XO\-BIXAHY EKHOR COHUEC'TIOX CODES ]'.]4'.] 



It is not unlikely that the near futin-e will see the development of 

 storage systems which will be able to store more than twcj states at every 

 basic storage location.- If such systems are developed, it seems likely 

 that they will be more erratic or noisy than binary storage systems, 

 since each location must store one of h signals instead of one of two. If a 

 cathode ray tube storage system were used, for example, different quan- 

 tities of charge would have to be distinguished; in a binary storage 

 system, only the presence or absence of charge must be detected. This 

 suggests that error correction codes may become essential with certain 

 types of non-binary storage systems. One object of this paper is to 

 develop codes for this purpose and to discover which number systems 

 are most easily correctable. 



Some investigations have been made on the use of computer sj^stems 

 using multi-state elements.* A switching algebra has been developed 

 similar to Boolean algebra for handling switching problems in terms of 

 multi-state elements. Single de\'ice ring counters (the cold cathode gas 

 stepping tube for example) alread}^ exist and might be useful in such 

 systems. But currently, only limited steps in this direction have been 

 made. Another object of this paper is to show the advantages and 

 problems of error correction codes in multi-state systems; it is not un- 

 reasonable to predict that error correction codes may be more necessary 

 in multi-state systems than in binary sj'stems. 



1.2 Geometric Concept of Error Correction Codes 



A geometric model of a code was suggested by R. W. Hamming' 

 which can be altered slightly to fit the non-binary case. For an n digit 

 message, a particular message is a point in n dimensional space. A 

 single error, however defined, will change the message, and will cor- 

 respond to another point in n dimensional space. The distance between 

 the original point and the new point is considered to be unity. Thus, 

 the distance d between the points corresponding to any two messages is 

 defined as the minimum number of errors which can convert the first 

 message into the second. 



With an error detection and/or correction code, the set of transmitted 

 messages is limited so that those which are correctly recei^•ed are recog- 

 nizable; those messages which are received with fewer than a given 

 number of errors are either corrected or the fact that they are wrong is 

 recognized and some other appropriate action (such as stopping a com- 

 puter) is taken. 



In the ease of binary codes, an error changes a 1 to or a to 1 . In 

 the non-binary case, two definitions of an error are possible and Mill he 



