Abstract:
Methods and apparatus for spreading and concentrating information are taught. The present invention relates to constant-weight encoding of data words on a parallel data line bus while allowing communication of information across sub-word paths.  
     In one embodiment, of the present invention, data transfer rates previously obtained only with differential architecture are achieved by only a small increase in line count above single ended architecture. For example, an 18-bit data word requires 22 encoded data lines for transmission, where previously, 16 and 32 lines would be required to transmit un-coded data with single-ended and differential architectures respectively. Constant-weight parallel encoding maintains constant current in the parallel-encoded data lines and the high and low potential driver circuits for the signal lines.

Description:
FIELD OF THE INVENTION  
         [0001]    The present invention relates to the field of information transmission systems. More particularly, in one implementation, the present invention relates to constant-weight encoding and decoding of data words on a parallel data line bus.  
         BACKGROUND OF THE INVENTION  
         [0002]    It is often desirable, in information transmission systems, to transform information into alternate forms. In one instance, a desirable form might be to encode data, to achieve a constant weight code, where each data word contained a constant number of data elements in each logic state. The data words could be binary words occupying two logic states, those of zero and one. This transformation enables the higher supply potential, Vdd, and the lower supply potential, Vss, to maintain constant current to the circuits that drive the signal lines and or constant current in the termination circuit, Vterm. In another instance it may be desirable to spread information from a data word into sub-words and the weight of the sub-words, where the weight of the sub-word is defined to be the number of data elements in each logic state. In a binary system the weight is the number of ones or zeros in a data word. In another instance it may be desirable to minimize the weight so that the power required to drive the parallel data line bus is minimized.  
           [0003]    Transmission of N-ary data, where there are more than the traditional two logic states of zero and one, does not impose the same constant weight criterion that is imposed on binary data in order to achieve the constant current conditions mentioned above. There are many combinations of N-ary weight vectors that provide the desired constant current state.  
           [0004]    Transmission of binary information on a parallel line bus requires the transmission of data words whose weight ranges from zero to the number of bits in the information word. Therefore, an eight-bit data word has a weight that ranges between zero and eight. Transmission of variable weight data presents problems to interconnect circuits, such as those used in a high-speed bus interface.  
           [0005]    In a single-ended parallel interface with line drivers, transmission of variable weight data creates a data-dependent current flow in the Vdd and Vss connections. This data-dependent current flow leads to timing losses. Vdd and Vss interconnect path inductances (L) exist between the voltage sources and the line driver higher and lower potential bias nodes, respectively. A changing current in this inductance path (di/dt) creates voltage variation according to v=Ldi/dt. These voltage changes compromise the integrity of the output signals as a function of data word value. Specifically, they delay high to low and low to high transitions of the signal lines (edges) and create uncertainty in edge location. Both of these effects compromise achievable system speeds. The larger Ldi/dt the greater the degree of the timing loss. If all lines change at once, or if more lines are present di/dt is increases.  
           [0006]    Path inductances (L) due to controllable design parameters is already at practical minimal limits. Differential architecture eliminates timing losses due to Vdd and Vss fluctuations, however, this comes at the expense of two lines per bit of information transmitted. It is desirable to obtain differential performance from single-ended architecture. Accordingly, constant-weight parallel encoding has been employed.  
           [0007]    Constant-weight parallel encoding can achieve constant Vdd and Vss current flow at the expense of a slight increase in the bus line count. It is possible to achieve many of the benefits of a differential system with only a small increase in lines over single-ended architecture. We will define constant weight parallel encoding to be the result of encoding a data word into an encoded data word with a constant number of data elements in each logic state independent of the input data word value. We further define balanced encoding to be constant-weight parallel encoding such that the number of data elements in each logic state is the same. Thus, a balanced 22-bit encoded data word would have 11 data element whose value was zero and 11 data elements whose value was one. An almost balanced parallel-encoded data word would be closer to the balanced case than the unbalanced case.  
           [0008]    As the data word length increases, the complexity of the encode process greatly increases the corresponding time to encode. To employ constant weight parallel encoding in a data transmission system, without compromising performance, requires efficient low latency coding methods. Design of efficient, low latency encoding/decoding methods has been an area of research for the past four decades. Attempts at solving this problem exist in the prior art.  
           [0009]    Tallini, G., et. al., “Design of Balanced and Constant Weight Codes for VLSI Systems,” IEEE, vol. 47 no. 5, Transactions on Computers May (1998) describes encoding techniques applied to the input word as a whole without partitioning the input word into sub-words. Computation time to encode over an entire word is much greater than the computation time to encode a sub-word, such undivided approaches result in high complexity encode functions. These techniques remain complex and are not easily reduced to low latency implementations on an integrated circuit chip. Burleson, W., et. al., “Bus-Invert Coding for Low-Power I/O,” IEEE, vol. 3, no. 1, Transactions on Very Large Scale Integrations (VLSI) Systems March (1995) describes an inversion method for encoding that divides the input word up into sub-words and then proceeds to encode the sub-words to minimize the variability in the weight. Tabor, J., “Noise Reduction Using Low Weight And Constant Weight Coding Techniques,” MS Thesis, Artificial Intelligence Lab, MIT, May (1990) also describes dividing the input word into sub-words, but does not achieve constant weight encoding.  
           [0010]    The prior-art techniques provide limited simplification of the problem. The prior-art techniques do not provide for sharing of information between sub-words or sub-word paths. Thus, it is desirable to provide efficient, low latency, encoding/decoding methodology that can be implemented with a minimum number of extra lines and encode/decode logic by sharing information between sub-words or sub-word paths to facilitate spreading information into the encoded sub-words as well as into the weight of the encoded sub-words.  
         SUMMARY OF THE INVENTION  
         [0011]    The present invention includes methods for spreading and concentrating information into encoded data sub-words and the weight of the encoded data sub-words. An embodiment of the present invention is directed to efficient apparatus and methods for constant-weight encoding of data that can be implemented with low latency in a data transmission system. Various embodiments of the present invention are described below.  
           [0012]    A method including dividing a data word into data sub-words onto sub-word paths; allowing communication between the sub-word paths; and encoding the data sub-words into encoded data sub-words; such that the information content of the data word is spread between the encoded data sub-words and the weight of the encoded data sub-words.  
           [0013]    Another method includes: allowing communication between sub-word paths; and decoding encoded data sub-words into data sub-words; such that the data sub-words form a data word whereby the information content of the data word is concentrated back into the data word.  
           [0014]    A preferred embodiment, of the present invention, includes a method of encoding a data word, whose data elements occupy at least a first logic state and a second logic state, the method includes: receiving data sub-words onto sub-word paths, the data sub-words comprising sets of data elements of the data word; and encoding the data sub-words into encoded data sub-words; such that the encoded data sub-words form an encoded data word wherein the information content of the data word is spread between the encoded data sub-words and the weight of the encoded data sub-words.  
