Abstract:
A method for storing data in a holographic storage medium comprises the steps of dividing a bit stream of binary data into data groups, encoding the data groups as binary patterns, and storing the binary patterns holographically as data pages. Each binary pattern comprises channel bits, wherein the number of “off” channel bits is greater than the number of “on” channel bits. To retrieve information from the storage medium, the medium is illuminated and resultant light signals are gathered. The light signals are converted to the binary patterns, and the binary patterns are converted to the data groups. Binary patterns stored in the storage medium only slightly perturb subsequent reading and writing of data, since the patterns have fewer “on” channel bits than “off” channel bits.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation-in-part application of U.S. patent application Ser. No. 09/224,031 filed Dec. 31, 1998, U.S. Pat. No. 6,064,586, which is hereby incorporated herein by reference. 
    
    
     GOVERNMENT RIGHTS 
     The present invention was made with government support under contract No. MDA-972-95-3-004, awarded by the ARPA/NSIC HDSS Consortium. The government has certain rights in this invention. 
    
    
     FIELD OF THE INVENTION 
     The present invention generally relates to holographic data storage, and more specifically to a method of encoding data for storage within a holographic storage medium. 
     BACKGROUND 
     Holographic data storage provides a promising technique for rapid and efficient data storage and retrieval. In holographic data storage systems, binary data is stored in a volume of a holographic storage medium. Holographic storage media contrast with, for example, conventional magnetic disks, where the data is stored on the surface of the storage material. 
     In a holographic storage system, the signal beam is generally monochromatic, and data is stored by passing a monochromatic signal beam through a spatial light modulator having transparent and opaque pixels representing data comprising a data page. The signal beam is then incident on the holographic storage. A reference beam is also incident upon the storage medium and interferes with the signal beam inside the storage medium. The index of refraction of the storage medium is perturbed by the resulting interference pattern, and a hologram representing a page of data is thereby stored. 
     The transparent and opaque pixels of the spatial light modulator correspond to “channel bits,” the bits that are actually stored in the storage medium. The transparent portions correspond to “on” channel bits, or binary 1&#39;s; opaque portions correspond to “off” channel bits, or binary 0&#39;s. The channel bits are typically an encoded version of the original binary bits to be stored in the medium. 
     To retrieve data from a holographic storage medium, the reference beam is incident upon the storage medium. The reference beam is partially diffracted by the perturbations in the index of refraction that correspond to the stored data. The diffracted portion of the reference beam is the image beam, which carries a holographic image of the data page stored. 
     More than one page of data may be holographically stored in the same volume of a holographic storage medium using well known multiplexing techniques. These techniques include angle multiplexing, where the reference beam has a different angle of incidence upon the storage medium for each page, and wavelength multiplexing, where the reference beam has a different wavelength for each page. 
     A primary difficulty with holographic storage systems is that the reading and writing signal-to-noise ratios worsen as the amount of data stored in the storage medium increases. A poor signal-to-noise ratio during reading results from the fact that the address beam or the image beam, or both, pass through regions of the storage medium where data that is not being accessed has been stored. These regions alter the signal beam or the address beam, or both. 
     A similar problem occurs during writing. After an initial amount of data is stored, future data storage requires that the signal beam passes through regions of the storage medium that already contain data. The signal beam is thereby partially scattered, resulting in a poor signal-to-noise ratio for writing data. The reference beam may also be perturbed by previously stored data during the writing process. 
     The more “on” channel bits that are stored in the storage medium, the worse the signal-to-noise ratios become, since it is the “on” channel bits that cause the unwanted dispersion of the light beams. Generally, a large number of “on” channel bits are usually stored in the storage medium. For balanced selection of channel bits, half of the channel bits are on, and half are off. 
     One solution to the problem is for the channel bits to contain an encoded version of the original binary data to be stored, wherein the code is selected to improve the signal-to-noise ratios. U.S. Pat. No. 5,450,218 by Heanue et al. discloses a method of encoding data that involves storing reference pixels as well as data pixels in the holographic medium. The data is then read by taking the difference between the signals generated from the reference and data pixels. 
     U.S. Pat. No. 5,510,912 by Blaum et al. and U.S. Pat. No. 5,727,226 by Blaum et al. disclose methods for modulating data so that “on” channel bits and “off” channel bits are distributed approximately uniformly throughout the page. Finally, U.S. Pat. No. 5,808,998 by Curtis et al. gives a method for encoding data that reduces the length of runs of same-state channel bits. 
     However, in all of these methods the number of “on” channel bits is still approximately equal to the number of “off” channel bits, so the problem of unwanted scattering of radiation beams during data storage and retrieval is not mitigated. 
     SUMMARY 
     Briefly and in general terms, the present invention provides a system and method for holographic data storage and retrieval having improved signal-to-noise ratios. With the present invention, digital data pages comprise sparse modulation codes, which have a smaller number of “on” channel bits than “off” channel bits on average. 
     In a presently preferred embodiment, by way of example and not necessarily by way of limitation, sparse modulation encoding for data storage is accomplished by dividing a bit stream of binary data into data groups, encoding these data groups as binary patterns, and storing the binary patterns as data pages in a holographic data storage medium. Each binary pattern comprises “on” channel bits and “off” channel bits. Generally, “on” channel bits are recorded when a recording signal originating from a signal beam is present, and “off” channel bits are recorded when the recording signal representing that bit is absent. Upon data retrieval, the binary patterns are converted to the data groups using lookup tables or conversion algorithms. Channel bits are represented by pixels of a page, and a number of pages are multiplexed in a single volume of the storage medium. 
     Generally, accumulated light exposure, or accumulated index of refraction perturbation, or both, due to recorded “on” channel bits decreases the signal-to-noise ratios for reading and writing data. For a system with a given dynamic range available for signal index perturbation, sparse modulation codes can reduce the accumulated light exposure, or the accumulated noise index of refraction perturbation, or both, thereby increasing the signal-to-noise ratios. For a given signal-to-noise ratio, the storage capacity of the storage medium is in general increased by the sparse modulation codes. 
     In one embodiment of the present invention, a lookup table is used to relate the data groups to the binary patterns. The k th  binary pattern comprises n k  channel bits, m k  of which are “on”. The k th  data group comprises b k  bits, and to ensure that the k th  binary pattern contains at least as much information as the k th  data group, n k -choose-m k ≧2 b     k   . A sparseness is defined as the average ratio of n k /m k . In the present modulation codes, the sparseness is greater than 2. In some embodiments, standard error-correction encoding and data interleaving techniques are used to process the binary data before it is modulation encoded. 
     To retrieve data that has been stored holographically, an address beam illuminates the storage medium. The address beam causes an image beam to emerge from the storage medium. Light signals from the image beam are detected. These detected signals are processed if necessary, and stored as a data array. Data sectors comprising a plurality of array elements are identified; the k th  data sector has n k  array elements and corresponds to the k th  binary pattern. In the preferred embodiment, the m k  largest array elements of the k th  data sector are identified as “on” channel bits, and the remaining n k −m k  array element values are identified as “off” channel bits; the k th  binary pattern is thereby recovered. Once the data sectors are converted to binary patterns, the binary patterns are converted to data groups of binary information. 
     Frequently, a diffraction efficiency of a holographic storage medium varies inversely with the square of the number of pages multiplexed. For such storage media, by example, sparse coding results in an increase in the number of pages that can be recorded. Sparse modulation codes can be chosen to have a sparseness that maximizes the storage capacity of these storage media. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 shows a bit stream to be stored. 
     FIG. 2 shows an example of a lookup table for relating data groups to binary patterns. 
     FIG. 3 shows an example of a page comprising binary patterns to be stored holographically. 
     FIG. 4 shows an apparatus for holographically storing a data page. 
     FIG. 5 shows an apparatus for retrieving a holographically stored page. 
     FIG. 6 is a schematic of a data array of signals corresponding to the stored page. 
     FIG. 7 is a graph of a storage capacity versus a sparseness for four different values a number of “on” channel bits per binary pattern. 
    
