Patent Publication Number: US-7908544-B2

Title: Extended convolutional codes

Description:
CROSS REFERENCE TO RELATED APPLICATION 
     This application claims the benefit of U.S. provisional application Ser. No. 60/575,318 filed May 28, 2004, and 60/582,612 filed Jun. 24, 2004 which is incorporated herein whole by reference. 
    
    
     FIELD 
     The invention relates to the field of error control coding for use in general communications, including but not limited to wire line, wireless, satellites, broadcasting, etc., and storage such as optical and magnetic, and many other fields. The present invention, in one particular aspect, relates to apparatuses and processes designed for use in wireless networks. More particularly, the present invention relates to techniques for modulation and coding, so that the data is protected to increase the coverage of a transmission system. 
     DESCRIPTION OF THE RELATED ART 
     In virtually all types of transmissions of data, coding is normally used to increase efficiency and provide error checking/error control. 
     Convolutional coding is a special case of error control-type coding. A quick review of convolutional coding reveals that in contrast with a block coder, which is a memory-less device, that uses fixed length blocks of information, a convolutional coder has a memory. In other words, the error coding performed in convolutional coding depends not only upon a set of input symbols being currently provided, but also takes into account some of the input symbols that were previously used and stored. The number of past information bits used is referred to as the constraint length. The more bits of past information (i.e. the higher is the value of a number representing the constraint length), the more efficient is the encoding. However, it is also true that as the number of bits of the constraint length increases, the more complex will be the decoder required for operation. Convolutional coding permits a fixed number of message symbols and permits an output of a fixed number of code symbols. 
     A convolutional encoder typically comprises modulo-2 adders and one or more delay elements.  FIG. 1   a  is an example of such. There are three memory registers  102 ,  104  and  106 . There is a first input bit Ui  108 , and three output bits (v 1 , v 2  and v 3  from modulo-adders  110 ,  112  and  114 ). As the number of output bits n=3, and the number of input bits k=1 (Ui  108 ), and the number of memory registers m=3 ( 102 ,  104 ,  106 ), the (n, k, m) parameters are (3, 1, 3). The k/n rate (input/output) is thus a ⅓ code rate. That means that each one input bit is coded into three output bits. Finally, the constraint length (L) is 2 (as U o  and U −1 ), which are also identified by shading, hold the past information bits. Thus the constraint length=2 L , and in this case equals four (2 2 ). When the input is one bit (k=1) code rates such as ⅓, ¼, ⅕, 1/9 can sometimes be called mother codes. Such single bit codes can be combined to produce punctured codes. Punctured codes are those with code rates other than 1/n. 
       FIG. 1   b  illustrates one example of error control coding by Shu Lin. The encoder  101  shown takes k-bit input information sequence U  103  and produces an n-bit sequence, which is codeword v  105 . Each encoded sequence v depends not only on the corresponding information u that is currently inputted, but also on m-1 previous message blocks (as the convolutional encoder has/accesses a memory). Hence, the encoder has a constraint length of m. In addition, a k-input, n-output encoder is also referred to as a rate k/n encoder, or a k-by-n matrix. 
     For example, a ½ rate encoder with generator of 133 8  and 171 8  and having a constraint length of 7 (shown in  FIG. 1   b ) is depicted with the following bit values so that 133 8  (1 011 011 in binary) and 171 8  (1 110 001 in binary) shown in  FIG. 1 . The binary numbers representing 133 8  and 171 8  are shown in  FIG. 1  as being arranged above and below the register memory boxes  107 , respectively. The two binary values can be considered a first output and a second output, respectively, of the encoder. 
     Coding is used by many protocols, including but not limited to the IEEE 802.11 standard of wireless transmission that has become extremely popular. 
     However, there still exists a need to improve the convolutional coding, in particular for use with the wire line transmission, satellite, broadcast, optical, and other types of transmission. In addition, for example, new “superframes” are disclosed in Philips, N. A. disclosure 779763/4 that comprise a plurality of packets, but are backward-compatible to permit the continued use of existing legacy devices. As there are many standard codes previously published for different rates, these new superframes, as well as new systems, cannot adopt a complete new code due to the need for backward compatibility. 
     BRIEF SUMMARY 
     The instantly claimed invention provides a method, system and apparatus for providing extended convolutional codes wherein a new code having one or more higher rates, after puncturing, that is backward compatible because it is extended from that existing code. The new code bears the minimum bit error rate given the generators operating at the higher rate. In addition, in one aspect of the invention, an array of services in the physical layer that are tailored for a wireless environment are provided so as to increase the coverage. For example, multiple concurrent services are enabled for scenarios, such as transmitting video contents and remote control information simultaneously that can take full advantage of extended convolutional codes to extend their wireless system coverage. 
    
    
     
       BRIEF DESCRIPTION 
         FIGS. 1   a  and  1   b  provide illustrations of convolutional encoders. 
         FIG. 2  provides a flowchart overview of a method according to the present invention. 
         FIG. 3  is a graphical illustration of comparative Bit error rates (BER) for different convolutional encoders. 
         FIG. 4  is an illustration providing detail about one possible arrangement of a system according to the present invention. 
     
    
    
     DETAILED DESCRIPTION 
     It is to be understood by persons of ordinary skill in the art that the following descriptions are provided for purposes of illustration, not for limitation. An artisan understands that there are many variations that lie within the spirit of the invention and the scope of the appended claims. Unnecessary details of known functions and operations may be omitted from the current description so as not to obscure the finer points of the present invention. 
     With regard to the present invention, according to the steps at least similar to those discussed in the flowchart, infra, the convolutional code is extended, which is heretofore unknown. For example, a ½ rate has 2 generators, and a ¼ rate has 4 generators. According to the present invention, for example, the code can be extended from 2 generators to 4 generators (or more, for example 2 generators to 50). It is to be stressed that the invention is not limited to a two generator to four generator extension, and covers extensions of any number to any larger number. 
     For further explanatory purposes,  FIG. 2  provides a flowchart illustrating one way the present invention may operate, and in order to follow the method more closely, TABLE 1 provides the data from a simulation to extend the codes according to the present invention. Thus, the method begins with a ½ convolutional code rate (CCR) (133, 171) with a constraint length m=7 that is extended to find a new convolutional code CC having a ¼ CCR (133, 171, xxx, xxx). The convolution code rate refers to the data compression achieved over a fixed length code. In addition, Table 1, which includes a listing of all the codes found, including the BER performance at SNR=0 db of the best extended convolutional code (133, 171, 135, 175) as compared with the optimal code of a known system (135, 135, 147, 164) at index 85. 
     The BER@SNR error count means, for example, in the case of BER6.40e-5@104 that the simulation has collected 104 errors and is showing the bit error rate of 6.4e-5. 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 Simulation Results 
               
            
           
           
               
               
               
               
            
               
                   
                   
                   
                 BER 
               
               
                   