           [0015]    Another embodiment, of the present invention, includes an encoder module, to encode a data word, whose data elements occupy at least a first logic state and a second logic state, the encoder module includes: at least two sub-word paths, each of the at least two sub-word paths to receive a data sub-word, including a set of data elements of the data word; and an encoder coupled with the at least two sub-word paths, the encoder to encode the data sub-word into an encoded data sub-word; such that encoded data sub-words form an encoded data word wherein the information content of the data word is spread between the encoded data sub-words and the weight of the encoded data sub-words.  
           [0016]    In another embodiment, the present invention provides a decoder module to decode an encoded data word, whose encoded data elements occupy at least a first logic state and a second logic state, the decoder module includes: at least two sub-word paths, each of said at least two sub-word paths to receive an encoded data sub-word, including a set of data elements of the encoded data word and to allow communication, between said at least two sub-word paths, such that information may be shared between the at least two sub-word paths; and a decoder coupled with the at least two sub-word paths, the decoder to decode the encoded data sub-word into a data sub-word; such that data sub-words form a data word.  
           [0017]    Yet another embodiment, of the present invention, includes a method of decoding an encoded data word, whose encoded data elements occupy at least a first logic state and a second logic state, the method includes: receiving encoded data sub-words onto sub-word paths, the encoded data sub-words including sets of data elements of the encoded data word; allowing communication between the sub-word paths, such that information may be shared between the sub-word paths; and decoding the encoded data sub-words into data sub-words; such that the data sub-words form a data word.  
           [0018]    Another preferred embodiment, of the present invention, is a data processing system including: at least two sub-word paths, each of said at least two sub-word paths to receive a data sub-word, comprising a set of data elements of a data word; an encoder coupled with the at least two sub-word paths, the encoder to encode the data sub-word into an encoded data sub-word; such that encoded data sub-words form an encoded data word wherein the information content of the data word is spread between the encoded data sub-words and the weight of the encoded data sub-words; a parallel encoded data line bus coupled with the at least two sub-word paths to receive the encoded data sub-words and to facilitate transmission of the encoded data sub-words; at least two sub-word paths coupled with the parallel encoded data line bus, each of the at least two sub-word paths to receive an encoded data sub-word, including a set of data elements of the encoded data word and allowing communication, between the at least two sub-word paths, such that information may be shared between the at least two sub-word paths; and a decoder coupled with the at least two sub-word paths, the decoder to decode the encoded data sub-word into the data sub-word; such that data sub-words form the data word.  
           [0019]    Another preferred embodiment, of the present invention, is a method for transmitting a data word in a data processing system, the method comprising: receiving data sub-words onto sub-word paths, the data sub-words comprising sets of data elements of the data word; encoding the data sub-words into encoded data sub-words, such that the encoded data sub-words form an encoded data word wherein the information content of the data word is spread between the encoded data sub-words and the weight of the encoded data sub-words; transmitting the encoded data sub-words over a parallel encoded data line bus; receiving the encoded data sub-words onto the sub-word paths, the encoded data sub-words comprising sets of data elements of the encoded data word; allowing communication, between the sub-word paths, such that information may be shared between the sub-word paths; and decoding the encoded data sub-words into the data sub-words; such that the data sub-words form the data word.  
       
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0020]    The present invention is illustrated by way of example and not limited in the Figures of the accompanying drawings, in which like references indicate similar elements.  
         [0021]    [0021]FIG. 1 depicts multiple sub-word paths and information sharing between sub-word paths, along with input data word to encoded data word flow.  
         [0022]    [0022]FIG. 2 applies the concept of shared information to the creation of a constant-weight encoded data word.  
         [0023]    [0023]FIG. 3 depicts a property of binary numbers, a binomial expansion, which shows the relationship between the number of elements in a binary word, n, the weight of the binary word, p, and the number of unique states available for the chosen weight p and binary word size n.  
         [0024]    [0024]FIG. 4 is a table, which summarizes relevant properties of binary numbers, showing the minimum number of extra lines required to achieve constant-weight coding.  
         [0025]    [0025]FIG. 5 is a code-weight vector tree that depicts parallel encoding within a sub-word, spreading information into the sub-words, and the resulting encoded word weights that result without sharing information across sub-words.  
         [0026]    [0026]FIG. 6 is a code-weight vector tree that depicts parallel encoding within sub-words, information sharing across sub-words, spreading information into the sub-words as well as the weight of the sub-word, and then using the shared information to achieve a constant-weight encoded data word.  
         [0027]    [0027]FIG. 7 shows the application of the present invention within a general purpose data processing system.  
         [0028]    [0028]FIG. 8 is a detail representation of two devices on a parallel data line bus employing the present invention.  
         [0029]    [0029]FIG. 9 shows the termination of the encoded parallel data lines according to the present invention, where less than two lines are required to transmit one bit of information.  
         [0030]    [0030]FIG. 10 depicts a prior art differential architecture termination of parallel data lines, where two lines are necessary to transmit one bit of information.  
         [0031]    [0031]FIG. 11 is an embodiment illustrating the best mode of the present invention as applied to an encoder for the specific case of encoding an 18-bit data word.  
         [0032]    [0032]FIG. 12 is a further detail of the encoding applied to sub-word a according to the present invention.  
         [0033]    [0033]FIG. 13 shows the truth tables employed by the encoder blocks shown in FIG. 12.  
         [0034]    [0034]FIG. 14 is a further detail of the encoding applied to sub-word b according to the present invention.  
         [0035]    [0035]FIG. 15 shows the truth tables employed by the encoder blocks shown in FIG. 14.  
         [0036]    [0036]FIG. 16 is a further detail of the encoding applied to sub-word c according to the present invention.  
         [0037]    [0037]FIG. 17 shows the truth tables employed by the encoder blocks shown in FIG. 16.  
         [0038]    [0038]FIG. 18 is a detail of the decoding applied to a generic sub-word x, where x refers to sub-word a, b, and c.  
         [0039]    [0039]FIG. 19 shows the truth tables employed by the decoder blocks shown in FIG. 18 for sub-word a.  
         [0040]    [0040]FIG. 20 shows the truth tables employed by the decoder blocks shown in FIG. 18 for sub-word b.  
         [0041]    [0041]FIG. 21 shows the truth tables employed by the decoder blocks shown in FIG. 18 for sub-word c.  
         [0042]    [0042]FIG. 22 is an alternative embodiment for encoding, using binomial coefficient matrix encoding, comparing the decimal value, binary value, and the encoded value for integer numbers ranging from zero to 19.  
         [0043]    [0043]FIG. 23 is an example of binomial coefficient matrix encoding for the decimal value 58.  
