Abstract:
A data coding method produces codewords with a scheme that changes for different codewords, and decodes codewords with a scheme that remains constant for all codewords. The coding method receives k user bits, codes the user bits to produce k+r output bits, corrupts any one of the output bits, and accurately reproduces at least k−r−1 user bits. The data coding method may code an input sequence (b 0 , b 1 , b 2 , . . . , b k−1 ) to produce a codeword c=(c(−r), c(−r+1), . . . , c(−1), c0, c1, . . . , c(k−1)), where for 0≦i≦k−1, symbols are determined according to the following:  
       c ( i ) f 0= c ( i )= b ( i )⊕ c ( i− 1)  f 1⊕ c ( i− 2) f 2⊕ . . . ⊕ c ( i−r ) fr    II  
     where c i  and b i  are corresponding user data and codeword bits at time i. In this case, for each of a plurality of input sequences, appropriate initial conditions (c(−r), . . . , c(−2), c(−1))=s(b) are independently selecting to satisfy a predetermined property in the codeword coded according to II. Codewords coded using the appropriate initial conditions are output. For each codeword, the appropriate initial conditions are appended to the codeword coded therefrom.

Description:
BACKGROUND OF THE INVENTION  
         [0001]    [0001]FIG. 1 is a schematic representation of a data system such as a magnetic recording medium. In FIG. 1, a sequence of user data is input to an Error Correcting Coder (ECC)  20 . An encoder  25  receives k-bit blocks of output b(i) from the ECC  20  and produces m-bit blocks c(i), which are referred to as codewords. The rate of the encoder  25  is thus k/m. The encoder  25  may include an application-specific integrated circuit (ASIC). The encoder  25  outputs the codeword c(i) to a 1/(1⊕D 2 ) precoder  30 . The ECC  20 , the encoder  25 , and the precoder  30  receive, encode, and process data in a digital domain. The encoder  25  and the precoder  30  may be combined into one control block capable of encoding and precoding the ECC output b(i), or the precoder may be omitted all together.  
           [0002]    The output of the precoder  30 , x(i) passes through a channel having one or more filters. FIG. 1 shows a cascade of channel filters  35 ,  40  denoted by (1−D 2 ) and (a+bD+cD 2 ). The output of the filters  35 ,  40  is corrupted by additive noise, n(i) such that the received sequence r(i) is defined by r(i)=z(i)+n(i). Based on the received sequence r(i), a Viterbi detector  50 , for example, generates a detected sequence {circumflex over (x)} (i), which is a reproduction of the input x(i) to the channel filters  35 ,  40 . Next, bits {circumflex over (x)} (i)s are filtered by a filter (1⊕D 2 )  55 , which is an inverse of the precoder  30 , to generate g(i). The filter (1⊕D 2 )  55  may be provided within the detector  50  as one unit. The output g(i) of the filter  55 , is decoded by a decoder  60  to produce a decoded sequence d(i), which is a reproduction of the ECC output sequence, b(i). An ECC decoder  65  receives the output sequence d(i).  
           [0003]    If x(i)≠{circumflex over (x)}(i), then it is determined that a channel error has occurred at location (time) i. Further, if b(i)≠d(i), then it is determined that a decoder error has occurred at location (time) i.  
           [0004]    In general, in magnetic recording, encoders and decoders perform one or more of the following tasks:  
           [0005]    1) enforcing Run Length Limited (RLL) conditions,  
           [0006]    2) enhancing system performance (distance), and  
           [0007]    3) providing clock recovery information (non-zero samples.)  
           [0008]    For task No. 3 mentioned above, there are clocks within the system that must be synchronized. A downstream clock must be expecting a symbol when it is received. Clock drift is a problem associated with the clocks not being synchronized. That is, when a part of the system is expecting a bit to be received, the bit is in the middle of transmission. To accomplish synchronization, a clock looks at the received non-zero symbols and synchronizes its pulses with the receipt of non-zero symbols. Hence, for the purpose of clock recovery, it is beneficial that there be a sufficiently large number of “non-zero”s traveling through the channel.  