                 # of paths w/ 
                   
                 @ 
               
               
                   
                 d_free = 20 at 
                   
                 SNR = 0 
               
               
                   
                 different 
                   
                 dB (ber 
               
               
                   
                 trace length 
                 # of paths @ different d_free 
                 @ err 
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 INDEX 
                 Generator (K = 7) 
                 7 
                 8 
                 9 
                 10 
                 11 
                 20 
                 21 
                 22 
                 23 
                 24 
                 25 
                 26 
                 27 
                 28 
                 29 
                 30 
                 31 
                 32 
                 cnt) 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 1 
                 133 
                 171 
                 135 
                 175 
                 0 
                 1 
                 3 
                 5 
                 7 
                 16 
                 22 
                 3 
                 7 
                 2 
                 0 
                 3 
                 1 
                 0 
                 1 
                 2 
                 0 
                 0 
                   
               
               
                 2 
                 133 
                 171 
                 175 
                 135 
                 0 
                 1 
                 3 
                 5 
                 7 
                 16 
                 22 
                 3 
                 7 
                 2 
                 0 
                 3 
                 1 
                 0 
                 1 
                 2 
                 0 
                 0 
               
               
                 3 
                 133 
                 171 
                 127 
                 175 
                 0 
                 1 
                 3 
                 5 
                 7 
                 16 
                 24 
                 0 
                 2 
                 6 
                 3 
                 3 
                 2 
                 0 
                 0 
                 1 
                 0 
                 0 
                 6.40e−5 @ 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 104 
               
               
                 4 
                 133 
                 171 
                 137 
                 165 
                 0 
                 1 
                 3 
                 5 
                 7 
                 16 
                 24 
                 0 
                 2 
                 6 
                 3 
                 3 
                 2 
                 0 
                 0 
                 1 
                 0 
                 0 
               
               
                 5 
                 133 
                 171 
                 165 
                 137 
                 0 
                 1 
                 3 
                 5 
                 7 
                 16 
                 24 
                 0 
                 2 
                 6 
                 3 
                 3 
                 2 
                 0 
                 0 
                 1 
                 0 
                 0 
               
               
                 6 
                 133 
                 171 
                 175 
                 127 
                 0 
                 1 
                 3 
                 5 
                 7 
                 16 
                 24 
                 0 
                 2 
                 6 
                 3 
                 3 
                 2 
                 0 
                 0 
                 1 
                 0 
                 0 
               
               
                 7 
                 133 
                 171 
                 123 
                 175 
                 1 
                 3 
                 5 
                 7 
                 9 
                 25 
                 0 
                 19 
                 0 
                 9 
                 0 
                 0 
                 0 
                 3 
                 0 
                 1 
                 0 
                 0 
               
               
                 8 
                 133 
                 171 
                 175 
                 123 
                 1 
                 3 
                 5 
                 7 
                 9 
                 25 
                 0 
                 19 
                 0 
                 9 
                 0 
                 0 
                 0 
                 3 
                 0 
                 1 
                 0 
                 0 
               
               
                 9 
                 133 
                 171 
                 117 
                 165 
                 1 
                 3 
                 5 
                 7 
                 9 
                 25 
                 0 
                 20 
                 0 
                 7 
                 0 
                 2 
                 0 
                 1 
                 0 
                 2 
                 0 
                 0 
                 6.42e−2 @ 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 102 
               
               
                 10 
                 133 
                 171 
                 165 
                 117 
                 1 
                 3 
                 5 
                 7 
                 9 
                 25 
                 0 
                 20 
                 0 
                 7 
                 0 
                 2 
                 0 
                 1 
                 0 
                 2 
                 0 
                 0 
               
               
                 11 
                 133 
                 171 
                 145 
                 175 
                 1 
                 3 
                 5 
                 7 
                 10 
                 26 
                 0 
                 19 
                 0 
                 8 
                 0 
                 0 
                 0 
                 2 
                 0 
                 1 
                 0 
                 1 
               
               
                 12 
                 133 
                 171 
                 175 
                 145 
                 1 
                 3 
                 5 
                 7 
                 10 
                 26 
                 0 
                 19 
                 0 
                 8 
                 0 
                 0 
                 0 
                 2 
                 0 
                 1 
                 0 
                 1 
               
               
                 13 
                 133 
                 171 
                 135 
                 165 
                 1 
                 2 
                 5 
                 8 
                 11 
                 27 
                 0 
                 17 
                 0 
                 6 
                 0 
                 3 
                 0 
                 3 
                 0 
                 1 
                 0 
                 0 
               
               
                 14 
                 133 
                 171 
                 165 
                 135 
                 1 
                 2 
                 5 
                 8 
                 11 
                 27 
                 0 
                 17 
                 0 
                 6 
                 0 
                 3 
                 0 
                 3 
                 0 
                 1 
                 0 
                 0 
               
               
                 15 
                 133 
                 171 
                 137 
                 151 
                 1 
                 3 
                 5 
                 8 
                 11 
                 28 
                 0 
                 15 
                 0 
                 7 
                 0 
                 5 
                 0 
                 1 
                 0 
                 0 
                 0 
                 1 
               
               
                 16 
                 133 
                 171 
                 151 
                 137 
                 1 
                 3 
                 5 
                 8 
                 11 
                 28 
                 0 
                 15 
                 0 
                 7 
                 0 
                 5 
                 0 
                 1 
                 0 
                 0 
                 0 
                 1 
               
               
                 17 
                 133 
                 171 
                 117 
                 135 
                 1 
                 3 
                 5 
                 8 
                 11 
                 28 
                 0 
                 16 
                 0 
                 6 
                 0 
                 3 
                 0 
                 3 
                 0 
                 1 
                 0 
                 0 
               
               
                 18 
                 133 
                 171 
                 125 
                 157 
                 1 
                 2 
                 5 
                 8 
                 12 
                 28 
                 0 
                 16 
                 0 
                 5 
                 0 
                 5 
                 0 
                 2 
                 0 
                 0 
                 0 
                 1 
               
               
                 19 
                 133 
                 171 
                 135 
                 117 
                 1 
                 3 
                 5 
                 8 
                 11 
                 28 
                 0 
                 16 
                 0 
                 6 
                 0 
                 3 
                 0 
                 3 
                 0 
                 1 
                 0 
                 0 
               
               
                 20 
                 133 
                 171 
                 157 
                 125 
                 1 
                 2 
                 5 
                 8 
                 12 
                 28 
                 0 
                 16 
                 0 
                 5 
                 0 
                 5 
                 0 
                 2 
                 0 
                 0 
                 0 
                 1 
               
               
                 21 
                 133 
                 171 
                 115 
                 137 
                 1 
                 3 
                 5 
                 8 
                 11 
                 28 
                 0 
                 17 
                 0 
                 3 
                 0 
                 6 
                 0 
                 2 
                 0 
                 1 
                 0 
                 0 
               
               
                 22 
                 133 
                 171 
                 115 
                 175 
                 1 
                 3 
                 5 
                 8 
                 11 
                 28 
                 0 
                 17 
                 0 
                 3 
                 0 
                 6 
                 0 
                 2 
                 0 
                 1 
                 0 
                 0 
               
