Abstract:
An apparatus has a conversion circuit, a precoder circuit, and a selection circuit. The conversion circuit converts user data b 1 , b 2 , b. . . b k  to a coded sequence c 0 , c 1 , c 2  . . . c q . The selection circuit selects c 0  in the coded sequence c 0 , c 1 , c 2  . . . c q  such that the output of the precoder circuit has less than a maximum number q of transitions. The conversion circuit may include an encoder circuit to convert user data b 1 , b 2 , b 3  . . . b k  to a sequence c 1 , c 2  . . . c q , and a transition minimization circuit to add c 0  to the sequence c 1 , c 2  . . . c q . The apparatus may have a circuit to add at least one additional bit, which may be a parity bit, to the coded sequence c 0 , c 1 , c 2  . . . c q .

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS  
   This application is a continuation of application Ser. No. 10/869,843, filed Jun. 18, 2004, is now U.S. Pat. No. 7,053,801, which is a continuation of application Ser. No. 10/253,911, tiled Sep. 25, 2002, now U.S. Pat. No. 6,788,223. 

   BACKGROUND OF THE INVENTION 
   1. Field of the Invention 
   The present invention relates to apparatus and methods to encode information to reduce a probability of errors in a transmission and/or a recording (storage) of the information. 
   2. Description of the Related Art 
   In magnetic recording, various sources of noise can corrupt accurate information (for example, thermal noise, interference, and media noise arising from sources such as jitter, DC erase noise, and pulse width/height modulation): Media noise is a dominant source of noise in many current recording systems. The media noise is usually treated as highly correlated non-stationary noise added to a read-back signal. “Transition jitter” is the dominant component of media noise and affects the position of transitions. 
   RLL Coding schemes use (d, k) constraints, which limit a minimum and a maximum run lengths of zeros, respectively, or alternatively, the schemes control high and low frequency contents of user data. Conventional high-rate RLL (0, k) codes are highly complex for circuit implementation and relatively “blind” in terms of error detection during a demodulation process. The d, k constraints include properties of the conventional codes exploitable for error control purposes. However, this specialized type of error is only a small subset of the total number of possible errors. 
   A construction of an encoder, which encodes arbitrary binary sequences into sequences, is needed that obeys a specific run-length-limited (RLL) constraint. It is important that the encoder encodes data at a high rate, that the decoder does not propagate channel errors, and that a complexity of encoding and decoding be low. 
   White noise is added to every symbol entering a channel in a magnetic recording medium. Media Noise, like white noise, is random. Unlike the white noise, the media noise is not added to every symbol. The media noise happens only when there is a transition on the input to the channel. For example, if we input 00010110, then we have media noise when the input changes from a “0” to a “1” and from a “1” to a “0”. The denser a signal is written onto the magnetic recording medium, the more severe media noise becomes. Thus, a recording density controls a ratio of media noise to white noise. For instance, a ratio of 50:50 may be one example. 
   Let n_j, n_w, and n_e to denote components of media noise, n, due to jitter, j, pulse width noise, w, and electronic noise, e, respectively.
 
 n=n   —   j+n   —   w+n   —   e+n ′, where,  n ′, represents all other noises.
 