           [0009]    In generating codewords c(i), the encoder  25  may consider clock recovery. In this case, the encoder  25  produces a sufficiently large number of “1”s for transmission through the channel. FIG. 2A shows an encoder map for an encoder having a rate 64/65 with 64 user bits being received and 65 coded bits c(i) being produced. The encoder map shown in FIG. 2A produces at least 33 “1”s in every 65 bit codeword. To do this, the encoder counts the number of “1”s in every 64 bit sequence of ECC output b(i). If the number of “1”s is more than 32, then the encoder adds a “0” at the end of the block as bit c 64 . Thus, (c 0 , c 1 , c 2 , . . . c 63 )=(b 1 , b 2 , b 3 , . . . b 64 ).  
           [0010]    When the encoder counts the number of “1”s in the 64 bit sequence of ECC output, it is possible that the number of “1”s will be less than or equal to 32. In this case, the 64 incoming bits are changed to the compliment thereof, such that a received “1” is changed to an output “0”, and a received “0” is changed to an output “1.” To produce the codeword, a “1” is added to the end as bit c 64  to the 64 bit block (c 0 , c 1 , c 2 , . . . c 63 ).  
           [0011]    The decoder takes the opposite action as the encoder. A map for the decoder is shown in FIG. 2B. With this decoder, if g 64  is a “0”, then the decoder outputs bits g 0  through g 63  as bits do through d 63 . On the other hand, if g 64  is a “1”, then the decoder outputs the compliment of bits g 0  through g 63  as bits do through d 63 .  
           [0012]    It should be apparent that bits c 64  and g 64  are very important. They determine the output for 64 bits of data. In traveling through the channel, the data can be corrupted. If g 64  is improperly received at the decoder, a 1 bit error will propagate into a 64 bit error. The operation of the decoder is depended on the map used by the encoder.  
         SUMMARY OF THE INVENTION  
         [0013]    One aspect of the invention relates to reducing error propagation. That is, if a single bit of data is corrupted in the channel or otherwise, it is desired that this single bit of data does not result in a much larger error.  
           [0014]    To achieve this and other objects, a data coding method produces codewords with a scheme that changes for different codewords, and decodes codewords with a scheme that remains constant for all codewords. The coding method receives k user bits, codes the user bits to produce k+r output bits, corrupts any one of the output bits, and accurately reproduces at least k−r−1 user bits.  
           [0015]    The data coding method may involve (a) coding user data bits (b 1 , b 2 , b 3 , . . . , b k ) to produce coded bits (c 1 , c 2 , c 3 , . . . , c k ) according to the following:  
           
         c 
         i 
         =bi⊕c 
         i−1  
       
           [0016]    where c i  and b i  are corresponding user data and coded bits, (b) determining if (a) produces more “1”s when the initial condition c 0  is “0” than when the initial condition c 0  is “1”, (c) producing coded bits (c 1 , c 2 , c 3 , . . . c k ) using the initial state resulting in the larger number of “1”s, and (d) producing coded bit c 0  equal to the initial state resulting in the larger number of “1”s. The user data bits (b 1 , b 2 , b 3 , . . . , b k ) may be reproduced from coded bits (c 0 , c 1 , c 2 , . . . , c k ) according to the following:  
             b   i   =c   i   ⊕c   i−1 .  
           [0017]    If an error occurs in bit c 0  before reproducing user data bits, at least k−4 user bits, more particularly at least k−3, and still more particularly at least k−2 user bits may be accurately reproduced.  
           [0018]    At least one additional bit may be appended to the coded bits (c 0 , c 1 , c 2  . . . c k ) to produce c 0  to c k+1 . The at least one additional bit may include a parity bit.  