               
                 23 
                 133 
                 171 
                 125 
                 173 
                 1 
                 2 
                 5 
                 8 
                 12 
                 28 
                 0 
                 17 
                 0 
                 5 
                 0 
                 3 
                 0 
                 2 
                 0 
                 1 
                 0 
                 1 
               
               
                 24 
                 133 
                 171 
                 137 
                 115 
                 1 
                 3 
                 5 
                 8 
                 11 
                 28 
                 0 
                 17 
                 0 
                 3 
                 0 
                 6 
                 0 
                 2 
                 0 
                 1 
                 0 
                 0 
               
               
                 25 
                 133 
                 171 
                 173 
                 125 
                 1 
                 2 
                 5 
                 8 
                 12 
                 28 
                 0 
                 17 
                 0 
                 5 
                 0 
                 3 
                 0 
                 2 
                 0 
                 1 
                 0 
                 1 
               
               
                 26 
                 133 
                 171 
                 175 
                 115 
                 1 
                 3 
                 5 
                 8 
                 11 
                 28 
                 0 
                 17 
                 0 
                 3 
                 0 
                 6 
                 0 
                 2 
                 0 
                 1 
                 0 
                 0 
               
               
                 27 
                 133 
                 171 
                 123 
                 137 
                 1 
                 3 
                 5 
                 8 
                 11 
                 28 
                 0 
                 19 
                 0 
                 3 
                 0 
                 0 
                 0 
                 6 
                 0 
                 1 
                 0 
                 0 
               
               
                 28 
                 133 
                 171 
                 137 
                 123 
                 1 
                 3 
                 5 
                 8 
                 11 
                 28 
                 0 
                 19 
                 0 
                 3 
                 0 
                 0 
                 0 
                 6 
                 0 
                 1 
                 0 
                 0 
               
               
                 29 
                 133 
                 171 
                 137 
                 145 
                 1 
                 3 
                 5 
                 8 
                 11 
                 28 
                 0 
                 19 
                 0 
                 3 
                 0 
                 0 
                 0 
                 6 
                 0 
                 1 
                 0 
                 0 
               
               
                 30 
                 133 
                 171 
                 145 
                 137 
                 1 
                 3 
                 5 
                 8 
                 11 
                 28 
                 0 
                 19 
                 0 
                 3 
                 0 
                 0 
                 0 
                 6 
                 0 
                 1 
                 0 
                 0 
               
               
                 31 
                 133 
                 171 
                 113 
                 175 
                 1 
                 3 
                 5 
                 8 
                 12 
                 29 
                 0 
                 15 
                 0 
                 5 
                 0 
                 5 
                 0 
                 2 
                 0 
                 0 
                 0 
                 1 
               
               
                 32 
                 133 
                 171 
                 151 
                 175 
                 1 
                 3 
                 5 
                 8 
                 12 
                 29 
                 0 
                 15 
                 0 
                 5 
                 0 
                 5 
                 0 
                 2 
                 0 
                 0 
                 0 
                 1 
               
               
                 33 
                 133 
                 171 
                 175 
                 113 
                 1 
                 3 
                 5 
                 8 
                 12 
                 29 
                 0 
                 15 
                 0 
                 5 
                 0 
                 5 
                 0 
                 2 
                 0 
                 0 
                 0 
                 1 
               
               
                 34 
                 133 
                 171 
                 175 
                 151 
                 1 
                 3 
                 5 
                 8 
                 12 
                 29 
                 0 
                 15 
                 0 
                 5 
                 0 
                 5 
                 0 
                 2 
                 0 
                 0 
                 0 
                 1 
               
               
                 35 
                 133 
                 171 
                 135 
                 171 
                 1 
                 3 
                 5 
                 8 
                 12 
                 29 
                 0 
                 16 
                 0 
                 4 
                 0 
                 3 
                 0 
                 4 
                 0 
                 1 
                 0 
                 0 
               
               
                 36 
                 133 
                 171 
                 171 
                 135 
                 1 
                 3 
                 5 
                 8 
                 12 
                 29 
                 0 
                 16 
                 0 
                 4 
                 0 
                 3 
                 0 
                 4 
                 0 
                 1 
                 0 
                 0 
               
               
                 37 
                 133 
                 171 
                 127 
                 165 
                 1 
                 2 
                 5 
                 9 
                 13 
                 30 
                 0 
                 10 
                 0 
                 9 
                 0 
                 5 
                 0 
                 3 
                 0 
                 0 
                 0 
                 0 
               
               
                 38 
                 133 
                 171 
                 165 
                 127 
                 1 
                 2 
                 5 
                 9 
                 13 
                 30 
                 0 
                 10 
                 0 
                 9 
                 0 
                 5 
                 0 
                 3 
                 0 
                 0 
                 0 
                 0 
               
               
                 39 
                 133 
                 171 
                 125 
                 167 
                 1 
                 2 
                 5 
                 9 
                 13 
                 30 
                 0 
                 12 
                 0 
                 6 
                 0 
                 6 
                 0 
                 2 
                 0 
                 1 
                 0 
                 0 
               
               
                 40 
                 133 
                 171 
                 167 
                 125 
                 1 
                 2 
                 5 
                 9 
                 13 
                 30 
                 0 
                 12 
                 0 
                 6 
                 0 
                 6 
                 0 
                 2 
                 0 
                 1 
                 0 
                 0 
               
               
                 41 
                 133 
                 171 
                 135 
                 135 
                 1 
                 2 
                 5 
                 9 
                 13 
                 30 
                 0 
                 15 
                 0 
                 5 
                 0 
                 0 
                 0 
                 6 
                 0 
                 0 
                 0 
                 1 
               
               
                 42 
                 133 
                 171 
                 115 
                 167 
                 1 
                 3 
                 6 
                 9 
                 12 
                 31 
                 0 
                 9 
                 0 
                 8 
                 0 
                 7 
                 0 
                 2 
                 0 
                 0 
                 0 
                 0 
               
               
                 43 
                 133 
                 171 
                 167 
                 115 
                 1 
                 3 
                 6 
                 9 
                 12 
                 31 
                 0 
                 9 
                 0 
                 8 
                 0 
                 7 
                 0 
                 2 
                 0 
                 0 
                 0 
                 0 
               
               
                 44 
                 133 
                 171 
                 147 
                 165 
                 1 
                 3 
                 6 
                 9 
                 12 
                 31 
                 0 
                 10 
                 0 
                 8 
                 0 
                 5 
                 0 
                 2 
                 0 
                 1 
                 0 
                 0 
               
               
                 45 
                 133 
                 171 
                 151 
                 167 
                 1 
                 3 
                 6 
                 9 
                 12 
                 31 
                 0 
                 10 
                 0 
                 8 
                 0 
                 5 
                 0 
                 2 
                 0 
                 1 
                 0 
                 0 
               
               
                 46 
                 133 
                 171 
                 163 
                 165 
                 1 
                 3 
                 6 
                 9 
                 12 
                 31 
                 0 
                 10 
                 0 
                 8 
                 0 
                 5 
                 0 
                 2 
                 0 
                 1 
                 0 
                 0 
               