     
    
     DETAILED DESCRIPTION  
       [0044]    [0044]FIG. 1 depicts multiple sub-word paths and information sharing between sub-word paths, along with input data word to encoded data word flow. With reference to FIG. 1, data word  2  is split into data sub-word  4  and data sub-word  6 . Data sub-word  4  travels along sub-word path  8  and data sub-word  6  travels along sub-word path  10 . Encoder  12  is connected with sub-word path  8  and encodes data sub-word  4 . Encoder  14  is connected with sub-word path  10  and encodes data sub-word  6 . It will be appreciated that many alternatives are possible, for example, encoder  12  could be comprised of a plurality of encoders or encoder  12  and encoder  14  could be a single encoder connected with both sub-word path  8  and sub-word path  10 . A plurality of encoders in contact with a sub-word path could encode the sub-word in parallel.  
         [0045]    Shared information  16  allows information to be shared between sub-word paths. Shared information  16  can occur anywhere along the sub-word paths. For example, shared information  16  could occur before encoder  12 , shared information  16  could occur between encoder  12  and encoder  14 . Shared information  16  could occur between the sub-word paths after the encoders. It will be appreciated that shared information  16  could occur between sub-word path  8  and sub-word path  10  in orders not specifically defined, the order does not limit the present invention.  
         [0046]    Data sub-word  4  travels along sub-word path  8  and is encoded by encoder  12 , encoded data sub-word  18  results from shared information  16  and encoder  12 . Data sub-word  6  travels along sub-word path  10  and is encoded by encoder  14 , encoded data sub-word  20  results from shared information  16  and encoder  14 . Encoded data sub-word  18  and encoded data sub-word  20  are combined to form encoded data word  22 . The encoding provided by encoder  12  and encoder  14  on data sub-word  4  and data sub-word  6  may result in encoded data word  22  being either, small-variance-weight, constant-weight, or balanced.  
         [0047]    [0047]FIG. 2 applies the concept of shared information to the creation of a constant-weight encoded data word. With reference to FIG. 2, an embodiment of the present invention provides constant-weight weight encoding  26 , accomplished with a plurality of encoders and a parity element that leads to a low latency logic integrated circuit implementation. Data word  2  is input on m input information word lines  28 . Input information word lines m  28  are divided between sub-word  1  lines m 1    30 , sub-word  2  lines m 2    32 , up to sub-word L lines m L    34 .  
         [0048]    Sub-word  1  lines m 1    30  connect with sub-word encoder  1   36 . Similarly, sub-word  2  lines m 2    32  connect with sub-word encoder  2   38  up to sub-word L lines m L    34  connecting with sub-word encoder L  40 . It will be appreciated that input data word  2  may be divided into a general number of sub-words as indicated by index L. It will also be appreciated by those of skill in the art that the sub-words need not contain the same number of data elements from input data word  2 , but can be of different size. The architecture shown in constant-weight weight encoding  26  is equivalent to the sub-word paths shown in FIG. 1 with the encoders connected with each sub-word path. Shared information  44  may be exchanged between sub-word  1  lines m 1    30 , sub-word  2  lines m 2    32 , sub-word  3  lines m 3    34 , sub-word encoder  1   36 , sub-word encoder  2   38 , sub-word encoder L  40 , and parity elements  42 .  
         [0049]    Sub-word encoder  1   36  has lines n 1    46  that are used to output the encoded data sub-word. Sub-word encoder  2   38  has lines n 2    48  that are used to output the encoded data sub-word. Sub-word encoder L  40  has lines n L   50  that are used to output the encoded data word. The sub-words from sub-word encoder  1   36 , sub-word encoder  2   38 , up to sub-word encoder L  40  are combined with parity element  42  to form encoded data sub-word  22 , which is output on total encoded lines  62 . Equation  64  shows the relations between total encoded line count  62 , and the line count from each sub-word encoder and parity lines  60 .  
         [0050]    [0050]FIG. 3 depicts a property of binary numbers, a binomial expansion, which shows the relationship between the number of elements in a binary word, n, the weight of the binary word, p, and the number of unique states available for the chosen weight p. With reference to FIG. 3, Binomial coefficient matrix  66  may be used for devising various methods of encoding data words. One such method, leads to a low latency logic implementation in an integrated circuit. According to this method, it is desirable to break up a long data word into shorter sub-words. Shorter sub-words can be encoded in less time than longer data words. With reference to binomial coefficient matrix  66 , it will be noted that an encoded word seven elements in length, n=7, has 35 states in which there are three ones, p=3 and 35 states in which there are four ones, p=4. Therefore, a total of 70 states exist in the encoded word with either three or four ones in each encoded data word.  
         [0051]    [0051]FIG. 4 is a table, which summarizes relevant properties of binary numbers, showing the minimum number of extra lines required to achieve constant weight coding. With reference to FIG. 4, table  68  indicates the minimum number of extra lines necessary for each input word length encoded. The last column titled “extra lines” represents encoding optimized to provide a minimum number of extra lines. The best mode of the present invention will add one extra line over the optimum shown in table  68  for encoding a binary word size of  18 , resulting in 22 encoded bits in the encoded data word. However, decreased encode time is achieved with the combination of smaller sub-word size and sharing of information across sub-word paths. Thus, the best mode of the present invention encodes three 6-bit input words with a total of 22 encoded lines, which is two lines less than the 24 required by binomially encoding three 6-bit input words.  
         [0052]    [0052]FIG. 5 is a code-weight vector tree that depicts parallel encoding within a sub-word, spreading information into the sub-words and the resulting encoded word weights that result without sharing information across sub-words. With reference to FIG. 5, code-weight vector tree  70  shows the encoded word weights that are possible when an 18-bit word is divided into three sub-words, each 6-bits in length, according to one embodiment of the present invention. Number of encoding weights applied to a sub-word  72  applies weights three and four to sub-words a, b, and c. Weights for sub-word a  74 , weights for sub-word b  76 , and weights for sub-word c  78  show the possible branches of the tree that result in encoded word weights  80 . Encoded word weights  80  include encoded word weights that range from nine to 12. Encoded word weight  82 , which results in a total encoded word weight of nine, results from encoded sub-word weights of three for each sub-word. Encoded word weight  96 , which results in a total encoded word weight of 12, results from encoded sub-word weights of four for each sub-word. The branches of the code-weight vector tree (FIG. 5) are listed as rows in Table 1 along with the corresponding reference numeral for the encoded word weight.  
         [0053]    Transmission of encoded word weights  80  results in a small-variance-weight encoding scheme. The small-variance-weight encoded words have a much narrower weight range than do the input data words. 18-bit input data word values range between all zeros and all ones, causing the word weight to ranging from zero to 18. The small-variance-weight encoded data word weights vary from nine to 12 for an encoded data word utilizing 21 lines. The reduced weight variance provides an improved solution to the problems described earlier with Vdd and Vss current fluctuations during transmission.  