           [0019]    According to a generalized data coding method, an input sequence (b 0 , b 2 , b 3 , . . . , b k−1 ) is coded to produce a codeword c=(c(−r), c(−r+1), . . . , c(−1), c 0 , c 1 , . . . , c(k−1)), where for 0≦i≦k−1, according to the following:  
             c ( i ) f 0 =c ( i )= b ( i )⊕ c ( i− 1) f 1 ⊕c ( i− 2) f 2 ⊕ . . . ⊕c ( i−r ) fr    II  
           [0020]    where c i  and b i  are corresponding user data and codeword bits at time i. For each of a plurality of input sequences, appropriate initial conditions (c(−r), . . . , c(−2), c(−1))=S(b) are independently selecting to satisfy a predetermined property in the codeword coded according to 11. The appropriate initial conditions (c(−r), . . . , c(−2), c(−1)) are thus appended to the codeword symbols (c0, c1, c2, . . . , c(k−1)). Codewords coded using the appropriate initial conditions are output. As a first alternative, the following may be used for the generalized data coding method:  
           k=64,  
           r=1,  
           f 1 =1=f r , and  
           [0021]    the predetermined property is producing codewords having more than 32 “1”s.  
           [0022]    As a second alternative, the following may be used for the generalized data coding method:  
           k=64,  
           r=2,  
           f 1 =1,  
           f 2 =1=f r , and  
           [0023]    the predetermined property is avoiding the following codewords:  
           w1=(1 1 1 . . . 1 1),  
           w2=(1 0 1 . . . 1 0), and  
           w3=(0 1 0 . . . 0 1).  
           [0024]    In addition to avoiding the codewords w1, w2, and w3, the predetermined property may also involve producing codewords having more than 22 “1”s.  
           [0025]    A pseudo random input vector h may be added mod. 2 to the input sequence before coding the input sequence. The pseudo random input vector h may be defined as follows:  
                                                   h=(h1 h2 . . . h64)=           (1 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0            1 0 1 0 0 1 1 1 1 0 1 0 0 0 1 1            1 0 0 1 0 0 1 0 1 1 0 1 1 1 0 1            1 0 0 1 1 0 1 0 1 0 1 1 1 1 1 1).                      
 
           [0026]    A computer readable medium can store a program to control a computer to perform the above method. A coding device may also have circuits to perform the method.  
       
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0027]    These and other objects and advantages of the present invention will become more apparent and more readily appreciated from the following description of the preferred embodiments, taken in conjunction with the accompanying drawings of which:  
         [0028]    [0028]FIG. 1 is a schematic representation of a data system such as a magnetic recording medium;  
         [0029]    [0029]FIGS. 2A and 2B show encoder and decoder maps for a related art system;  
         [0030]    [0030]FIGS. 3A and 3B show encoder and decoder maps, respectively, for one aspect of the present invention;  
         [0031]    [0031]FIGS. 4A and 4B are explanatory diagrams related to the encoder and decoder described in FIGS. 3A and 3B respectively; and  
         [0032]    [0032]FIG. 5 is a simplified block diagram for the purpose of explaining the encoder map shown in FIG. 3A and the example shown in FIGS. 4A and 4B. 
     
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT  
       [0033]    Reference will now be made in detail to the preferred embodiments of the present invention, examples of which are illustrated in the accompanying drawings, wherein like reference numerals refer to like elements throughout.  
         [0034]    Let vector, f=(f0, f1, . . . , fr), have coordinates in {0,1} such that for every i,  0 ≦i≦r, fi is 0 or 1. We assume f0=1. If f0=0, then f0 is eliminated from f. Now given a positive integer k, we define a rate k/(k+r) code with the encoder and decoder maps shown respectively in FIGS. 3A and 3B. In FIG. 3A, the phrase where for “0≦i≦k−1” means that the formula which follows at line  3  for c(i) is used only for determining c(0), c(1), c(2), . . . , c(k−1). In other words, the formula at line  3  is not used for determining c(−r), c(−r+1), . . . , c(−1).  