    
     DETAILED DESCRIPTION 
     I. Data Storage 
     A method and apparatus for holographic storage of binary data is described. In the preferred embodiment, the binary data is first processed using known techniques. The binary data is divided into data bytes, then additional error correction code bytes are added. A typical error correction code is the Solomon-Reed code. The result of the error correction encoding is a stream of code words, each code word comprising data bytes and error correction code bytes. Typically, each code word comprises 15 bytes total; however, any number of bytes is possible. 
     The bytes are then shuffled, or reordered, to make a new sequence of bytes. The bytes belonging to a given codeword are nonadjacent in the new sequence. This reordering reduces errors in data storage and retrieval since if errors are generated in a string of consecutively stored bytes, that string is not likely to contain any code word in its entirety. The defective string is more likely to contain portions of a number of code words, due to the reordering. Errors produced in these portions can generally be corrected using the error correction code. 
     FIG. 1 illustrates a bit stream  30  comprising bits  32  that are to be stored. In a preferred embodiment, bits  32  belong to bytes that have been error correction encoded and reordered as outlined above; in general, however, bit stream  30  is any desired stream of binary data. 
     Bit stream  30  is divided into a series of N data groups G 1 , G 2 , . . . G N . The k th  data group G k  comprises b k  bits. In some embodiments, all of the data groups have the same number of bits; that is, b k =b for 1≦k≦N, where b is an integer number. However, in general different data groups have different numbers of bits. 
     FIG. 1 depicts an example where b=3. Hence data groups G 1  and G 2 , illustrated in FIG. 1, each comprise three bits. In general, b is any desired number. 
     Once bit stream  30  is divided into the data groups, the data groups are converted into binary patterns  40 , as illustrated in FIG.  2 . Binary patterns  40  comprise channel bits  42 . The term “channel bits” is used to distinguish the components of binary patterns  40  from bits  32 . Binary patterns  40  are arrangements of “off” channel bits  42   a  and “on” channel bits  42   b . In some embodiments, the data groups are converted into the binary patterns  40  using a lookup table  46 , as illustrated in FIG.  2 . Lookup table  46  comprises a list of data groups and a list of binary patterns  40 ; lookup table  46  serves as a dictionary to translate the data groups into binary patterns  40 . In the example of FIG. 2, lookup table  46  correlates three-bit data groups to binary patterns  40  having two “on” channel bits  42   b  and four “off” channel bits  42   a . The process of converting bits  32  to binary patterns  40  is referred to as “modulation encoding” the bits. The equivalence between the binary patterns and the data groups, as exemplified by FIG. 2, is called a “modulation code.” 
     The association of the data groups to the binary patterns is arbitrary, and FIG. 2 shows only one possibility. The important point about the conversion of the data groups to the binary patterns is that each binary pattern has fewer “on” channel bits than “off” channel bits. 
     In general, the k th  data group G k  is converted into a k th  binary pattern having n k  channel bits. The k th  binary pattern has m k  “on” channel bits and n k −m k  “off” channel bits, where m k &lt;n k /2 and n k &gt;2. There are n k -choose-m k  different ways to arrange m k  “on” channel bits in a pattern of n k  channel bits, where            n   k          -        choose        -          m   k       =       (           n   k               m   k           )     =           n   k     !           m   k     !            (       n   k     -     m   k       )     !         .                              
     Since data group G k  has b k  bits, n k  and m k  must satisfy the condition n k -choose-m k ≧2 b     k   . This condition ensures that the k th  binary pattern contains at least as much information as the k th  data group G k . 
     In some embodiments, b k =b for each of the data groups, and n k =n and m k =m for each of the binary patterns. In other words, in these embodiments, all of the binary patterns have the same number of “on” channel bits and the same number of “off” channel bits. In one embodiment, for example, n=8 and m=3. However, in other embodiments the binary patterns have various numbers of “on” channel bits and “off” channel bits. 
     Binary patterns  40  are formatted onto a page  50 , as shown in FIG.  3 . Page  50  comprises page pixels  52 . Binary patterns  40  are formed as arrangements of opaque page pixels  52   a  representing “off” channel bits, and transparent page pixels  52   b  representing “on” channel bits. Page  50  may also comprise reference marks  54 , which can be used, for example, for alignment or diagnostic purposes during the readout of the data. Reference marks  54  comprise any desired arrangements of opaque and transparent page pixels. Although page  50  as shown in FIG. 3 has 484 page pixels  52 , in the preferred embodiment page  50  has approximately one million page pixels  52 . The figures, therefore, are purely schematic. 