               
                 47 
                 133 
                 171 
                 165 
                 147 
                 1 
                 3 
                 6 
                 9 
                 12 
                 31 
                 0 
                 10 
                 0 
                 8 
                 0 
                 5 
                 0 
                 2 
                 0 
                 1 
                 0 
                 0 
               
               
                 48 
                 133 
                 171 
                 165 
                 163 
                 1 
                 3 
                 6 
                 9 
                 12 
                 31 
                 0 
                 10 
                 0 
                 8 
                 0 
                 5 
                 0 
                 2 
                 0 
                 1 
                 0 
                 0 
               
               
                 49 
                 133 
                 171 
                 165 
                 165 
                 1 
                 2 
                 5 
                 9 
                 14 
                 31 
                 0 
                 10 
                 0 
                 8 
                 0 
                 5 
                 0 
                 2 
                 0 
                 0 
                 0 
                 1 
               
               
                 50 
                 133 
                 171 
                 167 
                 151 
                 1 
                 3 
                 6 
                 9 
                 12 
                 31 
                 0 
                 10 
                 0 
                 8 
                 0 
                 5 
                 0 
                 2 
                 0 
                 1 
                 0 
                 0 
               
               
                 51 
                 133 
                 171 
                 117 
                 153 
                 1 
                 3 
                 6 
                 9 
                 12 
                 31 
                 0 
                 11 
                 0 
                 8 
                 0 
                 4 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
               
               
                 52 
                 133 
                 171 
                 123 
                 157 
                 1 
                 3 
                 6 
                 9 
                 12 
                 31 
                 0 
                 11 
                 0 
                 7 
                 0 
                 4 
                 0 
                 3 
                 0 
                 1 
                 0 
                 0 
               
               
                 53 
                 133 
                 171 
                 123 
                 173 
                 1 
                 3 
                 6 
                 9 
                 12 
                 31 
                 0 
                 11 
                 0 
                 8 
                 0 
                 4 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
               
               
                 54 
                 133 
                 171 
                 145 
                 157 
                 1 
                 3 
                 6 
                 9 
                 12 
                 31 
                 0 
                 11 
                 0 
                 8 
                 0 
                 4 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
               
               
                 55 
                 133 
                 171 
                 153 
                 117 
                 1 
                 3 
                 6 
                 9 
                 12 
                 31 
                 0 
                 11 
                 0 
                 8 
                 0 
                 4 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
               
               
                 56 
                 133 
                 171 
                 157 
                 123 
                 1 
                 3 
                 6 
                 9 
                 12 
                 31 
                 0 
                 11 
                 0 
                 7 
                 0 
                 4 
                 0 
                 3 
                 0 
                 1 
                 0 
                 0 
               
               
                 57 
                 133 
                 171 
                 157 
                 145 
                 1 
                 3 
                 6 
                 9 
                 12 
                 31 
                 0 
                 11 
                 0 
                 8 
                 0 
                 4 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
               
               
                 58 
                 133 
                 171 
                 173 
                 123 
                 1 
                 3 
                 6 
                 9 
                 12 
                 31 
                 0 
                 11 
                 0 
                 8 
                 0 
                 4 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
               
               
                 59 
                 133 
                 171 
                 145 
                 173 
                 1 
                 3 
                 6 
                 9 
                 13 
                 32 
                 0 
                 11 
                 0 
                 6 
                 0 
                 4 
                 0 
                 2 
                 0 
                 1 
                 0 
                 1 
               
               
                 60 
                 133 
                 171 
                 173 
                 145 
                 1 
                 3 
                 6 
                 9 
                 13 
                 32 
                 0 
                 11 
                 0 
                 6 
                 0 
                 4 
                 0 
                 2 
                 0 
                 1 
                 0 
                 1 
               
               
                 61 
                 133 
                 171 
                 123 
                 167 
                 1 
                 3 
                 6 
                 10 
                 14 
                 34 
                 0 
                 0 
                 0 
                 19 
                 0 
                 0 
                 0 
                 4 
                 0 
                 0 
                 0 
                 0 
               
               
                 62 
                 133 
                 171 
                 135 
                 147 
                 1 
                 3 
                 6 
                 10 
                 14 
                 34 
                 0 
                 0 
                 0 
                 19 
                 0 
                 0 
                 0 
                 4 
                 0 
                 0 
                 0 
                 0 
               
               
                 63 
                 133 
                 171 
                 135 
                 163 
                 1 
                 3 
                 6 
                 10 
                 14 
                 34 
                 0 
                 0 
                 0 
                 19 
                 0 
                 0 
                 0 
                 4 
                 0 
                 0 
                 0 
                 0 
               
               
                 64 
                 133 
                 171 
                 145 
                 167 
                 1 
                 3 
                 6 
                 10 
                 14 
                 34 
                 0 
                 0 
                 0 
                 20 
                 0 
                 0 
                 0 
                 2 
                 0 
                 0 
                 0 
                 1 
               
               
                 65 
                 133 
                 171 
                 147 
                 135 
                 1 
                 3 
                 6 
                 10 
                 14 
                 34 
                 0 
                 0 
                 0 
                 19 
                 0 
                 0 
                 0 
                 4 
                 0 
                 0 
                 0 
                 0 
               
               
                 66 
                 133 
                 171 
                 163 
                 135 
                 1 
                 3 
                 6 
                 10 
                 14 
                 34 
                 0 
                 0 
                 0 
                 19 
                 0 
                 0 
                 0 
                 4 
                 0 
                 0 
                 0 
                 0 
               
               
                 67 
                 133 
                 171 
                 167 
                 123 
                 1 
                 3 
                 6 
                 10 
                 14 
                 34 
                 0 
                 0 
                 0 
                 19 
                 0 
                 0 
                 0 
                 4 
                 0 
                 0 
                 0 
                 0 
               
               
                 68 
                 133 
                 171 
                 167 
                 145 
                 1 
                 3 
                 6 
                 10 
                 14 
                 34 
                 0 
                 0 
                 0 
                 20 
                 0 
                 0 
                 0 
                 2 
                 0 
                 0 
                 0 
                 1 
               
               
                 69 
                 133 
                 171 
                 115 
                 157 
                 1 
                 3 
                 6 
                 10 
                 14 
                 34 
                 0 
                 2 
                 0 
                 15 
                 0 
                 4 
                 0 
                 0 
                 0 
                 2 
                 0 
                 0 
               
               
                 70 
                 133 
                 171 
                 115 
                 173 
                 1 
                 3 
                 6 
                 10 
                 14 
                 34 
                 0 
                 2 
                 0 
                 15 
                 0 
                 4 
                 0 
                 0 
                 0 
                 2 
                 0 
                 0 
               
               
                 71 
                 133 
                 171 
                 157 
                 115 
                 1 
                 3 
                 6 
                 10 
                 14 
                 34 
                 0 
                 2 
                 0 
                 15 
                 0 
                 4 
                 0 
                 0 
                 0 
                 2 
                 0 
                 0 
               