                                                           TABLE 1                           Code weight vectors from FIG. 5                        Encoded   Encoded word                   word   weight reference       Weight of   Weight of   Weight of   weights 80   numeral       Sub-word a   Sub-word b   Sub-word c   (FIG. 5)   (FIG. 5)                    3   3   3   9   82       3   3   4   10   84       3   4   3   10   86       3   4   4   11   88       4   3   3   10   90       4   3   4   11   92       4   4   3   11   94       4   4   4   12   96                  
 
         [0054]    However, further reduction in the variance of the encoded word weight can be achieved by employing shared information across sub-word paths in order to produce constant weight encoded data. By inspecting the range of encoded word weight, nine to 12, it is evident that by encoding two ones, and employing a single parity line, balanced constant-weight encoded data words will be achieved with weight of 11, on 22 total lines by allowing communication between sub-word paths.  
         [0055]    Each branch of code weight vector tree  70  encodes a constant number of states for an encoded sub-word of length 7 bits, 
         2 6 ·2 6 ·2 6 =262,144. 
         [0056]    Each code weight vector encodes the same number of states: 
         2 5 ·2 5 ·2 5 =2 15 . 
         [0057]    There are eight code weight vectors in the tree as indicated by the eight rows in table 1. Therefore the total number of states encoded is: 
         8·2 15 =218. 
         [0058]    No information is carried by the weight of the code at each level because at each level the code can be either three or four. Knowing the weight at any level in the tree does not help you determine how to encode the sub-word at any other level. Thus there are no constraints placed on the weight, except that they must be either three or four. The resulting code-weight vectors range from nine to 12 as shown in encoded word weights 80.  
         [0059]    The code variance at each level in the tree of FIG. 5 is three or four. If the variance is increased to two, three, four, or five, the total number of possible states becomes: 
         (2 4 =2 5 =2 4 ) 3 =844,736. 
         [0060]    Only 262,144 states of the possible 844,736 states are needed to encode 18-bit input numbers. The fully-expanded tree diagram would contain 84 nodes and the code-weight vectors encoded word weight would range from 6 to 15. However, we only need a subset of the code-weight vectors to cover 262,144 input states. The tree diagram of FIG. 6 contains 20 nodes, which are sufficient to cover 262,144 input states. Each code-weight vector encodes the following number of states: 
         code-weight vector {2,4,4} contains 16,384 states; 
         code-weight vector {3,3,5} contains 16,384 states; 
         code-weight vector {4,4,2} contains 16,384 states; 
         code-weight vector {5,3,3} contains 16,384 states; 
         code-weight vector {3,3,4} contains 32,768 states; 
         code-weight vector {3,4,4} contains 32,768 states; 
         code-weight vector {4,3,3} contains 32,768 states; 
         code-weight vector {4,3,4} contains 32,768 states; 
         code-weight vector {4,4,3} contains 32,768 states. 
         [0061]    The sum of these code-weight vectors contain 262,144 states and the weight is either 10 or 11. By adding a single parity bit, the almost-constant code of 10 or 11 can be made into a constant (and balanced) code of 11.  
         [0062]    [0062]FIG. 6 is a code-weight vector tree that depicts parallel encoding within sub-words, spreading information into the sub-words as well as the weight of the sub-word, information sharing across sub-words, and then using the shared information to achieve a balanced constant-weight encoded data word.  
         [0063]    With reference to FIG. 6, information carried in the individual sub-word weights is used to encode the other sub-words. For example, if sub-word a is 2, then sub-words b and c must have weight  4  as shown  10  FIG. 6.  
         [0064]    It will be appreciated by those with skill in the art that an encoded sub-word of weight three and length seven can be simply converted to a sub-word of weight four by inverting each of the output data elements and vice versa. Likewise, an encoded sub-word of weight two and length seven can be simply converted to a sub-word of weight five by the same inversion technique which also applies vice versa.  
         [0065]    According to the best mode of the present invention, the determination of the encoded sub-word weight is made by examining the most significant one or two bits of the individual sub-words. The encoding for sub-channel a based on the input bits d 0  . . . d 5  of FIG. 11 and the most significant bit from sub-word b (d 11 ) and the two most significant bits from sub-word c (d 17  and d 16 ). If the most significant bit of sub-words a, b, and c are all zero, then the first sixteen sub-words of the sub-word a are encoded with weight two and the second set of sixteen encoded sub-words of sub-word a are encoded with five. Otherwise, the first thirty-two encoded sub-words of sub-word a are of weight three and the next thirty-two encoded sub-words of sub-word a are of weight four. There is one exception in accordance with the code-weight vector tree diagram of FIG. 6. If the most significant bit of sub-word b and the two most significant bits of sub-word c are all one, then the second set of thirty-two states are of weight three.  
         [0066]    The encoding for the sub-words of sub-word b is based in the input bits d 6  . . . d 11  of FIG. 11 and the two most significant bits from sub-word a (d 5  and d 4 ) and the two most significant bits from sub-word c (d 17  and d 16 ). The first thirty-two encoded sub-words of sub-word b are weight three and the second set of thirty-two encoded sub-words of sub-word c are of weight four. There are two exceptions in accordance with the weight-code vector tree diagram of FIG. 6. The first exception is that if the most significant bit of sub-word a is one and the two most significant bits of sub-word c are one then the second-set of thirty-two sub-words of sub-channel c are of weight three. The second exception is that if the most significant bit of sub-word a is zero and the two most significant bits of sub-word c are zero then the first set of thirty-two sub-words of the sub-channel c are of weight four.  
         [0067]    From the previous discussion of FIG. 3 it will be noted that a seven element encoded word has 35 states in which there are three ones and 35 states in which there are four ones, always set high, thus a 6-bit data word can be encoded by using both sets of states. A 6-bit data word requires 64 states, leaving six states unused. The first 32 states of the 6-bit input word will be encoded with three ones and the second 32 states will be encoded with four ones. Exceptions to this encode scheme will occur when the three input sub-words result in weights of three or weights of four being generated simultaneously in each encoded sub-word. Alternative encoding is required to handle the encoded word weights of nine and 12 shown in FIG. 5 and Table 1. With reference to FIG. 6, code-weight vector tree  98  shows the encoded sub-word weights that are possible in the encoding architecture for an 18-bit input data word divided into three 6-bit sub-words.  
         [0068]    Weights for sub-word a  102  include encoding sub-word a with two ones or the inverse of two ones encoding which is five ones encoding along with three and four ones encoding. It will be appreciated by those of skill in the art that four ones encoding is the inverse of three ones encoding. Alternative encode paths  142  and  144 , for encoding two ones and encoding five ones respectively, in sub-word a, are alternative encodings that are done in conjunction with information sharing in order to reduce the weight variance of the encoded word weights that result in nine or 12. Selective use of alternative encode paths  142  and  144  result in paths  134  and  136  being bared from use in the alternative encoding scheme. Thus, additional encoding states exist for which are not used. Weights for sub-word b  104  are limited to three ones or four ones encoding. Weights for sub-word c  106  include two, three, four, and five ones encoding.  