         [0035]    [0035]FIGS. 4A and 4B are explanatory diagrams related to the encoder and decoder maps shown in FIGS. 3A and 3B respectively. Although FIG. 4A and 4B do not correspond exactly with any of the embodiments described below, it is believed they are helpful in understanding the general principles of the inventions. In FIG. 4A, reference numeral  405  represents a sequence of data, perhaps output from an error correction coder (ECC). This sequence is then coded according to 1/(1 ⊕ D). Because this coding method performs an exclusive-or function by comparing an incoming data bit with the previously output data bit, it is necessary to specify an initial condition. An initial condition “0” is used for coder  410 , and an initial condition of “1” is used for coder  415 . Thus, for bit b 1 , coder  410  performs an exclusive-or function with the initial condition of zero. With two “0”s, the one-and-only-one non-zero element function is not satisfied. Thus, a “0” is output as bit c1 in the data sequence  420 . For both the data sequence  420  and the data sequence  425 , bits c 1  through c 8  are determined in this manner. For bit c 0 , the initial condition is used. Thus, data sequence  420  has a c 0  of “0”, and data sequence  425  has a c 0  of “1”.  
         [0036]    A comparator  430  compares the number of “1”s in data sequence  420  with the number of “1”s in data sequence  425 . Data sequence  425  has 5 “1”s, whereas data sequence  420  only has four “1”s. Comparator  430  selects the data sequence with more “1”s and outputs that data sequence. Thus, for our example, data sequence  425  is output.  
         [0037]    A few things should be noted about the example encoder shown in FIG. 4A. First, the rate k/m is 8/9. Second, four “1”s are contained in data sequence  420 , and five “1”s are contained in data sequence for  425 . It is not a coincidence that the bits contained in the data sequences  420 ,  425  total 9, the sum of 4 and 5. In determining the number of “1”s produced, aspects of the invention rely upon this feature, as will become apparent later. Third, data stream  420  is the compliment of data stream  425 .  
         [0038]    [0038]FIG. 4B schematically shows what happens in a decoder provided downstream from the encoder shown in FIG. 4A. In FIG. 4B, data sequence (g 0 , g 1 , g 2 , . . . , g 8 ) corresponds with data sequence  425  (c 0 , c 1 , c 2 , . . . , c 8 ). The function performed in decoder is 1 ⊕ D. Thus, the decoder compares its g0 and g1 in an exclusive-or manner to produce decoded symbol d1. The decoder does not require any initial conditions because it is only comparing decoder inputs. If there is an error in bit g 4 , for example, that error will propagate to cause errors in bits d 4  and d 5 . However, unlike other methods which ensure a large number of “1”s, no single bit error can corrupt an entire codeword. In the related art example described above, it is crucial that bit g 8  be received correctly. This bit determines the entire codeword data sequence. This is not the case for the encoder and decoder shown in FIGS. 4A and 4B, respectively.  
         [0039]    [0039]FIG. 5 is a simplified block diagram for the purpose of explaining the encoder map shown in FIG. 3A and the example shown in FIGS. 4A and 4B. In FIG. 5, b 1  through b k  are the same as described previously. The filter function in box  505  considers the r previously output symbols. The vector R under the box  505  represents the initial conditions (v 1 , v 2 , v 3 , . . . v r ). The output from the encoder (c 0 , c 1 , c 2 , . . . , c m ) is formed by appending the selected initial conditions be (v r , v r−1 , v r−2 , . . . v 1 ) as shown to the mathematic outputs of box  505  (o 1 , o 2 , o 3 , . . . o k ). These outputs are produced using the selected initial conditions (v 1 , v 2 , v 3 , . . . v r ). For the example shown in FIGS. 4A and 4B, k=8, m=8, f=(f0, f1)=(1,1), R=(v 1 ), and v 1  is selected as shown in FIG. 4A to be “1”.  