     In one embodiment of the present invention, page  50  is displayed on a spatial light modulator  16 , as shown in FIG. 4. A signal beam  10 , generated by a monochromatic light source (not shown), passes through spatial light modulator  16 . Signal beam  10  subsequently strikes a holographic storage medium  20 . A reference beam  12  is also incident upon storage medium  20 . Standard optical components  13 , such as lenses, are used to deliver signal beam  10  to storage medium  20 . An interference pattern between signal beam  10  and reference beam  12  is stored in storage medium  20  according to known principles. In regions of constructive interference, the index of refraction of storage medium  20  is changed. Page  50  is thereby stored as a hologram. Using the above technique, a number of pages are stored in the same storage medium  20  by known multiplexing methods. 
     II. Data Retrieval 
     A system for holographic readout of data pages is shown in FIG.  5 . An address beam  15  is incident upon storage medium  20 , causing an image beam  22  to be emitted. Address beam  15  is identical to reference beam  12  in the embodiments of FIGS. 4 and 5. Image beam  22  is incident upon a detector  24 . An image of page  50  stored in storage medium  20  is thereby projected onto detector  24 . Standard optical components  21 , such as lenses, deliver image beam  22  to detector  24 . Detector  24  has detector pixels  26 , and is preferably a CCD detector. 
     Each detector pixel  26  produces an electrical signal proportional to a light intensity received from image beam  22 . That is, detector  24  converts light signals to electrical signals. The electrical signals are used to form a data array  60 , diagrammed in FIG. 6, that is stored in a standard electronic memory (not shown). Data array  60  comprises array elements  62  that correspond to page pixels  52  of page  50 . In the simplest embodiment, each one of the detector pixels  26  receives an image of one of the page pixels  52 , and the values of array elements  62  are proportional to the light signals received by the detector pixels  26 . 
     In another embodiment, each detector pixel  26  is one fourth the size of an image of one page pixel  52 . In this embodiment, each array element is an average of electrical signals received from four detector pixels. In still other embodiments, the electrical signals from detector pixels  26  are processed and then stored as data array  60 . Such processing may include, for example, a correction for a misalignment between detector  24  and image beam  22 , or corrections for aberrations caused by optical components  13  and  21 . Several such techniques for processing are known in the art; see, for example, U.S. Pat. No. 5,511,058 by Visel et al., “Distortion Correction of a Reconstructed Holographic Data Image.” Other techniques are disclosed in co-pending U.S. patent application “Method for Holographic Data Storage and Retrieval” by Ray Snyder and Andrew Daiber. The processing may carried out using digital electronics. 
     Electrical signals from detector pixels  26  are used to produce data array  60 , wherein the value of each array element  62  corresponds to a light signal generated by a page pixel  52  stored in storage medium  20 . Locations of N data sectors  64  are identified within data array  60 , as shown in FIG. 6, the k th  data sector comprising n k  array elements  62 . Each data sector  64  corresponds to one of the binary patterns  40 . Locating the data sectors is simplified in some embodiments by first locating array elements that correspond to reference marks  54  of page  50 ; see FIG.  3 . Since binary patterns  40  occur at known distances from reference marks  54  on page  50 , data sectors  64  in data array  60  occur at known locations with respect to detected images of reference marks  54 . 
     Each data sector  64  is converted to a binary pattern  40  as follows. The k th  data sector comprises n k  array elements. The m k  largest values of these n k  elements are identified as “on” channel bits; the remaining n k −m k  values are identified as “off” channel bits. The k th  data sector is thereby converted into the k th  binary pattern. 
     Typically the array elements detect gray scale intensity, meaning that they take on a range of values, for example between 1 and 100. Using the above technique of identifying the m k  largest array element values of the k th  data sector as binary 1&#39;s, and the remaining n k −m k  values as binary 0&#39;s, the gray code is converted into the binary patterns. 
     Binary patterns  40  are then converted into bits  32  by reversing the original technique of modulation encoding the data. Lookup table  46  is used to associate data groups G 1  . . . G N  to the N binary patterns  40 ; see FIG.  2 . If necessary, the data groups are then reordered, or unshuffled, and error-correction decoded. A plurality of pages may be read from storage medium  20  in this way. 
     In the preferred embodiment, a plurality of pages are stored in storage medium  20 , each page comprising binary patterns  40 . Because binary patterns  40  comprise fewer “on” channel bits than “off” channel bits, pages previously stored in storage medium  20  do not greatly affect signal beam  10  as page  50  is stored. Similarly, during data readout, pages not being accessed by address beam  15  minimally perturb image beam  22  in storage medium  20 . 
     In general, bits  32  are stored by first forming data groups G 1  . . . G N  from bits  32 , converting the data groups to the binary patterns, and storing the binary patterns in storage medium  20  as perturbations in an optical property of the storage medium. 
     To retrieve data recorded as pages in storage medium  20 , address beam  15  is incident upon the storage medium. Light signals are induced by address beam  15  and are detected. The detected signals are used to from data array  60  comprising data sectors  64 . The data sectors are converted to the binary patterns, and the binary patterns are converted into bits  32 . 