               
                 72 
                 133 
                 171 
                 173 
                 115 
                 1 
                 3 
                 6 
                 10 
                 14 
                 34 
                 0 
                 2 
                 0 
                 15 
                 0 
                 4 
                 0 
                 0 
                 0 
                 2 
                 0 
                 0 
               
               
                 73 
                 133 
                 171 
                 117 
                 155 
                 1 
                 3 
                 6 
                 10 
                 15 
                 35 
                 0 
                 0 
                 0 
                 18 
                 0 
                 0 
                 0 
                 3 
                 0 
                 0 
                 0 
                 1 
               
               
                 74 
                 133 
                 171 
                 151 
                 173 
                 1 
                 3 
                 6 
                 10 
                 15 
                 35 
                 0 
                 0 
                 0 
                 18 
                 0 
                 0 
                 0 
                 3 
                 0 
                 0 
                 0 
                 1 
               
               
                 75 
                 133 
                 171 
                 155 
                 117 
                 1 
                 3 
                 6 
                 10 
                 15 
                 35 
                 0 
                 0 
                 0 
                 18 
                 0 
                 0 
                 0 
                 3 
                 0 
                 0 
                 0 
                 1 
               
               
                 76 
                 133 
                 171 
                 173 
                 151 
                 1 
                 3 
                 6 
                 10 
                 15 
                 35 
                 0 
                 0 
                 0 
                 18 
                 0 
                 0 
                 0 
                 3 
                 0 
                 0 
                 0 
                 1 
               
               
                 77 
                 133 
                 171 
                 155 
                 171 
                 1 
                 3 
                 6 
                 10 
                 16 
                 36 
                 0 
                 0 
                 0 
                 17 
                 0 
                 0 
                 0 
                 2 
                 0 
                 0 
                 0 
                 2 
               
               
                 78 
                 133 
                 171 
                 171 
                 155 
                 1 
                 3 
                 6 
                 10 
                 16 
                 36 
                 0 
                 0 
                 0 
                 17 
                 0 
                 0 
                 0 
                 2 
                 0 
                 0 
                 0 
                 2 
               
               
                 79 
                 133 
                 171 
                 113 
                 157 
                 1 
                 3 
                 6 
                 11 
                 16 
                 37 
                 0 
                 0 
                 0 
                 14 
                 0 
                 0 
                 0 
                 5 
                 0 
                 0 
                 0 
                 1 
               
               
                 80 
                 133 
                 171 
                 157 
                 113 
                 1 
                 3 
                 6 
                 11 
                 16 
                 37 
                 0 
                 0 
                 0 
                 14 
                 0 
                 0 
                 0 
                 5 
                 0 
                 0 
                 0 
                 1 
               
               
                 81 
                 133 
                 171 
                 65 
                 137 
                 1 
                 3 
                 7 
                 11 
                 15 
                 37 
                 0 
                 10 
                 0 
                 3 
                 0 
                 5 
                 0 
                 1 
                 0 
                 1 
                 0 
                 0 
               
               
                 82 
                 133 
                 171 
                 137 
                 65 
                 1 
                 3 
                 7 
                 11 
                 15 
                 37 
                 0 
                 10 
                 0 
                 3 
                 0 
                 5 
                 0 
                 1 
                 0 
                 1 
                 0 
                 0 
               
               
                 83 
                 133 
                 171 
                 137 
                 152 
                 1 
                 3 
                 7 
                 11 
                 15 
                 37 
                 0 
                 10 
                 0 
                 3 
                 0 
                 5 
                 0 
                 1 
                 0 
                 1 
                 0 
                 0 
               
               
                 84 
                 133 
                 171 
                 152 
                 137 
                 1 
                 3 
                 7 
                 11 
                 15 
                 37 
                 0 
                 10 
                 0 
                 3 
                 0 
                 5 
                 0 
                 1 
                 0 
                 1 
                 0 
                 0 
               
               
                 85 
                 133 
                 171 
                 113 
                 173 
                 1 
                 3 
                 6 
                 11 
                 17 
                 38 
                 0 
                 0 
                 0 
                 13 
                 0 
                 0 
                 0 
                 4 
                 0 
                 0 
                 0 
                 2 
               
               
                 86 
                 133 
                 171 
                 117 
                 133 
                 1 
                 3 
                 6 
                 11 
                 17 
                 38 
                 0 
                 0 
                 0 
                 13 
                 0 
                 0 
                 0 
                 4 
                 0 
                 0 
                 0 
                 2 
               
               
                 87 
                 133 
                 171 
                 133 
                 117 
                 1 
                 3 
                 6 
                 11 
                 17 
                 38 
                 0 
                 0 
                 0 
                 13 
                 0 
                 0 
                 0 
                 4 
                 0 
                 0 
                 0 
                 2 
               
               
                 88 
                 133 
                 171 
                 173 
                 113 
                 1 
                 3 
                 6 
                 11 
                 17 
                 38 
                 0 
                 0 
                 0 
                 13 
                 0 
                 0 
                 0 
                 4 
                 0 
                 0 
                 0 
                 2 
               
               
                 89 
                 133 
                 171 
                 53 
                 175 
                 1 
                 3 
                 7 
                 11 
                 16 
                 38 
                 0 
                 10 
                 0 
                 1 
                 0 
                 5 
                 0 
                 2 
                 0 
                 1 
                 0 
                 0 
               
               
                 90 
                 133 
                 171 
                 65 
                 175 
                 1 
                 3 
                 7 
                 11 
                 16 
                 38 
                 0 
                 10 
                 0 
                 1 
                 0 
                 5 
                 0 
                 2 
                 0 
                 1 
                 0 
                 0 
               
               
                 91 
                 133 
                 171 
                 75 
                 135 
                 1 
                 3 
                 7 
                 11 
                 16 
                 38 
                 0 
                 10 
                 0 
                 2 
                 0 
                 4 
                 0 
                 1 
                 0 
                 2 
                 0 
                 0 
               
               
                 92 
                 133 
                 171 
                 126 
                 175 
                 1 
                 3 
                 7 
                 11 
                 16 
                 38 
                 0 
                 10 
                 0 
                 1 
                 0 
                 5 
                 0 
                 2 
                 0 
                 1 
                 0 
                 0 
               
               
                 93 
                 133 
                 171 
                 135 
                 75 
                 1 
                 3 
                 7 
                 11 
                 16 
                 38 
                 0 
                 10 
                 0 
                 2 
                 0 
                 4 
                 0 
                 1 
                 0 
                 2 
                 0 
                 0 
               
               
                 94 
                 133 
                 171 
                 135 
                 172 
                 1 
                 3 
                 7 
                 11 
                 16 
                 38 
                 0 
                 10 
                 0 
                 2 
                 0 
                 4 
                 0 
                 1 
                 0 
                 2 
                 0 
                 0 
               