   Components n_j and n_w are proportional to a number of pairs, (x(i), x(i+1)), that are (0, 1) or (1, 0). In other words, n_j and n_w, are proportional to a number of times there is a transition in the x sequence either from 0 to 1, or from 1 to 0. Because, n_j and n_w depend on input data, the error performance of the system can vary significantly with the data. Sequences, x, having few transitions will suffer less from, n_j and n_w, than those having many transitions. Accordingly, an encoder is needed to reduce media noise from being added to an input of the channel x(i). 
   SUMMARY OF THE INVENTION 
   Various objects and advantages of the invention will be set forth in part in the description that follows and, in part, will be obvious from the description, or may be learned by practice of the invention. 
   According to one aspect, an apparatus has a conversion circuit, a precoder circuit and a selection circuit. The conversion circuit converts user data b 1 , b 2 , b 3  . . . b k  to a coded sequence c 0 , c 1 , c 2  . . . c q . The precoder circuit having an initial state (s 2 ( 0 ), s 1 ( 0 )) produces an output x 0 , x 1 , x 2  . . . x q  from the coded sequence c 0 , c 1 , c 2  . . . c q  as follows: x(i)=c(i)⊕s 2 (i−2), where (x(−2), x(−1))=(s 2 ( 0 ), s 1 ( 0 )). 
   The selection circuit selects c 0  in the coded sequence C 0 , c 1 , c 2  . . . c q  such that the output x 0 , x 1 , x 2  . . . x q  of the precoder circuit has less than a maximum number q of transitions. 
   The conversion circuit may include an encoder circuit to convert user data b 1 , b 2 , b 3  . . . b k  to a sequence c 1 , c 2  . . . c q , and a transition minimization circuit to add c 0  to the sequence c 1 , c 2  . . . c q . 
   The apparatus may have a circuit to append the coded sequence c 0 , c 1 , c 2  . . . c q  by adding at least one additional bit to the coded sequence c 0 , c 1 , c 2  . . . c q  to produce a sequence c 0 , c 1 , c 2 , . . . , c q , c q+1 , . . . , c m . The at least one additional bit added to produce c 0 , c 1 , c 2 , . . . , c q , c q+1 , . . . , c m  may include a parity bit. 
   According to another aspect, a method for coding includes adding a single bit to a input sequence of length q, and producing an output sequence of length q+1 having t transitions such that for any input sequence, t is an integer less than or equal to one half the maximum number of transitions and is represented by the following formula: t≦q/2. 
   A computer readable medium may store a program for controlling at least one computer to perform the method. 
   These together with other aspects and advantages which will be subsequently apparent, reside in the details of construction and operation as more fully hereinafter described and claimed, reference being had to the accompanying drawings forming a part hereof, wherein like numerals refer to like parts throughout. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The above and other objects, features and other advantages of the present invention will be more clearly understood from the following detailed description taken in conjunction with the accompanying drawings, in which: 
       FIG. 1  is a diagram illustrating a configuration of a magnetic recording system of high rate coding for media noise, in accordance with an embodiment of the present invention; 
       FIG. 2  illustrates a first embodiment of a high rate coding method performed by the encoder of  FIG. 1 ; and 
       FIG. 3  illustrates a second embodiment of the high rate coding method performed by the encoder of  FIG. 1 . 
   

   DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
   Hereinafter, embodiments of the present invention will be described in detail with reference to the attached drawings. The present invention may, however, be embodied in many different forms and should not be construed as being limited to the embodiments set forth herein; rather, these embodiments are provided so that the present disclosure will be thorough and complete, and will fully convey the concept of the invention to those skilled in the art. 
   In an embodiment according to the present invention, referring to  FIG. 1 , a user bit sequence, b(i)&#39;s is encoded, for instance, by a rate k/m encoder  25  to produce bits, c(i)&#39;s. In one exemplary embodiment, an Error Correcting Code (ECC) may generate the user bit sequence, b(i)&#39;s. The encoder  25  receives k-bit blocks and produces m-bit blocks. The encoder  25  may include an application-specific integrated circuit (ASIC). The m-bit blocks are called codewords and m is called codeword length. The encoder  25  outputs the c(i) to a 1/(1⊕D 2 ) precoder  30 . The encoder  25  and the precoder  30  receive, encode, and process data in a digital domain. In an alternative embodiment, the encoder  25  and the precoder  30  may be combined into one control block capable of encoding and precoding the user bit sequence, b(i)&#39;s. 
   Thus, as shown in  FIG. 1 , the output, x(i)&#39;s, of the precoder  30  pass through a cascade of channel filters denoted by (1−D 2 )  35  and (a+bD+cD 2 )  40 . At the output of the filters, data z(i) is corrupted by additive noise, n&#39;s, r(i)=z(i)+n(i). Based on a received sequence, r(i)&#39;s, a Viterbi detector  50  generates, {circumflex over (x)}(i)&#39;s, which are reproductions of x(i)&#39;s. Next, bits {circumflex over (x)}(i)&#39;s are filtered by a filter (1⊕D 2 )  55 , which is an inverse of the precoder  30 , to generate ĉ(i)&#39;s. In an alternative embodiment, the filter (1⊕D 2 )  55  may be provided with the Viterbi detector  50  as one unit. The ĉ(i)&#39;s, are decoded by a decoder  60  to produce, {circumflex over (b)}(i)&#39;s, which are reproductions of the user bit sequence, b(i)&#39;s. In one exemplary embodiment, an ECC decoder may receive the reproductions of the user bit sequence, {circumflex over (b)}(i)&#39;s. Further, if x(i)≠{circumflex over (x)}(i), then it is determined that a channel error occurred at time i. Further, if b(i)≠{circumflex over (b)}(i), then it is determined that a decoder error occurred at time i. 
   As previously set forth, the encoder  25  outputs the c(i)&#39;s to the 1/(1⊕D 2 ) precoder  30 . The precoder  30  has at time, i, a state s(i)=(s 2 (i), s 1 (i)), an input, c(i), and an output x(i), where x(i)=c(i)⊕s 2 (i). The state, s(i), is updated for time i+1, for instance, as follows: 
   s(i+1)=(s 2 (i+1),s 1 (i+1)), where s 2 (i+1)=s 1 (i) and s 1 (i+1)=x(i). In an embodiment where the precoder  30  comprises 1/(1⊕D), the precoder  30  would have at time, i, a state s(i), an input, c(i), and an output x(i), where x(i)=c(i)⊕s(i). The state, s(i), is updated for time i+1, for instance, as follows: s(i+1)=x(i). 
   In addition to the user bit sequence, b(i)&#39;s, the encoder  25  may use a state, s(i)=(s 2 (i),s 1 (i)), of the precoder  30  to generate c(i)&#39;s, which will be explained in more detail below. The precoder  30  is a finite state component and includes a memory to store the state, s(i). Initially, a first state (s 2 ( 0 ), s 1 ( 0 )) is preset to an initial value of, for instance, (s 2 ( 0 ), s 1 ( 0 ))=(0, 0). In an alternative embodiment, the precoder  30  may be provided as 1/(1⊕D), where initially a first state, s( 0 ), is preset to an initial value of, for instance, s( 0 )=0. 
   For instance, assuming that the preset state values of x(i−2) at i=0 and 1 are set to “0” and the input to the precoder  30 , c(i)&#39;s, include the following: c( 0 )=0, c( 1 )=0, c( 2 )=1, c( 3 )=1, and c( 4 )=0. The output, x(i)&#39;s, of the precoder  30 , would provide the relationship as shown in Table 1. 
   
     
       
             
             
             
           
         
             
                 
               TABLE 1 
             
             
                 
                 
             
             
                 
               Output of Encoder 
               1/(1⊕ D 2 ) precoder 
             
             
                 
                 
             
           
           
             
                 
               c(0) = 0 
               x(0) = (c(0) ⊕ x(−2)) = (0 ⊕ 0) = 0 
             
             
                 
               c(1) = 0 
               x(1) = (c(1) ⊕ x(−1)) = (0 ⊕ 0) = 0 
             
             
                 
               c(2) = 1 
               x(2) = (c(2) ⊕ x(0)) = (1 ⊕ 0) = 1 
             
             
                 
               c(3) = 1 
               x(3) = (c(3) ⊕ x(1)) = (1 ⊕ 0) = 1 
             
             
                 
               c(4) = 0 
               x(4) = (c(4) ⊕ x(2)) = (0 ⊕ 1) = 1 
             
             
                 
               c(5) = 1 
               x(5) = (c(5) ⊕ x(3)) = (1 ⊕ 1) = 0 
             
             
                 
               c(6) = 0 
               x(6) = (c(6) ⊕ x(4)) = (0 ⊕ 1) = 1 
             
             
                 
                 
             
           
        
       
     