         [0069]    Information sharing across sub-words reduces the weight variance of the encoded data words and ultimately made constant. For the case of each sub-word simultaneously encoding into a weight of three, the 32 states in sub-word a, are split into two cases with 16 states in each. The first case (states 0 to 15) is encoded using two ones. According to the best mode of the present invention, this case was chosen to occur when the second most significant bit (MSB) of sub-word a is equal to zero. Many others ways of making this choice exist. Encoded sub-words b and c are inverted which changes the weight of sub-words b and c from three to four. The resulting code weight vector {2,4,4} is shown in code-weight vector tree  98  resulting in sub-word weight sum  108  of ten resulting in encoded word weight  114  equal to eleven by setting parity bit value  110  to one. The inversion of the pre-balanced encoded sub-words b and c is called post inversion (PI).  
         [0070]    The fact that the weight of encoded sub-word a equals two imparts the information that on decode, the inversion of encoded sub-words b and c must be considered. Thus, information is shared across sub-word paths in terms of the weight of the sub-word.  
         [0071]    The second case for each sub-word simultaneously encoding into a weight of three (states 16 to 31) requires sub-word a to be encoded with five ones. In the best mode, of the present invention, this case occurs when the second MSB of sub-word a is equal to one. The resulting code-weight vector {5,3,3} is shown in code-weight vector tree  98  resulting in sub-word weight sum  108  of eleven resulting in encoded word weight  132  equal to eleven, by setting parity bit value  110  to zero. No PI of encoded sub-word b or c is required.  
         [0072]    It will be appreciated that the case of each sub-word encoding into a weight of three ones occurs when the first MSB of each sub-word equals zero. When the second MSB of encoded sub-word a equals zero encoded sub-words b and c must be decoded accordingly.  
         [0073]    The special case of each encoded sub-word resulting in a weight of four is treated with alternative encoding  138  and alternative encoding  140  applied to sub-word c. In the best mode, of the present invention, the second MSB of sub-word c is used to split the encoding for sub-word c into the two cases. When the second MSB of sub-word c equals zero the first 16 states (0 to 15) are encoded using two ones, alternative encoding  140 , and when the second MSB of sub-word c equals one, the second 16 states (16 to 31) are encoded using five ones, alternative encoding  138 .  
         [0074]    The first case using alternative encoding  140  results in code-weight vector {4,4,2} as shown in code weight vector tree  98  resulting in sub-word weight sum  108  equal to ten resulting in encoded word weight  130  equal to 11 by setting parity bit value  110  to one. No PI of encoded sub-word a or b is required.  
         [0075]    The second case, when the second MSB of sub-word c equals one and sub-word c is encoded using five ones, alternative encoding  138 , results in the need to invert encoded sub-words a and b. The resulting code weight vector {3,3,5} is shown in code-weight vector tree  98  resulting in sub-word weight sum  108  of eleven resulting in encoded word weight  116  equal to eleven by setting parity bit value  110  to zero. On decode, the weight of sub-word c equal to five will provide the information that encoded sub-words a and b need to be decoded accordingly because of the PI previously applied.  
         [0076]    The shared information that is occurring in the encoding process just described is the first and second MSB of sub-words a and c, as well as the first MSB of sub-word b.  
         [0077]    Alternative encoding is not required for combinations of mixed sub-word weights of three and four. Thus, encoded word weight  118 , encoded word weight  120 , encoded word weight  122 , encoded word weight  124 , encoded word weight  126 , and encoded word weight  128 , result from weight vectors that do not require alternative encoding.  
         [0078]    The branches of the code-weight vector tree (FIG. 6) are listed as rows in Table 1 long with the corresponding reference numeral for the encoded word weight. Where post inversion is required, it is so indicated with the symbol PI next to the appropriate sub-word weight. It is evident from Table 2 that the weight variance of encoded data words has been reduced to range between 10 and 11. The single parity line is used to produce the desired constant-weight encoded data words.  
                                     TABLE 2                           Code weight vectors from FIG. 6                            Encoded word                   Encoded   weight                   word   reference       Weight of   Weight of   Weight of   weights 112   numeral       Sub-word a   Sub-word b   Sub-word c   (FIG. 6)   (FIG. 6)               2     4 PI     4 PI   10   114         3 PI     3 PI   5   11   116       3   3   4   10   118       3   4   3   10   120       3   4   4   11   122       4   3   3   10   124       4   3   4   11   126       4   4   3   11   128       4   4   2   10   130       5   3   3   11   132                  
 
         [0079]    It will be appreciated by those of skill in the art that the embodiment of the present invention just described, is not limited to three sub-words, but is generally applicable, as shown in FIG. 2, to a general number of sub-words, L, indicated by sub-word encoder L  40  and a general number of parity lines  60 .  
         [0080]    [0080]FIG. 7 shows the application of the present invention within a general-purpose data processing system. With reference to FIG. 7, general-purpose data processing system  142  might include printer  144 , pointer  146 , and keyboard  148 , connected to south bridge  150  via bus  150   a . North bridge  156  is connected to south bridge  150  via bus  160   b,  memory  152  via bus  152   a,  graphics  154  via bus  160   a,  and processor  158  via bus  160 . Two devices on the bus  162  are shown employing an embodiment of the present invention. Encoded data bus  160  is shown between processor  158  and north bridge  156 . The present invention may be used in other locations within general-purpose data processing system  142 , for example bus  160   a  and  160   b  are examples of other locations in which the present invention could be employed. The present invention can be used in any situation in which data transmission occurs, the bus locations mentioned with respect to FIG. 7 are merely illustrative and are not to be construed in a limiting sense.  
         [0081]    [0081]FIG. 8 is a detail representation of two devices on a parallel data line bus employing the present invention as seen previously in FIG. 7. With reference to FIG. 8, two devices on the bus  162  are shown employing an embodiment of the present invention. Input data word  164  could be an 18-bit data word, as previously discussed, entering encoder  166  of device- 1   156 . Input data word  164  would be encoded by encoder  166  and be transmitted by transmitter  168  onto parallel encoded data line bus  170  to device- 2   158 , also connected with parallel encoded data line bus  170 . Device- 2   158  may have receiver  172  and decoder  174  configured to receive and decode the encoded data word, thus outputting the data word at data output  176 .  
         [0082]    Each of the devices may employ the reciprocal ability to both receive data words as input, encode, transmit onto the parallel data lines, decode and output the data word as shown in FIG. 8.  
