1380 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1957 



Table XV - — Reed-Muller Codes — 256 Digit Message 



systematic codes are conveniently applicable only to the correction of 

 single errors and a few special cases of multiple errors. 



The Reed-Muller^*' codes are not systematic codes, ("systematic" 

 being used in the narrow sense indicated above, not in the sense of Ham- 

 ming"), but offer the advantage that multiple error correction is rela- 

 tively straightforward. For this reason, it is desirable to find some way 

 of adapting the binary Reed-Muller codes for correcting a number of 

 small errors in non-binary codes. 



To explain the nature of the Reed-Muller codes completely is beyond 

 the scope of this paper; a list of their important features is sufficient. 

 This is: 



1. The length of a message is 2 binary digits for the simpler versions 

 of the code. 



2. If Cc represents the number of combinations of d items taken 

 c at a time, and Cf = dl/[cl(d — c)!], then 2^ — ^T=o Ck-i information 

 digits may be transmitted correctly in a message containing 2 digits, 

 if no more than 2'" — 1 errors occur in the messages; 2'" errors are de- 

 tected but they are not always correctable. The Reed-Muller codes for 

 correcting a large number of errors will frequently correct more than 

 2" — 1 errors, and will always correct 2'" — 1 or fewer errors. 



These values are given for a 256 digit message in Table XV. 



3. Each digit of the transmitted message is a parity check of a group 

 of digits from the information source ; the message cannot be broken down 

 into information digits and check digits. 



4. The decoding is accomplished by a number of majority decisions 

 among different groups of message digits. 



A technique will be described for using a Reed-Muller code efficiently 

 to correct a number of small (±1) errors for any code base h that is a 

 multiple of 2, and also, at a small sacrifice of efficiency, a number of larger 

 errors. 



A theorem, stating that any code which is generated by a set of parity 