   
   Thus, a every time the input bit, c(i), to the precoder  30  is a “1”, the output value of the output bit, x(i), of the precoder  30  equals the compliment of x(i−2). For other instances, when c(i)=0, the output bit, x(i), of the precoder  30  is x(i−2). In the alternative, if the 1/(1⊕D) precoder is used, then, each time the input bit, c(i), to the precoder  30  is “1”, the output bit, x(i), of the precoder  30  is the compliment of x(i−1). For other instances, when c(i)=0, the output bit, x(i), of the 1/(1⊕D) precoder is x(i−1). 
   Although the reproductions of the user bit sequence, {circumflex over (b)}(i)&#39;s, should be same as the user bit sequence, b(i)&#39;s, and the input to the precoder  30 , c(i)&#39;s, should be same as the output of the inverse of the precoder, ĉ(i)&#39;s, the equality is not always possible because noise, such as media noise, is added to the output of the filters  35  and  40 , z(i)&#39;s. Jitter noise and/or pulse width noise happens only when there is a transition on the input to the channels  35  and  40 . Accordingly, one way to reduce the noise is to reduce a number of transitions occurring at the input of the channels  35  and  40 . 
     FIG. 2  illustrates a first embodiment of a high rate coding method performed by the encoder  25  to generate a least number of transitions at the output of the precoder  30 , and thus, at the input of the channels  35  and  40 . At operation  100 , a rate k/q code is implemented where the encoder  25  would receive the user bit sequence b(i)&#39;s as k-bit blocks to generate the input sequence to the precoder  30  as c 1 , c 2 , . . . , c q . The operation  100  would resolve, for instance, one or more of RLL conditions, distance enhancement, clock recovery information, etc. At operation  110 , the additional bit, c 0 , is added at a beginning of the input sequence c 1 , c 2 , . . . , c q  to generate a codeword c 0 , c 1 , c 2 , . . . , c q . At operation  120 , the additional bit, c 0 , is set to be equal to a value of “0” or “1” to produce a least number of transitions at the output, x(i), of the precoder  30  corresponding to the codeword c 0 , c 1 , c 2 , . . . , c q . 
   In an exemplary embodiment, x′=(x′( 0 ), x′( 1 ), . . . , x′(q)) is the output of the precoder  30  and input of the channels  35  and  45 , having an initial state, s=(s 2 , s 1 ) and input (0, c 1 , c 2 , . . . , c q ). Further, let x″=(x″( 0 ), x″( 1 ), . . . , x″(q)) be the output of the precoder  30  and input of the channels  35  and  45 , having the initial state, s=(s 2 , s 1 ) and input (1, c 1 , c 2 , . . . , c q ). Then, a maximum number of transitions, q, has the following relationship:
 
(Number of transitions in  x ′)+(Number of transitions in  x ″)= q. 
         where: x′(2i+1)=x″(2i+1) 0≦i≦(q−1)/2, and
           x′(2i)=(1−x″(2i)) 0≦i≦q/2.   
               

   Accordingly, the codeword, (c 0 , c 1 , c 2 , . . . , c q ) generated according to the exemplary embodiment above will produce no more than q/2 transitions at the output of the precoder, such as one-half the maximum number of transitions. 
   For illustrative purposes, Table 2 illustrates x′ and Table 3 illustrates x″, where the preset state values of s( 0 )=(s 2 ( 0 ), s 1 ( 0 ))=(0, 0). The inputs of the precoder  30  range from c( 0 ) to c( 9 ). Accordingly, the maximum number of transitions, q, would be 9. 
   
     
       
             
             
             
           
         
             
                 
               TABLE 2 
             
             
                 
                 
             
             
                 
               Output of Encoder 
               1/(1 ⊕ D 2 ) precoder 
             
             
                 
                 
             
           
           