         [0083]    [0083]FIG. 9 shows the termination of the encoded parallel data lines according to the present invention, where less than two lines are required to transmit one bit of information. With reference to FIG. 9, encoded line termination  178  is shown connecting transmit device  180  and receive device  182 . Encoded data line  184 ,  186 ,  188 , and  190  allow transmission of the encoded data word between transmit device  180  and receive device  182 . Line drivers  184   a,    186   a,    188   a,  and  190   a  drive the encoded data lines. As previously discussed in the embodiment of the present invention directed to the encoding of an 18-bit data word, 22 total encoded data lines were used. 22 encoded data lines represents four more lines than would be required by single-ended architecture and 14 less lines than would be required by differential architecture.  
         [0084]    [0084]FIG. 10 depicts a prior art differential architecture termination of parallel data lines, where two lines are necessary to transmit one bit of information. With reference to FIG. 10, differential line termination  192  shows the termination necessary for two lines to connect transmit device  194  and receive device  196 . Differential line  198  and differential line  200  are required to transmit one bit of information in a binary system between transmit device  194  and receive device  196 .  
         [0085]    [0085]FIG. 11 illustrates the best mode of the present invention applied to the specific case of encoding an 18-bit information word. With reference to FIG. 11, encoder  166  is shown in greater detail. Data word  202  is divided into sub-word a  204 , sub-word b  206 , and sub-word c  208 . Sub-word a  204  is encoded by encoder a  210  resulting in encoded sub-word a  212 . Sub-word a  204  includes input data lines Da 0  to Da 5 , encoded data sub-word a  212  includes encoded data lines Ea 0  to Ea 6 , thus six lines of input data are encoded onto seven encoded lines. Sub-word b  206  is encoded by encoder b  214  resulting in encoded sub-word b  216 . Sub-word b  206  includes input data lines Db 0  to Db 5 , encoded data sub-word b  216  includes encoded data lines Eb 0  to Eb 6 . Sub-word c  208  is encoded by encoder c  218  resulting in encoded sub-word c  220 . Sub-word c  208  includes input data lines Dc 0  to Dc 5 , encoded data sub-word c  220  includes encoded data lines Ec 0  to Ec 6 .  
         [0086]    In FIG. 11, a sub-word path may be conceptualized as the path taken by the data sub-word from data word  202  on the input side of the encoder to the output side of the encoder, where the encoded sub-words merge together to form encoded data word  227 . Shared information  222  flows between sub-word paths and parity logic  224 . Based on shared information  222 , parity logic  224  sets parity bit  226  to balance encoded data word  227 . Encoded sub-word a  212 , encoded sub-word b  216 , encoded sub-word c  218 , and parity bit  226  form encoded data word  227 .  
         [0087]    Alternative encoding, according to the best mode implementation for an 18-bit data word, requires shared information  222  to provide the value of the first and second MSB of sub-word a  204  and sub-word c  208  and the first MSB of sub-word b  206  on each sub-word path and at parity logic  224  to balance encoded data word  227 . Post Inversion (PI) was applied in two cases; case one is the situation where the first MSB of each sub-word equals zero (Da 5 , Db 5 , Dc 5 ) and the second MSB of sub-word a  204  equals zero (Da 4 ), then encoded sub-word b  216  and encoded sub-word c  220  are inverted; case two is the situation where the first MSB of each sub-word (Da 5 , Db 5 , Dc 5 ) equals one and the second MSB of sub-word c  208  equal one (Dc 5 ), then encoded sub-word a  212  and encoded sub-word b  216  are inverted.  
         [0088]    [0088]FIG. 12 is a further detail of the encoding applied to sub-word a  204  by sub-word encoder  210  according to the present invention. With reference to FIG. 12, sub-word a  204  is encoded in parallel by Ga encoder block  228 , Ha encoder block  230 , Fa encoder block  232 , Ja encoder block  234 , and Ka encoder block  236 . Shared information  222  a is used by, parity logic  224  to set the state of parity bit  226 , mux logic  240  to signal mux  242 , and inversion logic  238  to perform post inversion for the cases requiring alternate encoding. Mux  242  together with mux logic  238  selects the encoded sub-word from Ga encoder block  228 , Ha encoder block  230 , Fa encoder block  232 , ja encoder block  234 , or Ka encoder block  236  that is transmitted as encoded sub-word a  212 .  
         [0089]    [0089]FIG. 13 shows the truth tables employed by the encoder blocks shown in FIG. 12. With reference to FIG. 13, mux truth table  240   a  displays the logic used by mux logic  240  (FIG. 12). Mux truth table  240   a  includes determination of the special cases requiring alternative encoding, the first MSB of each sub-word (Da 5 , Db 5 , Dc 5 ) are considered in mux truth table  240   a.    
         [0090]    Distinct patterns between numbers of input least significant bits (LSBs) in the sub-words and numbers of LSBs in the encoded sub-words are used to form five unique patterns of encoding that are incorporated into the encoding blocks which have a measure of similarity across sub-words. The five distinct patterns result in Ga truth table  228   a,  Ha truth table  230   a,  Fa truth table  232   a,  Ja truth table  234   a,  and Ka truth table  236   a.  Ga truth table  228   a  is used by Ga encoder block  228  (FIG. 12). Ha Truth table  230   a  is used by Ha encoder block  230  (FIG. 12). Fa truth table  232   a  is used by Fa 1 ,  2  encoder block  232  (FIG. 12). These three encoder blocks map the two least significant bits LSBs of sub-word a  204  (FIG. 12) into the five LSBs of the encoded sub-word. The two MSBs of the encoded sub-word may be chosen by considering a combination of the bits in sub-word a  204  (FIG. 12) and shared information  222   a  (Db 5 , Dc 5 , Dc 4 ).  
         [0091]    Ja truth table  234   a  is used by Ja encoder block  234  (FIG. 12), and Ka truth table  236   a  is used by Ka encoder block  236  (FIG. 12). These two encoder blocks map the three LSBs of sub-word a  204  (FIG. 12) into the five LSBs of the encoded sub-word.  
         [0092]    Inversion logic truth table  238   a  is used by inversion logic  238  (FIG. 12) to invert encoded sub-word a  212  when alternate encoding is performed. It will be appreciated by those of skill in the art that post inversion may be performed after mux  242  (FIG. 12) as a particular application is considered. The present invention is not limited by the order of mux  242  and the inversion of encoded sub-word a  212 .  
         [0093]    [0093]FIG. 14 is a further detail of the encoding applied to sub-word b  206  by sub-word encoder  214  according to the present invention. With reference to FIG. 14, sub-word b  206  is encoded in parallel by Gb encoder block  244 , Hb encoder block  246 , Fb 1 ,  2  encoder block  248 , Jb encoder block  250 , and Kb encoder block  252 . Shared information  222   b  is used by mux logic  256  to signal mux  258  and inversion logic  254  to perform post inversion for the cases requiring alternate encoding. Mux  258  together with mux logic  256  selects the encoded sub-word from Gb encoder block  244 , Hb encoder block  246 , Fb 1 ,  2  encoder block  248 , Jb encoder block  250 , or Kb encoder block  252  that is transmitted as encoded sub-word b  216 .  