               
                 95 
                 133 
                 171 
                 152 
                 175 
                 1 
                 3 
                 7 
                 11 
                 16 
                 38 
                 0 
                 10 
                 0 
                 1 
                 0 
                 5 
                 0 
                 2 
                 0 
                 1 
                 0 
                 0 
               
               
                 96 
                 133 
                 171 
                 172 
                 135 
                 1 
                 3 
                 7 
                 11 
                 16 
                 38 
                 0 
                 10 
                 0 
                 2 
                 0 
                 4 
                 0 
                 1 
                 0 
                 2 
                 0 
                 0 
               
               
                 97 
                 133 
                 171 
                 175 
                 53 
                 1 
                 3 
                 7 
                 11 
                 16 
                 38 
                 0 
                 10 
                 0 
                 1 
                 0 
                 5 
                 0 
                 2 
                 0 
                 1 
                 0 
                 0 
               
               
                 98 
                 133 
                 171 
                 175 
                 65 
                 1 
                 3 
                 7 
                 11 
                 16 
                 38 
                 0 
                 10 
                 0 
                 1 
                 0 
                 5 
                 0 
                 2 
                 0 
                 1 
                 0 
                 0 
               
               
                 99 
                 133 
                 171 
                 175 
                 126 
                 1 
                 3 
                 7 
                 11 
                 16 
                 38 
                 0 
                 10 
                 0 
                 1 
                 0 
                 5 
                 0 
                 2 
                 0 
                 1 
                 0 
                 0 
               
               
                 100 
                 133 
                 171 
                 175 
                 152 
                 1 
                 3 
                 7 
                 11 
                 16 
                 38 
                 0 
                 10 
                 0 
                 1 
                 0 
                 5 
                 0 
                 2 
                 0 
                 1 
                 0 
                 0 
               
               
                 101 
                 135 
                 135 
                 147 
                 163 
                 1 
                 3 
                 6 
                 11 
                 18 
                 39 
                 0 
                 0 
                 0 
                 10 
                 0 
                 0 
                 0 
                 7 
                 0 
                 0 
                 0 
                 1 
                 2.6e−4 @ 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 108 
               
               
                 102 
                 133 
                 171 
                 133 
                 171 
                 1 
                 3 
                 6 
                 11 
                 18 
                 39 
                 0 
                 0 
                 0 
                 10 
                 0 
                 0 
                 0 
                 7 
                 0 
                 0 
                 0 
                 1 
               
               
                 103 
                 133 
                 171 
                 171 
                 133 
                 1 
                 3 
                 6 
                 11 
                 18 
                 39 
                 0 
                 0 
                 0 
                 10 
                 0 
                 0 
                 0 
                 7 
                 0 
                 0 
                 0 
                 1 
               
               
                 104 
                 133 
                 171 
                 57 
                 165 
                 1 
                 3 
                 7 
                 12 
                 17 
                 40 
                 0 
                 2 
                 0 
                 9 
                 0 
                 5 
                 0 
                 0 
                 0 
                 1 
                 0 
                 0 
               
               
                 105 
                 133 
                 171 
                 75 
                 127 
                 1 
                 3 
                 7 
                 12 
                 17 
                 40 
                 0 
                 2 
                 0 
                 9 
                 0 
                 5 
                 0 
                 0 
                 0 
                 1 
                 0 
                 0 
               
               
                 106 
                 133 
                 171 
                 127 
                 75 
                 1 
                 3 
                 7 
                 12 
                 17 
                 40 
                 0 
                 2 
                 0 
                 9 
                 0 
                 5 
                 0 
                 0 
                 0 
                 1 
                 0 
                 0 
               
               
                 107 
                 133 
                 171 
                 127 
                 172 
                 1 
                 3 
                 7 
                 12 
                 17 
                 40 
                 0 
                 2 
                 0 
                 9 
                 0 
                 5 
                 0 
                 0 
                 0 
                 1 
                 0 
                 0 
               
               
                 108 
                 133 
                 171 
                 136 
                 165 
                 1 
                 3 
                 7 
                 12 
                 17 
                 40 
                 0 
                 2 
                 0 
                 9 
                 0 
                 5 
                 0 
                 0 
                 0 
                 1 
                 0 
                 0 
               
               
                 109 
                 133 
                 171 
                 165 
                 57 
                 1 
                 3 
                 7 
                 12 
                 17 
                 40 
                 0 
                 2 
                 0 
                 9 
                 0 
                 5 
                 0 
                 0 
                 0 
                 1 
                 0 
                 0 
               
               
                 110 
                 133 
                 171 
                 165 
                 136 
                 1 
                 3 
                 7 
                 12 
                 17 
                 40 
                 0 
                 2 
                 0 
                 9 
                 0 
                 5 
                 0 
                 0 
                 0 
                 1 
                 0 
                 0 
               
               
                 111 
                 133 
                 171 
                 172 
                 127 
                 1 
                 3 
                 7 
                 12 
                 17 
                 40 
                 0 
                 2 
                 0 
                 9 
                 0 
                 5 
                 0 
                 0 
                 0 
                 1 
                 0 
                 0 
               
               
                 112 
                 133 
                 171 
                 55 
                 175 
                 1 
                 3 
                 7 
                 12 
                 19 
                 42 
                 0 
                 0 
                 0 
                 11 
                 0 
                 0 
                 0 
                 3 
                 0 
                 0 
                 0 
                 1 
               
               
                 113 
                 133 
                 171 
                 132 
                 175 
                 1 
                 3 
                 7 
                 12 
                 19 
                 42 
                 0 
                 0 
                 0 
                 11 
                 0 
                 0 
                 0 
                 3 
                 0 
                 0 
                 0 
                 1 
               
               
                 114 
                 133 
                 171 
                 175 
                 55 
                 1 
                 3 
                 7 
                 12 
                 19 
                 42 
                 0 
                 0 
                 0 
                 11 
                 0 
                 0 
                 0 
                 3 
                 0 
                 0 
                 0 
                 1 
               
               
                 115 
                 133 
                 171 
                 175 
                 132 
                 1 
                 3 
                 7 
                 12 
                 19 
                 42 
                 0 
                 0 
                 0 
                 11 
                 0 
                 0 
                 0 
                 3 
                 0 
                 0 
                 0 
                 1 
               
               
                 116 
                 133 
                 171 
                 55 
                 137 
                 1 
                 3 
                 7 
                 13 
                 20 
                 44 
                 0 
                 0 
                 0 
                 7 
                 0 
                 0 
                 0 
                 5 
                 0 
                 0 
                 0 
                 1 
               
               
                 117 
                 133 
                 171 
                 132 
                 137 
                 1 
                 3 
                 7 
                 13 
                 20 
                 44 
                 0 
                 0 
                 0 
                 7 
                 0 
                 0 
                 0 
                 5 
                 0 
                 0 
                 0 
                 1 
               
               
                 118 
                 133 
                 171 
                 137 
                 55 
                 1 
                 3 
                 7 
                 13 
                 20 
                 44 
                 0 
                 0 
                 0 
                 7 
                 0 
                 0 
                 0 
                 5 
                 0 
                 0 
                 0 
                 1 
               