             
                 
               c(0) = 0 
               x′(0) = (c(0) ⊕ x′(−2)) = (0 ⊕ 0) = 0 
             
             
                 
               c(1) = 1 
               x′(1) = (c(1) ⊕ x′(−1)) = (1 ⊕ 0) = 1 
             
             
                 
               c(2) = 1 
               x′(2) = (c(2) ⊕ x′(0)) = (1 ⊕ 0) = 1 
             
             
                 
               c(3) = 0 
               x′(3) = (c(3) ⊕ x′(1)) = (0 ⊕ 1) = 1 
             
             
                 
               c(4) = 1 
               x′(4) = (c(4) ⊕ x′(2)) = (1 ⊕ 1) = 0 
             
             
                 
               c(5) = 0 
               x′(5) = (c(5) ⊕ x′(3)) = (0 ⊕ 1) = 1 
             
             
                 
               c(6) = 1 
               x′(6) = (c(6) ⊕ x′(4)) = (1 ⊕ 0) = 1 
             
             
                 
               c(7) = 0 
               x′(7) = (c(7) ⊕ x′(5)) = (0 ⊕ 1) = 1 
             
             
                 
               c(8) = 0 
               x′(8) = (c(8) ⊕ x′(6)) = (0 ⊕ 1) = 1 
             
             
                 
               c(9) = 0 
               x′(9) = (c(9) ⊕ x′(7)) = (0 ⊕ 1) = 1 
             
             
                 
                 
             
           
        
       
     
   
   
     
       
             
             
             
           
         
             
                 
               TABLE 3 
             
             
                 
                 
             
             
                 
               Output of Encoder 
               1/(1⊕ D 2 ) precoder 
             
             
                 
                 
             
           
           
             
                 
               c(0) = 1 
               x″(0) = (c(0) ⊕ x″(−2)) = (1 ⊕ 0) = 1 
             
             
                 
               c(1) = 1 
               x″(1) = (c(1) ⊕ x″(−1)) = (1 ⊕ 0) = 1 
             
             
                 
               c(2) = 1 
               x″(2) = (c(2) ⊕ x″(0)) = (1 ⊕ 1) = 0 
             
             
                 
               c(3) = 0 
               x″(3) = (c(3) ⊕ x″(1)) = (0 ⊕ 1) = 1 
             
             
                 
               c(4) = 1 
               x″(4) = (c(4) ⊕ x″(2)) = (1 ⊕ 0) = 1 
             
             
                 
               c(5) = 0 
               x″(5) = (c(5) ⊕ x″(3)) = (0 ⊕ 1) = 1 
             
             
                 
               c(6) = 1 
               x″(6) = (c(6) ⊕ x″(4)) = (1 ⊕ 1) = 0 
             
             
                 
               c(7) = 0 
               x″(7) = (c(7) ⊕ x″(5)) = (0 ⊕ 1) = 1 
             
             
                 
               c(8) = 0 
               x″(8) = (c(8) ⊕ x″(6)) = (0 ⊕ 0) = 0 
             
             
                 
               c(9) = 0 
               x″(9) = (c(9) ⊕ x″(7)) = (0 ⊕ 1) = 1 
             
             
                 
                 
             
           
        
       
     