         [0040]    The following are the properties associated with the general encoding and decoding scheme described above. First, the encoder and decoder above are completely characterized by vector f and map S. If the vector f and the map S are known, any sequence can be coded. Therefore, we sometimes represent the encoder and decoder respectively using notations, E(f, S) and D(f, S). These notations emphasize the underlying structure of the encoder and the decoder.  
         [0041]    Second, despite what is stated above, the decoder is actually independent of map S. However, this is not true for the encoder. For every pair of distinct maps, S′ and S″, in general, E(f, S′) is not the same as E(f, S″).  
         [0042]    Third, if r is small, then the above encoder and decoder pair will have short error propagation. From the decoder description above, each channel error (c(i)≠g(i)) can cause at most r+1 decoder errors.  
         [0043]    Fourth, let R0, R1, . . . , Rr be the following initial state vectors (v 1 , v 2 , v 3 , . . . , v r ) used to map from {0,1} k  to {0,1} r . For every, b in {0,1} k ,  
           R 0( b )=(0 0 . . . 0)  
           R 1( b )=(1 0 . . . 0)  
           R 2( b )=(0 1 . . . 0)  
           Rr ( b )=(0 0 . . . 1)  
         [0044]    Now given a map, S, and a vector, b in {0,1} k , we have  
           E ( f, S )( b )= E ( f, R 0)( b )⊕ v   1   *E ( f, R 1)(0) ⊕ v   2   *E ( f, R 2)(0)⊕ . . .  v   r   E ( f, Rr )(0),  
         [0045]    where 0 represents an all-zero vector having length k in {0,1} k  and (v 1 , v 2 , v 3 , . . . , v r ) is the initial condition vector R selected to produce for the codeword b being considered.  
         [0046]    We define a set F to be a collection comprising, E(f, Ri)(0)&#39;s, for 1≦i≦r. As can be seen from the above, the output is a linear combination of terms. This provides a short cut to determine the output for any input sequence (b 0 , b 2 , b 3 , . . . , b k−1 ) and any initial condition vector R=(v 1 , v 2 , v 3  . . . , v r ). As described above (see FIG. 5, for example), the appropriate initial condition vector may be selected after knowing what would be produced from all possible initial conditions. With the above property, given any input sequence (b 0 , b 2 , b 3 , . . . , b k−1 ) and any set of initial conditions R=(v 1 , v 2 , v 3 , . . . , v r ), the output can be calculated by:  
         [0047]    1) calculating output for the input sequence (b 0 , b 2 , b 3 , . . . , b k−1 ) for an all-zero vector of initial conditions R=R0;  
         [0048]    2) calculating the output for an all-zero input sequence for each of the non-zero initial condition terms. That is, to determine the output for initial conditions R=(1, 0, 0, 1), the output for an all-zero input sequence vector would be calculated for initial conditions R=(1, 0, 0, 0) and R=(0, 0, 0, 1); and  
         [0049]    3) taking the linear combination of the terms produced from items 1) and 2) above according to the equation.  
         [0050]    If it were necessary to calculate the output for all possible initial condition vectors (v 1 , v 2 , v 3 , . . . v r ), this would require 2 r  calculations for every input sequence (b 0 , b 1 , b 2 , . . . , b k−1 ). With the above property, all that is necessary is to perform one calculation for the given input sequence and an all-zero initial condition vector and calculate the output for an all-zero input sequence and the initial condition vectors corresponding to the non-zero terms. Thus, instead of performing 2 r  calculations, only r+1 calculations or fewer are necessary. Further, the outputs for item (2) above do not depend on the input sequence, and thus only need to be calculated once after knowing the length k.  