               
                 119 
                 133 
                 171 
                 137 
                 132 
                 1 
                 3 
                 7 
                 13 
                 20 
                 44 
                 0 
                 0 
                 0 
                 7 
                 0 
                 0 
                 0 
                 5 
                 0 
                 0 
                 0 
                 1 
                 1.37e−4 @ 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 113 
               
               
                   
               
            
           
         
       
     
     At step  210 , there is a finding of a set of codes (Set A) that have the maximal-minimal free distance. In other words, for example, the algorithm searches the free distances of all codes having the generators of (133,171), and is not catastrophic, by going through all of the possible combinations. Of all the free distances for each code, the minimum free distance, d min,free , is found for each code. 
     Table 1, supra, shows 119 codes that have the maximal d min,free =20. The free distance represents the error correcting capability of the code. The maximal minimal free distance is the minimal distance between any two codeword sequences having the same number of bits. Thus, in this particular case there are 119 codes found that have a free distance of 20. 
     At step  220 , a set of codes (subset of A) is found from (Set A) that have the smallest number of paths at the free distance  20 . In this particular case, 6 codes were found that have a total number of 16 paths, as shown for index numbers 1-6 and going across to the column with a “20” distance, each index# has a quantity of 16 in the free distance quantity of 20 free paths each. The quantity 16 is further analyzed by looking at the trace length from 7 to 11, and it turns out they all have the same characteristics 0, 1, 3, 5, 7. It should be noted that the trace length is equivalent to the number of inputs. In other words, there is no output sequence that has d_free=20 for 7 input bits; there is 1 sequence that has d_free=20 for 8 input bits, and 3 sequences having d_free=20 for 9 input bits; etc. 
     At step  230 , the code having the smallest number of paths at free distance  21 ,  22 ,  23  is selected. Since all 6 codes have the same number  16 , we compare the next one-21. Codes  1  and  2  only have 22 while codes  3 - 6  have 24. The codes  1 , 2 , which are equivalent, are selected. In this particular example, the selection of (133, 171, 135, 175) is considered to be the best candidate for the extended convolution codes of (133, 171) at the ¼ rate. 
     At step  240  (which is not shown in  FIG. 2  as it is optional), the code&#39;s performance is verified at the rate of ⅓. We go through the same steps for the rate of ⅓ and, in this case, (133, 171, 135) happens to be the best code with the best known free distance of 15 as shown in boldface in the worksheet in Table 2. 
     
       
         
           
               
               
               
               
               
             
               
                   
                 TABLE 2 
               
               
                   
                   
               
               
                   
                 1 st  Code 
                 2 nd  Code 
                 ⅓ CC 
                 Free Distance 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                   
                 133 
                 171 
                 1 
                 11 
               
               
                   
                 133 
                 171 
                 2 
                 11 
               
               
                   
                 133 
                 171 
                 3 
                 12 
               
               
                   
                 133 
                 171 
                 4 
                 11 
               
               
                   
                 133 
                 171 
                 5 
                 12 
               
               
                   
                 133 
                 171 
                 6 
                 12 
               
               
                   
                 133 
                 171 
                 7 
                 12 
               
               
                   
                 133 
                 171 
                 10 
                 11 
               
               
                   
                 133 
                 171 
                 11 
                 12 
               
               
                   
                 133 
                 171 
                 12 
                 12 
               
               
                   
                 133 
                 171 
                 13 
                 13 
               
               
                   
                 133 
                 171 
                 14 
                 12 
               
               
                   
                 133 
                 171 
                 15 
                 13 
               
               
                   
                 133 
                 171 
                 16 
                 12 
               
               
                   
                 133 
                 171 
                 17 
                 12 
               
               
                   
                 133 
                 171 
                 20 
                 11 
               
               
                   
                 133 
                 171 
                 21 
                 12 
               
               
                   
                 133 
                 171 
                 22 
                 12 
               
               
                   
                 133 
                 171 
                 23 
                 13 
               
               
                   
                 133 
                 171 
                 24 
                 12 
               
               
                   
                 133 
                 171 
                 25 
                 13 
               
               
                   
                 133 
                 171 
                 26 
                 13 
               
               
                   
                 133 
                 171 
                 27 
                 12 
               
               
                   
                 133 
                 171 
                 30 
                 12 
               
               
                   
                 133 
                 171 
                 31 
                 13 
               
               
                   
                 133 
                 171 
                 32 
                 13 
               
               
                   
                 133 
                 171 
                 33 
                 12 
               
               
                   
                 133 
                 171 
                 34 
                 12 
               
               
                   
                 133 
                 171 
                 35 
                 14 
               
               
                   
                 133 
                 171 
                 36 
                 12 
               
               
                   
                 133 
                 171 
                 37 
                 12 
               
               
                   
                 133 
                 171 
                 40 
                 11 
               
               
                   
                 133 
                 171 
                 41 
                 12 
               
               
                   
                 133 
                 171 
                 42 
                 12 
               
               
                   
                 133 
                 171 
                 43 
                 13 
               
               
                   
                 133 
                 171 
                 44 
                 12 
               
               
                   
                 133 
                 171 
                 45 
                 13 
               
               
                   
                 133 
                 171 
                 46 
                 13 
               
               
                   
                 133 
                 171 
                 47 
                 14 
               
               
                   
                 133 
                 171 
                 50 
                 12 
               
               
                   
                 133 
                 171 
                 51 
                 13 
               
               
                   
                 133 
                 171 
                 52 
                 13 
               
               
                   
                 133 
                 171 
                 53 
                 14 
               
               
                   
                 133 
                 171 
                 54 
                 13 
               
               
                   
                 133 
                 171 
                 55 
                 14 
               
               
                   
                 133 
                 171 
                 56 
                 12 
               
               
                   
                 133 
                 171 
                 57 
                 13 
               
               
                   
                 133 
                 171 
                 60 
                 12 
               
               
                   
                 133 
                 171 
                 61 
                 13 
               
               
                   
                 133 
                 171 
                 62 
                 13 
               
               
                   
                 133 
                 171 
                 63 
                 14 
               
               
                   
                 133 
                 171 
                 64 
                 13 
               
               
                   
                 133 
                 171 
                 65 
                 14 
               
               
                   
                 133 
                 171 
                 66 
                 12 
               
               
                   
                 133 
                 171 
                 67 
                 13 
               
               
                   
                 133 
                 171 
                 70 
                 12 
               
               
                   
                 133 
                 171 
                 71 
                 14 
               
               
                   
                 133 
                 171 
                 72 
                 14 
               
               
                   
                 133 
                 171 
                 73 
                 13 
               
               
                   
                 133 
                 171 
                 74 
                 12 
               
               
                   
                 133 
                 171 
                 75 
                 13 
               
               
                   
                 133 
                 171 
                 76 
                 12 
               
               
                   
                 133 
                 171 
                 77 
                 12 
               
               
                   