   
   As shown in Table 2, if the additional bit c( 0 ) added to the input of the precoder  30 , in accordance with an embodiment of the present invention, is set to equal to zero, then the output of the precoder  30 , x′(i), transitions three times. Specifically, as a first transition, the output of the precoder  30  transitions from x′( 0 )=0 to x′( 1 )=1. Subsequently, as a second transition, the output of the precoder  30  transitions from x′( 3 )=1 to x′( 4 )=0. As a third transition, the output of the precoder  30  transitions from x′( 4 )=0 to x′( 5 )=1. 
   In contrast, as shown in Table 3, if the additional bit c( 0 ) added to the input of the precoder  30 , in accordance with an embodiment of the present invention, is set to equal to one, then the output of the precoder  30 , x″(i), transitions six times. Specifically, as a first transition, the output of the precoder  30  transitions from x″( 1 )=1 to x″( 2 )=0. As a second transition, the output of the precoder  30  transitions from x″( 2 )=0 to x″( 3 )=1, and as a third transition, the output of the precoder  30  transitions from x″( 5 )=1 to x″( 6 )=0. Subsequently, as a fourth transition, the output of the precoder  30  transitions from x″( 6 )=0 to x″( 7 )=1, and as a fifth transition, the output of the precoder  30  transitions from x″( 7 )=1 to x″( 8 )=0. Finally, as a sixth transition, the output of the precoder  30  transitions from x″( 8 )=0 to x″( 9 )=1. Accordingly, to reduce the number of transitions at the output of the precoder  30 , to thereby resolve, for instance, the reduction of media noise, the additional bit, c( 0 ), would be best set to equal to zero. In an alternative embodiment, two additional bits may be used at the beginning of the input sequence to a precoder  30  of c 1 , c 2 , . . . , c q  to significantly reduce a number of transitions at the input of channel filters  35  and  40  in the magnetic recording medium. 
   For instance, for a rate of 80/81, the encoder  25  of the first embodiment receives 80 bits, b=(b( 01 )–b( 80 )), and generates a codeword, c=(c( 00 ) c( 0 /) . . . c( 81 )), where (c( 01 ) . . . c( 81 )), is a codeword generated by a code, C, in response to (b( 01 )–b( 80 )), and where, c( 00 ), is obtained based on operation  120  of the first embodiment. In turn, the decoder  60  receives 82 bits, {circumflex over ( c )}=[ĉ( 0 ) ĉ( 1 ) . . . ĉ( 80 ) ĉ( 81 )], and generates, {circumflex over ( b )}=[{circumflex over (b)}( 1 ) {circumflex over (b)}( 2 ) . . . {circumflex over (b)}( 80 )], where, {circumflex over ( b )}, is generated by the code, C, decoder in response to, ĉ( 1 ) . . . ĉ( 80 ) ĉ( 81 ). The C code improves the RLL conditions, the distance enhancement, and/or the clock recovery information. Details of the C code are set forth in the U.S. patent application titled “MODULATION CODING BASED ON AN ECC INTERLEAVE STRUCTURE,” filed concurrently herewith, the disclosure of which is incorporated herewith by reference. 
     FIG. 3  illustrates a second embodiment of a high rate coding method performed by the encoder  25  to generate the reduced number of transitions at the output of the precoder  30 , and thus, at the input of the channels  35  and  40 . Appendix A of the present application illustrates a pseudo code for the first and second embodiments illustrating the addition of the additional bit, c 0 , the determination of the reduced number of transitions, the generation and decoding of c i , and an addition of a parity bit, c m ; The method of the second embodiment, in addition to reducing the number of transitions at the output of the precoder  30  or at the input of the channels  35  and  40  to reduce the media noise, inserts the parity bit, c m , to force an even parity structure at the output of the precoder  30 . 
   In particular, operations  200 ,  210 , and  220  of  FIG. 3  are same as operations  100 ,  110 , and  120 , respectively, of  FIG. 2  accordingly, the detailed description of the operations provided above is incorporated herein. At operation  230 , a systematic code is applied with a rate (q+1)/m to generate a codeword c(i)=(c 0 , c 1 , c 2 , . . . , c q , c q+1 , . . . , c m ). Specifically, at least one bit (c q+1 , . . . , c m ) is added at the end of the codeword (c 0 , c 1 , c 2 , . . . , c q , c q+1 , . . . , c m ). If c 0  is calculated and inserted after the parity bit c m  is added, some of the parity properties of the codeword may be corrupted. By adding the parity bit after c 0  has been determined, it is possible to accurately count the number of “1”s. In an alternative embodiment, the at least one bit (c q+1 , . . . , c m ), may be added at some middle point within the codeword. 
   One example of exactly one bit c q+1 , where c q+1  is the parity bit, is as follows: given 64 user bits b=(b( 1 ), b( 2 ), . . . b( 63 ), b( 64 )), and state, s=(s 2 , s 1 ), of the precoder  30 , the encoder  25  produces a 67 bit codeword,
 
 c =[c(0) c(1) . . . c(65) c(66)],
 
where (c( 1 ) . . . c( 65 )) is a codeword generated by the code C, in response to b. Bit, c( 0 ), is generated as follows:
         c( 0 )=0 if number of transitions of x′=(x′( 0 ), . . . , x′( 65 ))≦33, and   c( 0 )=1 if number of transitions of x″=(x″( 0 ), . . . , x″( 65 ))≦33,
 
where, as before, x′, is the output of, 1/(1⊕D 2 ), precoder  30  having an initial state, s=(s 2 , s 1 ), and input ( 0 , c( 1 ), c( 2 ), . . . , c( 65 )). Further, x″, is the output of the precoder having initial state, s=(s 2 , s 1 ), and input (1, c( 1 ), c( 2 ), . . . , c( 65 )). The above description of c( 0 ) is valid due to the following relationship:
 