         [0051]    First Embodiment:  
         [0052]    The first embodiment is analogous to the example illustrated in FIGS. 4A and 4B, and has the following properties:  
         [0053]    1) k=64 In the example shown in FIGS. 4A and 4B, k=8,  
         [0054]    2) f1=(1, 1), (r=1) The notation f1 indicates that this is the f vector for the first embodiment. With the f vector f1=(1,1), the encoder uses the formula 1/(1 ⊕ d). r=1 indicates that only one initial condition is necessary, and this follows from the vector f, and  
         [0055]    3) map S1 is defined as follows,  
                 R        (     b   _     )       =          0     ,     if                   E        (     f1   ,   R0     )                       (     b   _     )                   has                 more                 than                 32                 ones     ,   and                 =          1     ,     otherwise   .                                 
 
         [0056]    Now, encoder and decoder, E(f1, S1) &amp; D(f1, S1) have the following properties:  
         [0057]    i) Rate=k/(k+r)=64/65  
         [0058]    ii) F1={E(f1, R1)(0)}={(1 1 . . . 1)} Thus, if an all-zero input sequence (b 0 , b 1 , b 2 , . . . b k−1 ) is received and processed with R=(v 1 )=R1=(0) according to the first embodiment, an all one sequence is produced as the output.  
         [0059]    iii) Short Error Propagation—Channel errors influence only two (r+1) consecutive data bits.  
         [0060]    iv) For every b in {0,1} 64 , the number of ones in c=E(f1,S1)(b) is at least 33. In fact, E(f1, S1) generates the same codewords as the encoder in the related art example. However, in the related art example, a channel error for one symbol could propagate to an error in 64 bits. For the first embodiment, a one bit error can propogate to at the most 2 bits using E(f1,S1).  
         [0061]    v) As stated in iv), encoder E(f1, S1) generates at least 33 ones in every codeword. This property is very useful for clock recovery. Previously, it was stated that for the example shown in FIGS. 4A and 4B, one of the possible outputs is the compliment of the other possible output. For the first embodiment, there are only two possibilities for the initial condition vector R, either R=(1) or R=(0). Above property ii) tells us that if the input sequence (b 0 , b 1 , b 2 , . . . , b k−1 ) is an all-zero vector, and the initial condition vector is defined as R=(1) then the output is an all one vector (1, 1, 1, . . . 1). This is useful in evaluating the linear combination discussed above, which is reproduced below. In the linear combination  
           E ( f, S )( b )= E ( f, R 0)( b )⊕ v   1   *E ( f, R 1)(0) ⊕ v   2   *E ( f, R 2)(0)⊕ . . . ⊕ v   r   *E ( f, Rr )(0),  
         [0062]    To find the output produced for a given input vector (b) when v 1 =0, only the first term on the right side of the equation is used. That is, if v 1 =0, the output is determined by E(f, R0) (b). On the other hand, if v1=1, the first and second terms are used. The subsequent terms are not present since r=1 and thus V 1 =r 1 . The first term is the same as produced from v 1 =0. The second term produces all one vector according to property ii). Performing mod. 2 addition, the output when v 1 =1 is the compliment of the output when v 1 =0.  
         [0063]    Second Embodiment:  
         [0064]    In the second embodiment of the proposed method, we let  
         [0065]    1) k=64,  
         [0066]    2) f2=(1, 1, 1), (r=2), Thus, we use 1/(1+D+D 2 ) and need two initial conditions.  
         [0067]    3) a subset, W={w1, w2, w3}, of {0,1} 66 , be defined by  
         [0068]    w1=(1 1 1 . . . 1 1),  
         [0069]    w2=(1 0 1 . . . 1 0),  
         [0070]    w3=(0 1 0 . . . 0 1), and  
         [0071]    [0071] 4 ) map S2 to be defined as  
                   S2        (     b   _     )       =            R0   =     (     0                 0     )         ,     if                   E        (     f2   ,   R0     )                       (     b   _     )                   is                 not                 in                 W     ,   and                              =            R2   =     (     0                 1     )         ,     otherwise   .                                   