     In FIG. 2, lookup table  46  is used to associate binary patterns  40  to the data groups G k . Lookup table  46  has 2 b     k    entries, where b k  is the largest integer that satisfies 2 b     k   ≦n k -choose-m k . Therefore, a number of possible binary patterns are unused. The number of unused binary patterns U is given by: U=n k -choose-m k −2 b     k   . These unused binary patterns constitute a waste of possible data storage space. 
     In one embodiment, the unused binary patterns are used advantageously for error detection. If one data sector  64  is converted into a binary pattern that does not appear on lookup table  46 , an error has been detected in that data sector. 
     In another embodiment, it is desired to waste as little data storage space as possible, and so to reduce the number of unused binary patterns U to zero. This reduction can be achieved by using a finite state machine to convert the data groups to the binary patterns and vice versa. In essence, the finite state machine uses a different lookup table for each data group. A k th  lookup table is used to convert the k th  data group to the k th  binary pattern. The structure of the k th  lookup table depends on the value of the (k−1) th  data group G k−1 . 
     The modulation codes discussed above are called “sparse” because they have more “off” channel bits than “on” channel bits. A pattern sparseness s k  is defined as n k /m k . Pattern sparseness s k  is the ratio of the total number of channel bits to the number of “on” channel bits of the k th  binary pattern. In the present method and system, s k ≧2 for every k. A sparseness s is defined as the average pattern sparseness:        s   =       1   N                       ∑     k   =   1     N                     s   k                                
     In one embodiment, all of the binary patterns comprise n channel bits; that is, n k =n for all k. In this embodiment, a desired sparseness s can be achieved by allowing m k  to vary with k. As an example, consider an embodiment where n=8. If m k =4 for all k, s=2. If m k =3 for all k, s=2.66. To achieve an s=2.44, two thirds of the binary patterns have m k =3, and one third of the binary patterns have m k =4. 
     In another embodiment, m k =m for all k, and a desired sparseness s is achieved by allowing n k  to vary with k. 
     III. Optimum Modulation Codes 
     In the present sparse modulation codes, s&gt;2. Because s&gt;2, the pages not being accessed by address beam  15  interfere less with image beam  22  than would pages without sparse modulation coding. However, as s increases, less data may be stored per page, and therefore more pages must be stored in storage medium  20 . This tradeoff is examined below. 
     In the following discussion, an embodiment where m k =m for all k is assumed. That is, it is assumed that all of the binary patterns have the same number of “on” channel bits. A code rate r is defined as the number of bits  32  stored per page pixel, and has an upper bound r max  given by                r   max     =       1   sm                       log   2          (         sm           m         )                 (   1   )                                
     In the foregoing embodiments using a lookup table, the code rate r is equal to the integer portion of r max . In these embodiments, r=floor[r max ], where the floor function returns the largest integer that is less than or equal to r max . In other embodiments using a finite state machine to convert between the data groups and the binary patterns, the code rate r approaches r max , and is some of these embodiments, r=r max . 
     A transfer rate T is the rate at which data is transferred to and from the storage medium. There are P page pixels per page stored in storage medium  20 , and detector  24  receives image beam  22  for an amount of time t. The transfer rate T is therefore equal to:              T   =     rP   t             (   2   )                                
     If the transfer rate T is held constant for different code rates r, then time t is proportional to r: 
       t r   (3) 
     A storage capacity C is the total amount of information that can be stored in storage medium  20 . If M pages can be stored in storage medium  20 , C=r P M. Therefore:              M   ∝     C   r             (   4   )                                
     A diffraction efficiency η diff  is defined as the intensity of image beam  22  divided by the intensity of address beam  15  during data readout. For many materials, the diffraction efficiency varies inversely with the square of the number of pages stored in storage medium  20 :                η   diff     ∝     1     M   2               (   5   )                                
     Eq. (5) holds, for example, when storage medium  20  is a photorefractive material, since a given data recording event partially erases data previously recorded. Eq. (5) also holds when storage medium  20  is a photopolymer, since the dynamic range of the photopolymer must be budgeted among the pages. Substituting Eq. (4) in Eq. (5),                η   diff     ∝       (     r   C     )     2             (   6   )                                
     Assuming that a certain threshold number of photons is needed for detector  24  to register an “on” channel bit, the number of photons per “on” channel bit that reach detector  24  is equal to a constant for different code rates:              const   =           η   diff       P   /   s          t     ∝           (     r   /   C     )     2       1   /   s                     r               (   7   )                                
     where Eqs. (6) and (3) have been used in the last step. Solving Eq.(7) for storage capacity C, 
     