Number of transitions of ( x ′)+Number of transitions of ( x ″)=65
       

   Subsequently, bit c( 66 ) is generated as follow: 
             c   ⁡     (   66   )       =       (   binary   )     ⁢     (         ∑     i   =   0     15     ⁢     c   ⁡     (       4   ⁢   i     +   1     )         +       ∑     i   =   0     15     ⁢     c   ⁡     (       4   ⁢   i     +   2     )         +     c   ⁡     (   65   )       +     s   ⁢           ⁢   1       )             
where bit c( 66 ) is such that x( 0 )+x( 1 )+ . . . +x( 66 ) has even number of ones (even parity.)
 
   Accordingly, the bit, c(m), is such that x( 0 )+x( 1 )+ . . . +x(m) has an even number of ones (i.e., even parity). Thus, the second embodiment of the present invention provides flexibility to allow resolving parity issues. Specifically, in one exemplary embodiment, the codeword or output of the encoder  25 , c(i)&#39;s, generated at operation  200  has an original even parity at the output of the precoder  30 . At operation  220 , by allowing the addition of the bit, c m , after the value of the additional bit, c 0 , is determined, the even parity of the codeword, (c 0 , c 1 , c 2 , . . . , c q , c q+1 , . . . , c m ), may be achieved. In alternative embodiments, additional bits may be added to the codeword (c 0 , c 1 , c 2 , . . . , c q , c q+1 , . . . , c m ) for other purposes. 
   One of the many advantages of the methods of  FIGS. 2 and 3 , in accordance with an embodiment of the present invention, is that the method reduces an average media noise. Another of the many advantages is that the method of  FIGS. 2 and 3  does not permit sequences, c, that generate a lot of transitions in x. 
   Although the method in accordance with an embodiment of the present invention is described in the context of a 1/(1⊕D 2 ) precoder, the application of the method is not limited to 1/(1⊕D 2 ) precoder. For instance, for a 1/(1⊕D) precoder, the embodiments below reduce (in average) the number of transitions at the output of the precoder  30 , thus, controlling media noise. First embodiment, a 1/(1⊕D) term is added to the code—effectively making the precoder look like 1/(1⊕D 2 ). Second embodiment, it must be noted that when precoder is 1/(1⊕D), a “1” in c(i)&#39;s causes a transition in x(i)&#39;s. Therefore, in construction of a code, operations  120  and  220  of  FIGS. 2 and 3  are changed, respectively, as follows, after inserting, c 0 , and modify c 1 , . . . , c q  to generate a codeword (c 0 , c 1 , . . . , c q ) as follows:
 
( c   0   , c   1   , c   2   , . . . , c   q )=(0 , c   1   , . . . , c   q ), if  c   1   + . . . +c   q   ≦└q/ 2┘, and
 
( c   0   , c   1   , c   2   , . . . , c   q )=(1−0, 1 −c   1 , . . . , 1 −c   q ), otherwise.
 
   The present invention has been described with respect to a system and method performing high rate coding by adding one additional bit to a beginning of the input sequence to a precoder as (c 1 , c 2 , . . . , c q ) and controlling a value of the additional bit to significantly reduce a number of transitions at an input of channel filters in a magnetic recording medium to reduce an amount of noise. 
   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. The processes can also be distributed via, for example, downloading over a network such as the Internet. Although the system of the present invention has been described in view of a magnetic recording medium, the system may be incorporated and applied to other communication systems. 
   The many features and advantages of the invention are apparent from the detailed specification and, thus, it is intended by the appended claims to cover all such features and advantages of the invention that fall within the true spirit and scope of the invention. Further, since numerous modifications and changes will readily occur to those skilled in the art, it is not desired to limit the invention to the exact construction and operation illustrated and described, and accordingly all suitable modifications and equivalents may be resorted to, falling within the scope of the invention.