138G THE BELL SYSTEM TECHNICAL JOUKXAL, NOVEMBER 1957 



vantage of the fact that all transmitted messages are not mutilatable to 

 an equal number of correctable received messages. 



From the point of view of deriving lower limits to the maximum effi- 

 ciency of a codebook technique, such a consideration is vital. Except for 

 a few relati^'ely trivial cases, no codes have been found which take sig- 

 nificant advantage of the above consideration, for deriving such a 

 limit.* 



IX. CONCLUSION 



In this paper, techniques have been presented for deriving error cor- 

 rection codes for non-binary systems. None of the methods presented 

 are overly complicated, nor do they recjuire excessive storage capacity 

 for either the encoding or decoding and correction system. 



The codes are sufficiently simple so that their use with a non-binary 

 storage system may be considered, and the development of such a 

 storage system should not be stopped because a system without flaws or 

 not subject to noise cannot be realized. 



An important disadvantage of using error correction codes with such 

 a system is the time recjuirement. Correction usually requires a signifi- 

 cant amount of time. This is probably one reason why the Hamming 

 Code is not used more extensively. The more advanced and complicated 

 codes, such as the Reed-Muller Codes, suffer particularly from the 

 amount of time required for a correction. The codes described in this 

 paper are therefore probably best suited to medium or low speed stor- 

 ages, which are not read too frecjuently. 



A study of this type may be of some interest to those who have been 

 considering the use of multi-state devices for building switching systems 

 and computers, since this paper represents a stvidy of a typical problem. 

 Certain lessons may be derived from this study: 



1. Restriction to a single number base for all operations is a severe 

 handicap. The more advanced codes presented in this paper, require 

 extensive use of different number base operations. The abiUty, inside 

 the computer, to change number bases for different operations, may well 

 be useful. 



2. Different problems are best solved using different number bases. 

 For example, the use of an even number base is desirable for multiple 

 small error correction codes, while the use of a prime number base is 

 desirable for correcting single large errors. It is the author's opinion that 



* Note that this restriction has less significance in the case of binary codes. In 

 a symmetrical channel with only two available signals, each value of a digit ma\' 

 be changed in as many ways, namely, one, as every other. 