       
           C=Kr   3/2   s   1/2   (8) 
       
     
     where K is a constant. To evaluate K, the maximum possible value of C is defined as one. This maximum occurs when r=1 and s=2. Therefore K=1/{square root over (2)}, and              C   =       1     2                       r                  3   /   2              s     1   /   2                 (   9   )                                
     Eq. (9) shows how storage capacity C depends on the code rate r and the sparseness s of a desired modulation code. 
     In the embodiments with r=r max , Eq. (9) expresses, in conjunction with Eq. (1), how the storage capacity C varies with the number of “on” channel bits per binary pattern, m, and the sparseness s. The storage capacity resulting from Eq. (9) and r=r max  is shown in FIG. 7 for several values of m. A first curve A 1  gives C as a function of s for m=1. Curves A 2 , A 3 , and A 4  give the storage capacity for m=2, m=4, and m=20, respectively. 
     It is preferred to maximize the storage capacity. As can be seen from FIG. 7, the maximum storage capacity is obtained for s≈4 when m=1, and as m grows larger, the storage capacity maximum shifts toward s≈3. 
     To be more precise, the values of s that maximize C when r=r max  are given in Table 1 for values of m from 1 to 10. As m tends toward infinity, the optimal value of s approaches 2.85. As described earlier, a desired value of s may be achieved for a given value of m by allowing n k  to vary with k. 
     