                 133 
                 171 
                 100 
                 11 
               
               
                   
                 133 
                 171 
                 101 
                 12 
               
               
                   
                 133 
                 171 
                 102 
                 12 
               
               
                   
                 133 
                 171 
                 103 
                 13 
               
               
                   
                 133 
                 171 
                 104 
                 12 
               
               
                   
                 133 
                 171 
                 105 
                 13 
               
               
                   
                 133 
                 171 
                 106 
                 13 
               
               
                   
                 133 
                 171 
                 107 
                 14 
               
               
                   
                 133 
                 171 
                 110 
                 12 
               
               
                   
                 133 
                 171 
                 111 
                 12 
               
               
                   
                 133 
                 171 
                 112 
                 13 
               
               
                   
                 133 
                 171 
                 113 
                 14 
               
               
                   
                 133 
                 171 
                 114 
                 13 
               
               
                   
                 133 
                 171 
                 115 
                 14 
               
               
                   
                 133 
                 171 
                 116 
                 14 
               
               
                   
                 133 
                 171 
                 117 
                 14 
               
               
                   
                 133 
                 171 
                 120 
                 12 
               
               
                   
                 133 
                 171 
                 121 
                 13 
               
               
                   
                 133 
                 171 
                 122 
                 13 
               
               
                   
                 133 
                 171 
                 123 
                 14 
               
               
                   
                 133 
                 171 
                 124 
                 13 
               
               
                   
                 133 
                 171 
                 125 
                 14 
               
               
                   
                 133 
                 171 
                 126 
                 14 
               
               
                   
                 133 
                 171 
                 127 
                 13 
               
               
                   
                 133 
                 171 
                 130 
                 13 
               
               
                   
                 133 
                 171 
                 131 
                 14 
               
               
                   
                 133 
                 171 
                 132 
                 14 
               
               
                   
                 133 
                 171 
                 133 
                 13 
               
               
                   
                 133 
                 171 
                 134 
                 12 
               
               
                   
                 133 
                 171 
                 135 
                 15 
               
               
                   
                 133 
                 171 
                 136 
                 13 
               
               
                   
                 133 
                 171 
                 137 
                 14 
               
               
                   
                 133 
                 171 
                 140 
                 12 
               
               
                   
                 133 
                 171 
                 141 
                 13 
               
               
                   
                 133 
                 171 
                 142 
                 13 
               
               
                   
                 133 
                 171 
                 143 
                 14 
               
               
                   
                 133 
                 171 
                 144 
                 13 
               
               
                   
                 133 
                 171 
                 145 
                 14 
               
               
                   
                 133 
                 171 
                 146 
                 14 
               
               
                   
                 133 
                 171 
                 147 
                 14 
               
               
                   
                 133 
                 171 
                 150 
                 13 
               
               
                   
                 133 
                 171 
                 151 
                 14 
               
               
                   
                 133 
                 171 
                 152 
                 14 
               
               
                   
                 133 
                 171 
                 153 
                 13 
               
               
                   
                 133 
                 171 
                 154 
                 12 
               
               
                   
                 133 
                 171 
                 155 
                 13 
               
               
                   
                 133 
                 171 
                 156 
                 13 
               
               
                   
                 133 
                 171 
                 157 
                 14 
               
               
                   
                 133 
                 171 
                 160 
                 12 
               
               
                   
                 133 
                 171 
                 161 
                 14 
               
               
                   
                 133 
                 171 
                 162 
                 14 
               
               
                   
                 133 
                 171 
                 163 
                 14 
               
               
                   
                 133 
                 171 
                 164 
                 14 
               
               
                   
                 133 
                 171 
                 165 
                 15 
               
               
                   
                 133 
                 171 
                 166 
                 13 
               
               
                   
                 133 
                 171 
                 167 
                 14 
               
               
                   
                 133 
                 171 
                 170 
                 12 
               
               
                   
                 133 
                 171 
                 171 
                 13 
               
               
                   
                 133 
                 171 
                 172 
                 13 
               
               
                   
                 133 
                 171 
                 173 
                 14 
               
               
                   
                 133 
                 171 
                 174 
                 12 
               
               
                   
                 133 
                 171 
                 175 
                 14 
               
               
                   
                 133 
                 171 
                 176 
                 12 
               
               
                   
                 133 
                 171 
                 177 
                 12 
               
               
                   
                   
               
            
           
         
       
     
     Thus, from steps  210 - 240  it can be concluded for that particular example, that for the mother code rate of ½, m-7, (133, 171), the best ¼ extended convolutional code is (133, 171, 135, 175) and the best ⅓ code, as punctured from the ¼ code is (133, 171, 135). 
       FIG. 3  provides a graphical illustration comparing the CC at the ½ error rate  301 , the standard ¼ rate  305 , and the optimal extended ¼ rate  307  according to the present invention. The graph shows that at 0 db the error rate of the optimal extended ¼ rate  307  is significantly smaller than the standard ¼ and ½ rates previously shown. The ratio used is Eb/No, and the comparison of the three items on the graph is particularly shown at 0 db, that the BER is a much smaller number for the optimal extended ¼ rate at 0 db than the standard ¼ rate or the ½ rate shown. 
       FIG. 4  provides an illustration of an extended convolutional coding system and apparatus according to the present invention. In this particular case, there is a wireless network running under a wireless network protocol  401 . It could be under 802.11, Bluetooth, ATM, etc., or although not shown in this illustration, the network could be wired such as under 802.3. In the network, the server transmits  405  to and from the clients  410 ,  415  and  420 , typically using packets or frames. These packets or frames could be conventional standardized packets, or new superframes having about 16 packets, as disclosed in the Philips disclosure 779763/4 that is previously referred to in the specification. Both the server and the clients either are in communication therewith, or have within their units, respective encoder/decoder modules  405 ,  406 ,  407  and  419 . The units would permit, before the transmission of a frame/packet or super frame, modulation and coding, preferably forward error correction, using convolutional coding and decoding. 
     Reference module  405   a  is a box representation of just such a convolution encoder/decoder module. Reference  405   b , on the other hand, is a representation of an extended convolutional encoder/decoder module. The extended convolutional encoder/decoder may have additional memory registers, modulo-adders, etc. In any even, as discussed in the example of one way the convolutional code could be extended, it is possible, for example that a mother code rate ½ (133,171) and a constraint length of 7 can be extended to a new rate ¼ CC with (133, 171, 135, 175). In other words, instead of two bits per input, the extended coding can provide 4 bits per input, or a punctured ⅓ rate from the ¼ code is (133, 171, 135) thus drastically improving the Bit Error Rate performance at predetermined SNR values (typically 0 db). 
     Various modifications to the above invention can be made by persons of ordinary skill in the art that do not depart from the spirit of the invention, or the scope of the appended claims. For example, in no way is the extended convolutional coding limited to ½ and ¼ rates, as it can be used on many other rates that drastically depart from values such as ½ and ¼ rates, (could be, for example 1/85, 7/68, etc.) including those that are not mother code rates, and have something other than one input that is not punctured.