NON-BINARY ERROR CORRECTION CODES 1385 



The number of different messages that can be produced by the en- 

 coder must be no greater than b"/w, subject to the above restriction, 6" 

 representing the maximum numbei- of messages that the decoder may 

 receive as an input. If only systematic and semi-systematic codes are 

 considered, the number of messages is Umited to multiples of powers of 

 b and of the information component base /3 of mixed digits. The number 

 of check states must be at least as large as w, so that w different correc- 

 tors may be calculated and associated with w different corrections. 



Subject to the above restrictions, the following statements may be 

 made. 



1. The systematic single small error correction codes derived using 

 the rules of Section III are the most efficient systematic single small 

 error correction codes possible. For those codes in which the two sides 

 of inecjuality (8a) are equal, no code, not even a non-systematic code, is 

 more efficient. 



2. The systematic single unrestricted error correction codes deri\'ed 

 using the rules of Section 4.1 are the most efficient systematic single 

 unrestricted error correction codes. For those codes in which the two 

 sides of inequality (31) are equal, no code is more efficient. 



3. No codes are more efficient than those semi-systematic codes, 

 derived using the rules of Section III, for which the two sides of in- 

 equalities (28) and (29) are equal and mi = 0. It is difficult to make 

 more general statements about semi-systematic codes, because spe- 

 cial techniques (such as those of Section VI), not all of which are known, 

 may be used with these codes. 



For multiple error correction codes, other techni(|ues are both simpler 

 and more efficient than the straight systematic and semi-systematic 

 techniques described in Sections III, lY and Y. One such scheme has 

 been described in detail in Section VL No codes have been found which 

 approach the limit set by iv, but the codes described in Section 6.2 are 

 moderately efficient. 



Throughout this paper, all techniques which in\-olve vast complica- 

 tions at the expense of slight additional efficiency have been avoided. 

 Codebook methods are always possible. If a technique is almost as com- 

 plicated as a codebook technique with only slightly greater efficiency 

 than a simple technique, the simple technique would always be used in 

 practice, and the codebook satisfies the mathematical and theoretical 

 requirements. In a sense, a really complicated technique is only useful 

 for deriving a better lower limit for the maximum efficiency of a code- 

 book code. In the non-binary case, howe^'er, a codebook system is con- 

 siderably more efficient than any code system which does not take ad- 