       
         
               
               
               
             
           
               
                   
                 TABLE 1 
               
               
                   
                   
               
             
             
               
                   
                 m = 1 
                 s = 4.48 
               
               
                   
                 m = 2 
                 s = 3.81 
               
               
                   
                 m = 3 
                 s = 3.55 
               
               
                   
                 m = 4 
                 s = 3.42 
               
               
                   
                 m = 5 
                 s = 3.33 
               
               
                   
                 m = 6 
                 s = 3.27 
               
               
                   
                 m = 7 
                 s = 3.22 
               
               
                   
                 m = 8 
                 s = 3.18 
               
               
                   
                 m = 9 
                 s = 3.15 
               
               
                   
                  m = 10 
                 s = 3.13 
               
               
                   
                 m → ∞ 
                 s → 2.85 
               
               
                   
                   
               
             
          
         
       
     
     In some embodiments, a lookup table such as lookup table  46  is used to convert between the data groups and the binary patterns. In these embodiments, r=floor[r max ]. Eq. (9) can then be used again to determine the storage capacity as a function of m and s, and to find the value of s that optimizes C for different values of m. The resulting optimum sparsenesses are shown in Table 2 for m=1 to 10. For values of m greater than 10, the storage capacity is maximized when s=3, with the exception of m=13, for which the maximum storage capacity occurs at s=4. 
     
       
         
               
               
               
             
           
               
                   
                 TABLE 2 
               
               
                   
                   
               
             
             
               
                   
                 m = 1 
                 s = 4 
               
               
                   
                 m = 2 
                 s = 6 
               
               
                   
                 m = 3 
                 s = 3 
               
               
                   
                 m = 4 
                 s = 5 
               
               
                   
                 m = 5 
                 s = 3 
               
               
                   
                 m = 6 
                 s = 4 
               
               
                   
                 m = 7 
                 s = 4 
               
               
                   
                 m = 8 
                 s = 3 
               
               
                   
                 m = 9 
                 s = 3 
               
               
                   
                  m = 10 
                 s = 3 
               
               
                   
                 m → ∞ 
                 s → 3 
               
               
                   
                   
               
             
          
         
       
     
     In the preferred embodiments, the number m of “on” channel bits per binary pattern is chosen, the sparseness s is selected to maximize the storage capacity C, and a modulation code having the optimum sparseness s for the chosen m is used to encode the data groups for storage. 
     It will be clear to one skilled in the art that the above embodiments may be altered in many ways without departing from the scope of the invention. For example, in some embodiments binary patterns  40  do not have the appearance shown in FIG.  2 . Binary patterns  40  have any form appropriate to the storage medium. In some embodiments, “on” channel bits  42   b  and “off” channel bits  42   a  are round, rather than rectangular. Any shape is possible for “on” channel bits  42   b  and “off” channel bits  42   a , and binary patterns  40  may comprise any geometrical arrangement of channel bits. Accordingly, the scope of the invention should be determined by the following claims and their legal equivalents.