         [0072]    Thus, if the output produced for the given input sequence (b 0 , b 2 , b 3 , . . . , b k−1 ) and the initial condition vector R=(0,0) is not one of w1, w2, w3, then we use R=(0,0)  
         [0073]    Encoder and decoder, E(f2, S2) &amp; D(f2, S2) have the following properties:  
         [0074]    i) Rate=k/(k+r)=64/66  
         [0075]    ii) F2={E(f2, R1)(0), E(f2, R2)(0)}={ (1 0 1 1 0 1 . . . 1 0 1 ), (0 1 1 0 1 1 . . . 0 1 1)}Note that the first two elements in both of the possible outputs are the initial condition vector. Thus, for E(f2,R1), (c 0 , c 1 )=(1,0)=R1. Although R1 is not used as a possible initial condition vector R according to the map for the second embodiment, this is an important property.  
         [0076]    iii) Short Error Propagation—Channel errors influence only three (r+1) consecutive data bits.  
         [0077]    iv) For every b in {0,1} 64 , the codeword c=E(f2,S2)(b) is not in W. This property is true since based on the above description we have  
         [0078]    E(f2, S2) (b)=E(f2, R0)(b), if E(f2, R0) (b) not in W, and  
         [0079]    E(f2, S2) (b)=E(f2, R0)(b)⊕E(f2, R2)(b)=E(f2, R0)(b)⊕(0 1 1 0 1 1 . . . 0 1 1), otherwise.  
         [0080]    In the above two equations for property iv), a few things should be considered referring to the general linear combination discussed previously and reproduced below.  
           E ( f, S )( b )= E ( f, R 0)( b )⊕ v   1   *E ( f, R 1)(0) ⊕ v   2   *E ( f, R 2)(0)⊕ . . . ⊕ v   r   *E ( f, Rr )(0)  
         [0081]    By definition, we know that if R=(v 1 , v 2 )=(0,0) produces an output which is not in w, then we use R0 for R. If v 1 =v 2 =0, then all terms on the right side of the linear combination disappear except for the first term.  
         [0082]    If the initial conditions R=R0=(0,0) produce an output which is in W, then we use the initial conditions R=R2=(0,1). In this case, v 1 =0 and v 2 =1. To find the output at R2, we can use the general linear combination. Because v 1 =0 and v 2 =1, the first and third terms remain on the right side of the linear combination. The first term E(f, R0)(b) is known to be one of the w1 through w3. The third term E(f, R2)(0) is known as described in property ii) above. The linear combination of these two vectors produces an output not in W.  
         [0083]    Output sequences generated with vectors in set W are not desirable for some channels because they might produce long error events. For these channels, property iv) contributes to limiting the length of the error events.  
         [0084]    Third Embodiment:  
         [0085]    The third embodiment is similar to the second embodiment. List 1)-5) below describes the third embodiment.  
         [0086]    1) k=64,  
         [0087]    2) f3=f2=(1,1,1), (r=2),  
         [0088]    3) set, W, is defined as in the second embodiment,  
         [0089]    4) let set, N, be a subset of {0,1} 66  comprising all vectors having less than 22 ones, and  
         [0090]    5) map S3 is defined by  
                   S3        (     b   _     )       =            R0   =     (     0                 0     )         ,       if                   E        (       f3        (   D   )       ,   R0     )                       (     b   _     )                   not                 in                 W     ⋃   N     ,   and                              =            R1   =     (     1                 0     )         ,     otherwise   .                                 
 
         [0091]    Thus, before we select R=R0=(0,0), we confirm that the output thereby produced is not in W and has at least 22 “1”s. Encoder and decoder, E(f3, S3) &amp; D(f3, S3) have the following properties:  
         [0092]    i) Rate=k/(k+r)=64/66  
         [0093]    ii) F3=F2  
         [0094]    iii) Short Error Propagation—Channel errors influence only (r+1) three consecutive data bits.  
