principles of the invention, and how to implement it, we now describe one particular embodiment—the 50/51 code—in detail. Two illustrative codeword arrangements for the 50/51 code are shown below in Table 2.  
                                                 TABLE 2                       Arrangements of 50/51 Codeword                                10   3   10   3   10   3   10   2       A   y 0     B   y 1     C   y 2     E   y 3         B   z 1     C   z 0     A   z 3     D   z 2                    
 
         [0056]    In Table 2, the first row shows the number of bits (total 51) and the second row illustrates one example of mapping the input ECC symbols, consisting of five 10-bit ECC symbols A B C D E, to the codeword. The third row illustrates an alternative mapping ECC symbol to the codeword. Many other mappings can be used as will be explained later. The 51-bit codeword is formed as follows.  
         [0057]    One of the ECC symbols, say D for illustration, is encoded using a rate 10/11 base code to form four nibbles y 0  y 1  y 3  y 4 . The unencoded symbols (ABCE) and the base code nibbles are interleaved, with one of the nibbles between each two unencoded symbols in a preferred embodiment. The particular mapping or sequence of this interleaving can vary, as explained later, but one example is shown in the middle row of Table 2 above. The 10-bit symbol D thus is absent from the middle row as that data is reflected in the base code nibbles. Another example of interleaving base code nibbles and unencoded ECC symbols, still within the scope of the invention, is shown for illustration in the bottom row of the table. In this example, ECC symbol E is encoded to form nibbles z 0  to z 3 , z 2  being the two-bit nibble in this case. Other variations in mapping can be used subject to the constraints described below.  
         [0058]    In general, it is preferable to divide the base code word into nibbles of roughly equal length. For example, in the preferred embodiment, the nibbles are 3,3,3,2 bits long. One alternative arrangement might be 2,2,2,5 bit nibbles. As noted earlier, it nibble lengths vary, k 1  is determined by the two consecutive nibbles whose lengths have the largest sum, including taking into account the concatenation of codewords. Thus for 2,2,2,5 bit nibbles, k 1 =10+7−1=16. In the 2,2,2,5 bit nibble example, the number of available codewords =3 3  * (2 5 −1)=27*31=837 which is less than the required 2 10  or 1024. In the preferred embodiment, for nibbles 3,3,3,2 bits long, the maximum run length is 10+6−1=15.  
         [0059]    Encoder For the Rate 50/51 Code  
         [0060]    The rate 50/51 encoding is done in several steps: First, receiving a block of input data consisting of a series of five 10-bit ECC symbols; and second, selecting one of the series of input symbols for rate 10/11 encoding, while leaving the four non-selected 10-bit input symbols unencoded. Referring now to FIG. 4, the first step involves separating the 10-bit symbol D[0:9] into four nibbles, D 0 =D[0:2], D 1 =D[3:5], D 2 =D[6:8], D 3 =D[9]. Next, the encoder determines if any of the 3-bit nibbles are all zero, f i =!(D i [0]+D i [1]+D i [2]), for i=0, 1, 2, where “!”is the logical NOT operator and “+”denotes OR operation. Based on this information and the value D 3  (=D[9]) the encoder calculates the encoded nibbles x 0 , x 1 , x 2  (3 bits each) and x 3  (2 bits) using a selected rate 10/11 base code such as that illustrated in Table 3 below.  
         [0061]    The final encoding step flips (complements) an all-ones nibble if the preceding bit is a one. Let A[0:9], B[0:9], C[0:9] and E[0:9] be the encoded symbols. The corresponding final encoding step is illustrated in FIG. 5A, whereby:  
         [0062]    If x 0 =111 and A[9]=1, then y 0 =000, else y 0 =x 0    
         [0063]    If x 1 =111 and B[9]=1, then y 1 =000, else y 1 =x 1    
         [0064]    If x 2 =111 and C[9]=1, then y 2 =000, else y 2 =x 2    
         [0065]    If x 3 =11 and E[9]=1, then y 3 =000, else y 3 =x 3    
         [0066]    This process is illustrated in FIG. 4.  
                                                 TABLE 3                           10/11 Base Code Table            f 0     f 1     f 2     D 3     x 0     x 1     x 2     x 3                 0   0   0   d/c   D 0     D 1     D 2     D 3  1       0   0   1   1   100   D 0     D 1     10       0   0   1   0   101   D 0     D 1     10       0   1   0   1   010   D 0     D 2     10       0   1   0   0   011   D 0     D 2     10       1   0   0   1   111   D 1     D 2     10       1   0   0   0   001   D 1     D 2     10       0   1   1   1   110   D 0     011   10       0   1   1   0   110   D 0     010   10       1   0   1   1   110   D 1     001   10       1   0   1   0   110   D 1     111   10       1   1   0   1   110   D 2     100   10       1   1   0   0   110   D 2     101   10       1   1   1   1   110   001   110   10       1   1   1   0   110   100   110   10                  
 
         [0067]    In the above table “d/c” denotes “don&#39;t care”.  
         [0068]    Other equivalent rate 10/11 codes can be used. The corresponding first decoding step is illustrated in FIG. 5B thus:  
         [0069]    First decoding step:  
         [0070]    If y 0 =000, then x 0 =111, else x 0 =y 0    
         [0071]    If y 1 =000, then x 1 =111, else x 1 =y 1    
         [0072]    If Y 2 =000, then x 2 =111, else x 2 =y 2    
         [0073]    If y 3 =00, then x 3 =11, else x 3 =y 3    
         [0074]    Rate 10/11 Subcode  
         [0075]    Finally, it should be noted that the rate 10/11 base code described herein can itself be used as an RLL code by not interleaving any unencoded symbols. To illustrate, one symbol D can be encoded as shown in Table 3, and then the final encoding step is changed in accordance with the following logic, as shown in FIG. 6:  
         [0076]    If x 0 =111, and y 3 [1]=1, then y 0 =000, else y 0 =x 0    
         [0077]    If x 1 =111, and y 0 [2]=1, then y 1 =000, else y 1 =x 1    
         [0078]    If x 2 =111, and y 1 [2]=1, then y 2 =000, else y 2 =x 2    
         [0079]    If x 3 =11, and y 2 [2]=1, then y 3 =00, else y 3 =x 3    
         [0080]    where y 3 [1] is the last bit of the previous codeword.  
         [0081]    If no precoder is used, the maximum number of consecutive zeros is 5. This aspect of the invention provides an RLL (0,k) code with k=4. When a 1/1⊕D precoder is used, the maximum number of consecutive zeros or ones is 6, thus k=5. With this precoder, the largest Nyquist sequence has a length of 6.  
         [0082]    It will be obvious to those having skill in the art that many changes may be made to the details of the above-described embodiment of this invention without departing from the underlying principles thereof. The scope of the present invention should, therefore, be determined only by the following claims.