         [0094]    [0094]FIG. 15 shows the truth tables employed by the encoder blocks shown in FIG. 14. With reference to FIG. 15, mux truth table  256   b  displays the logic used by mux logic  256  (FIG. 14). Gb truth table  244   b  is used by Gb encoder block  244  (FIG. 14). Hb truth table  246   b  is use by Hb encoder block  246  (FIG. 14) and Fb truth table  248   b  is used by Fb 1 ,  2  encoder block  248  (FIG. 14). These three truth tables employ the same mapping between the two LSBs of the data sub-word and the five LSBs of the encoded sub-word as was used for sub-word a encoder blocks and truth tables.  
         [0095]    Jb truth table  250   b  is used by Jb encoder block  250  (FIG. 14) and Kb truth table  252   b  is used by Kb encoder block  252  (FIG. 14). These two truth tables employ the same mapping between the three LSBs of the data sub-word and the five LSBs of the encoded sub-word as was used for sub-word a encoder blocks and truth tables.  
         [0096]    Inversion logic truth table  254   b  is used by inversion logic  254  (FIG. 14) to invert encoded sub-word b  216  when alternate encoding is performed. It will be appreciated by those of skill in the art that post inversion may be performed after mux  258  (FIG. 14) as a particular application is considered. The present invention is not limited by the order of mux  258  and the inversion of encoded sub-word b  216 .  
         [0097]    [0097]FIG. 16 is a further detail of the encoding applied to sub-word c  208  by sub-word encoder  218  according to the present invention. With reference to FIG. 16, Sub-word c  208  is encoded in parallel by Gc encoder block  260 , Hc encoder block  262 , Fc 1 ,  2  encoder block  264 , Jc encoder block  266 , and Kc encoder block  268 . Shared information  222  c is used by mux logic  272  to signal mux  274  and inversion logic  270  to perform post inversion for the cases requiring alternate encoding. Mux  274  together with mux logic  272  selects the encoded sub-word from Gc encoder block  260 , Hc encoder block  262 , Fc 1 ,  2  encoder block  264 , Jc encoder block  266 , or Kc encoder block  268  that is transmitted as encoded sub-word c  220 .  
         [0098]    [0098]FIG. 17 shows the truth tables employed by the encoder blocks shown in FIG. 16. With reference to FIG. 17, mux truth table  272   c  displays the logic used by mux logic  270  (FIG. 16). Mux truth table  272  c includes determination of the special cases requiring alternative encoding, the first MSB of each sub-word (Da 5 , Db 5 , Dc 5 ) is considered in mux truth table  272   c.    
         [0099]    Gc truth table  260   c  is used by Gc encoder block  260  (FIG. 16). Hc truth table  262   c  is use by Hc encoder block  262  (FIG. 16) and Fc truth table  264   c  is used by encoder block Fc 1 ,  2   264  (FIG. 16). These three truth tables employ the same mapping between the two LSBs of the data sub-word and the five LSBs of the encoded sub-word as was used for sub-word a encoder blocks and truth tables.  
         [0100]    Jc truth table  266   c  is used by Jc encoder block  266  (FIG. 16) and Kc truth table  268   c  is used by Kc encoder block  268  (FIG. 16). These two truth tables employ the same mapping between the three LSBs of the data sub-word and the five LSBs of the encoded sub-word as was used for sub-word a encoder blocks and truth tables.  
         [0101]    Inversion logic truth table  270   c  is used by inversion logic  270  (FIG. 16) to invert encoded sub-word c  220  when alternate encoding is performed. It will be appreciated by those of skill in the art that post inversion may be performed after mux  274  (FIG. 16) as a particular application is considered. The present invention is not limited by the order of mux  274  and the inversion of encoded sub-word c  220 .  
         [0102]    [0102]FIG. 18 is a detail of the decoding applied to a generic sub-word x, where x refers to sub-word a, b, and c. It will be appreciated that the advantage taken of the similarity existing across sub-word encoding is also taken during encoded sub-word decoding. The five LSBs rendered by each encoder block (i.e., Ga encoder block  228 , Ha encoder block  230 , Fa encoder block  232 , Ja encoder block  234 , and Ka encoder block  236  (FIG. 12), that are used in each sub-word, are unique. This property is used for designing the decoding logic.  
         [0103]    Decoder  174  (FIG. 8) is shown in greater detail in FIG. 18. With reference to FIG. 18, when a valid five LSB line state is detected on one of the decoder blocks (G′ decode block  278 , H′ decode block  280 , F′ decode block  282 , J′ decode block  284 , or K′ decode block  286 ) it will be the only valid input line state over all decoder blocks. Detection of this valid input line state drives decode mux  292  which selects the decode block that produced the valid line state.  
         [0104]    [0104]FIG. 19 shows the truth tables employed by the decoder blocks shown in FIG. 18 for encoded sub-word (x=a)  276 . With reference to FIG. 19, reverse truth tables are created for decoding, G′ decode truth table  278   a  is used to decode the encoding performed by Ga encode truth table  228   a  (FIG. 13). H′ decode truth table  280   a  is used to decode the encoding performed by Ha encode truth table  230   a  (FIG. 13). F′ decode truth table  282   a  is used to decode the encoding performed by Fa encode truth table  232   a  (FIG. 13). J′ decode truth table  284   a  is used to decode the encoding performed by ja encode truth table  234   a  (FIG. 13). K′ decode truth table  286   a  is used to decode the encoding performed by Ka encode truth table  236   a  (FIG. 13). In a similar way decode truth tables are created for sub-words b and c based on the corresponding encoding truth tables.  
         [0105]    The conventions used in sub-word a decode truth table  296  (FIG. 19), sub-word b decode truth table  298  (FIG. 20), and sub-word c decode truth table  300  (FIG. 21) are as follows:  
                                                       A • B   Both A and B are true;           A + B   either A or B or both A and B are true;           xor   Only A or B are true, but not both A and B are true;           A —     Invert the value of A.                      
 
         [0106]    Returning to the example above, where mux  292  (FIG. 18) selected the appropriate decode block, for the given unique line state, the decoded sub-word is determined from the corresponding truth table entries using the conventions previously listed.  
         [0107]    For example, if the valid line state caused mux  292  (FIG. 18) to select the first entry in G′ decode truth table  278   a,  the decoded value returned would be “0 0 0 Ea 5   — 0 0, ” where Ea 5 _indicates the inverse of the encoded bit.  