         [0095]    iv) For every b in {0,1} 64 , the codeword c=E(f3,S3)(b) is not in W∪N. This property is true since  
           E ( f 3,  S 3)( b )= E ( f 3,  R 0)( b ), if  E ( f 3( D ),  R 0)( b ) not in W∪N, and  
           E ( f 3,  S 3)( b )= E ( f 3,  R 0)( b )⊕ E ( f 3,  R 0)(0)= E ( f 3,  R 0)( b )⊕(1 0 1 1 0 1 . . . 1 0 1), otherwise.  
         [0096]    We note that the third embodiment not only satisfies property iv) of the second embodiment but also every codeword generated by encoder, E(f3, S3), has at least 22 ones. Therefore, encoder of the third embodiment, E(f3, S3), contributes more to providing clock recovery information than the encoder of the second embodiment, E(f2, S2).  
         [0097]    Forth Embodiment:  
         [0098]    In many systems, the error performance depends on the user data; some user data on the average performs better than other user data. In general, this dependency is due to 1) the encoder map, 2) the properties of the channel noise, and 3) the properties of the channel filter. Since in the present coding method, there is a strong connection between the structure of a codeword and its corresponding user data, the error performance of a system based on the present method might depend on user data as well.  
         [0099]    In such systems where error performance depends on user data, the impact of user data is minimized by modifying the encoder and the decoder as follows. A predetermined pseudo random vector, h, is added to both the data at the input to the encoder and to the output of the decoder. The forth embodiment encoder and decoder, E and D, use the earlier embodiments as follows.  
         [0100]    Encoder:  
         [0101]    First, encoder, E, accepts 64 bits, b. Next it adds (bit-wise mod 2) a constant vector h to b, thereby producing p=b+h. Then, for a fixed i, 1≦i≦3, the encoder applies the map, E(fi, Si), (see previous embodiments) to p, generating codeword c. Thus, after performing mod. 2 addition, one of the first three embodiments is used.  
         [0102]    Decoder:  
         [0103]    Decoder, D, receives m bits, g, then it applies the map, D(fi, Si), (where i defines the map of one of the previous embodiments—the same embodiment as used for the encoder map) to generate vector, q (for i=1, m=65, and for i=2&amp;3, m=66). Finally, the decoder adds the vector h to the vector produced by the map D(f i , S i ) to produce bits, d.  
         [0104]    In one example, the vector h may be defined as follows:  
                                                   h=(h1 h2 . . . h64)=           (1 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0            1 0 1 0 0 1 1 1 1 0 1 0 0 0 1 1            1 0 0 1 0 0 1 0 1 1 0 1 1 1 0 1            1 0 0 1 1 0 1 0 1 0 1 1 1 1 1 1).                      
 
         [0105]    In addition to adding the vector h, encoders and decoders designed based on the present method can be modified as follows. Additional bits can be added to each codeword to force properties such as:  
         [0106]    1) Parity structures in codewords, or  
         [0107]    2) Parity structures at the output of a precoder in recording systems that use a precoder between the encoder and the channel.  
         [0108]    These modified encoders and decoders contribute to enhancing system performance (distance). The additional bits can be added to the sequence either before or after the encoder. Of course, if the additional bits are added after the encoder, then they would need to be removed before the decoder. Where the additional bits are added (before or after the encoder), depends on whether the encoding and decoding method will corrupt the additional bits. For example, a parity bit considers the number of “1”s in the codeword after the precoder. The number of “1”s and hence the parity bit cannot be determined until after the codeword is produced. Thus, the parity bit should be added after the encoder and removed before the decoder.  
         [0109]    The system implementing the method described above includes permanent or removable storage, such as an application specific integrated circuit (ASIC), magnetic and optical discs, RAM, ROM, etc. on which the process and data structures of the present invention can be stored and distributed. Also, the system implementing the method described above applies not only to magnetic recording, but so to various other communication systems. The processes can also be distributed via, for example, downloading over a network such as the Internet.  
         [0110]    The invention has been described in detail with particular reference to preferred embodiments thereof and examples, but it will be understood that variations and modifications can be effected within the spirit and scope of the invention.