NON-BIXARY ERROR CORRECTION CODES 1387 



number bases which are the product of several small factors are best. 

 Suggested values are six, ten and twelve. Number bases with two differ- 

 ent prime factors, may offer an advantage, since they permit simple 

 translation and change of number base among at least three different 

 numbers. 



In the comparison between binary and non-binary error correction 

 codes, the following observations may be made: 



1. Keeping the amount of information per message fixed, a binary 

 single error correction code is less efficient than a non-binary single 

 small error correction code, provided b, the channel base, is greater than 

 three, but is more efficient than a non-binary single unrestricted error 

 correction code. 



2. Non-binary codes are slightly more complicated to implement than 

 binary codes; this applies to multiple error correction codes as well as to 

 single error correction codes. The amount of added complication is in no 

 case really extensive. 



It was initially hoped that this study might also produce some addi- 

 tional binary error correction techniciues. One such technique was dis- 

 covered: the use of a systematic ecjuivalent Reed-]\Iuller code to ap- 

 proach error free coding (see Section VII) . 



Finally, the author wishes to express the hope that further work on 

 non-binary systems will be encouraged by this study. 



ACKNOWLEDGEMENTS 



This work was performed under the part-time Graduate Study Plan 

 of Bell Telephone Laboratories at the Columbia University School of 

 Engineering under the guidance of Prof. L. A. Zadeh. The author wishes 

 to acknowledge the help of Prof. Zadeh, both in the selection of a dis- 

 sertation topic and in the subsequent guidance of the study. In addition, 

 a number of helpful discussions with C. Y. Lee and A. C. Rose helped 

 to guide the research into a study of the most significant problems in 

 the field. 



REFERENCES 



1. R. W. Hamming, Error Detecting and Error Correcting Codes, B.S.T.J., 29, 



p. 147-160, April, 1950. 



2. Of Current Interest, Elec. Engg., p. 871, Sept., 1956. 



3. C.Y.Lee and W.H.Clien, Several-Vahiod Combinational Switching Circnit.s, 



Trans. A.I.E.E., 75, Part I, p. 278-28:3, Jiilv, 1956. 



4. D. Slepian, A Class of Binary Signaling Alphabets, B. S.T.J. , 35, p. 203-234, 



Jan., 1956. 