         [0108]    The code mappings are chosen such that there is symmetry with respect to inversion. An encoded sub-word can be decoded as described or if the encoded sub-word is inverted it can be decoded as described and then inverted and the correct decoded sub-word will be obtained. Thus, “invert-decode-invert” yields the same result as “decode.” 
         [0109]    With reference to FIG. 19, inversion logic  290   a  inverts decoded sub-word (x=a)  294  (FIG. 18) when the weight of encoded sub-word c  220  (FIG. 16) equals five by signaling invert decoded result  290  (FIG. 18).  
         [0110]    [0110]FIG. 20 shows the truth tables employed by the decoder blocks shown in FIG. 18 for encoded sub-word b (x=b)  276 . With reference to FIG. 20, reverse truth tables are created for decoding, G′ decode truth table  278   b  is used to decode the encoding performed by Gb encode truth table  244   b  (FIG. 15). H′ decode truth table  280   b  is used to decode the encoding performed by Hb encode truth table  246   b  (FIG. 15). F′ decode truth table  282   b  is used to decode the encoding performed by Fb encode truth table  248   b  (FIG. 15). J′ decode truth table  284   b  is used to decode the encoding performed by Jb encode truth table  250   b  (FIG. 15). K′ decode truth table  286   b  is used to decode the encoding performed by Kb encode truth table  252   b  (FIG. 15).  
         [0111]    Inversion logic  290   b  inverts decoded sub-word (x=b)  294  (FIG. 18) when the weight of encoded sub-word c  220  (FIG. 16) equals five or if the weight of sub-word a  212  (FIG. 12) equals two.  
         [0112]    [0112]FIG. 21 shows the truth tables employed by the decoder blocks shown in FIG. 18 for encoded sub-word (x=c)  276 . In a similar way decode truth tables are created for sub-word c based on the corresponding encoding truth tables. With reference to FIG. 21, reverse truth tables are created for decoding, G′ decode truth table  278   c  is used to decode the encoding performed by Gc encode truth table  260   c  (FIG. 17). H′ decode truth table  280   c  is used to decode the encoding performed by Hc encode truth table  262   c  (FIG. 17). F′ decode truth table  282   c  is used to decode the encoding performed by Fc encode truth table  264   c  (FIG. 17). J′ decode truth table  284   c  is used to decode the encoding performed by Jc encode truth table  266   c  (FIG. 17). K′ decode truth table  286   c  is used to decode the encoding performed by Kc encode truth table  268   c  (FIG. 17).  
         [0113]    Inversion logic  290   c  inverts decoded sub-word (x=c)  294  (FIG. 18) when the weight of encoded sub-word a  212  (FIG. 12) equals two.  
         [0114]    [0114]FIG. 22 is an alternative embodiment for encoding binary numbers, with reference to FIG. 22, binomial coefficient matrix encoding  302  displays the results of binomial encoding, comparing decimal value  304 , binary value  306 , and encoded value  308  for integer numbers ranging from zero to 19.  
         [0115]    [0115]FIG. 23 is an alternative embodiment of encoding binary numbers using the principle of binomial expansion. With reference to  3102  of FIG. 23, the following recurrence relation hold true for all binomial coefficients. 
         ( n ) p =( n− 1) p +( n− 1) p−1 . 
         [0116]    n is the length of the encoded codeword and p is the number of ones in the encoded codeword. The recurrence relation partitions the span of an n-bit codeword weight p into two contiguous sub-ranges. The codebook interpretation is the following. The first sub-range are those codewords that have a zero in the n th  bit position and p ones in the remaining n−1 bit positions. There are (n−1) p  such codewords in the n-bit codeword of constant weight p ones in the remaining n−1 bit position. There are (n−1) p  such codewords in the n-bit codeword of constant weight p. The second range defines those codewords that have a one in the n th  bit position and p−1 ones in the remaining n−1 bit positions. There are (n−1) p−1  of these codewords in the n bit codeword of constant weight p.  
         [0117]    To convert the information content of a binary number to an n-bit codeword of constant weight p, the binary number is compared to (n−1) p . If it is smaller, then the n th  bit is set to zero. If it is greater, then the n th bit is set to one. The procedure is applied recursively with three restrictions.  
         [0118]    1. For the case where the n th  bit is set to one, the original binary number must be numerically adjusted downward into a number range for which it can be computed as an n−1 bit codeword of constant weight p−1. This numerical adjustment is done by subtracting (n−1) p .  
         [0119]    2. The algorithm does not work for the all-zero binary codeword. To compensate, all binary numbers can be increased by one, or the all-zero codeword can be mapped to an unused binary value such as (n) p+1    
         [0120]    3. 35 (n) 0  equal one for any non-negative integer value of n.  
         [0121]    4. (n) p  equals zero when p is larger than p.  
         [0122]    To speed up the conversion process, it is desirable to pre-compute the binomial coefficients and store them in a table or codebook such as the one shown in  310  of FIG. 23. To encode the 6-bit binary number 57 into a 8-bit constant-weight code of weight four, the process is as follows. The first step is to compare 58 to (7) 4  which is found in the table of FIG. 23 at  312  and returns the numerical value 35. The first bit is then set to one and the algorithm is applied recursive by comparing 58−35+23 to (6) 3  Note that the original number 58 is down shifted to 23, and the codeword length and number of ones remaining in the codeword are both reduced by one.  
         [0123]    The value of (6) 3  is looked up in the codebook at  320 . The numerical value is 20. The second bit is then set to one and the algorithm is applied recursive by comparing 23−20=3 to (5) 2 . The next three recursion are for (5) 2 , (4) 2 , and ( 3)   2 , found at locations  326 ,  330 , and  334  in the codebook. For each recursion, the values are larger than 3, so the next three bits are all zero. For (2) 2  at  338  in the codebook, the value is 1 which is less than 3. The new number is 3−1=2, the codeword length and the number of ones remaining in the codeword are reduced by one and the sixth bit is set to one. The next value to compare is (1), at  349  which is greater than 2. The new number is 2−1=1, the codeword length and the number of ones remaining in the codeword are reduced by one, and the seventh bit is set to one. The next value to compare is (0) 1  which is equal than 1. The eighth bit is set to zero and the algorithm stops. The conversion of the numerical value 58 results in a codeword of 11000110 as shown in  310   b  of FIG. 32. Note that this value equals the sum of the codebook entries when the number was greater than the codebook entry, 
         11000110=(7) 4 +(6) 3 +(2) 2 +(1) 1 =35+20+2+1=58. 
         [0124]    In the foregoing specification, the invention has been described with reference to specific embodiment thereof. It will be, however, evident that various modifications and changes may be made thereto without departing from the broader scope and spirit of the invention. The specification and drawings are, accordingly, to be regarded in an illustrative rather than restrictive sense.