NON-BINARY ERROR CORRECTION CODES 1383 



error free transmission, some iterative coding procedure may be used. 

 This problem has been solved by Elias.'^ His methods are directly appli- 

 cable to non-binary codes, since nothing restricts the digits to binary 

 values. L I 



In order to minimize the complexity of an iterative coding procedure, 

 systematic codes are desirable. The advantages of the Reed-Muller code 

 are significant however, especially for the case of a relatively noisy 

 channel. A sound procedure for a binary channel would therefore be to 

 use SERM codes, (see Fig. 3) ; such codes are more efficient than iterated 

 Hamming Codes in a relatively noisy channel. 



VIII. SUMMARY AND ANALYSIS : . 



Many codes have been presented in this paper, all constructed by 

 some combination of procedures involving linear congruence or modulo 

 equations. 



In most cases, more efficient codes exist. Exhaustive procedures exist 

 for deriving maximum efficiency codes, although the codes derived in 

 this manner usually require an extensive codebook, both at the encoder 

 and at the decoder. Even for simple single error correction binary codes, 

 the most efficient code is not always a systematic code. For example, 

 the best systematic single error correction binary code working with an 

 eight digit message has only 16 different allowable messages; it is known'* 

 that a non-systematic code with at least 19 allowable messages exists. 



In the case of non-binary codes, the situation is somewhat worse. 

 Very few of the codes given in this paper take advantage of the fact that, 

 for most situations, a digit that is incorrectly received as or 6 — 1 is 

 usually corrected only in one direction and no need exists to specif \'^ 

 whether the correction is ±1. Most of the codes are arranged so that 

 any received digit may be corrected either positively or negativel}'. Xo 

 codes have been found which take full advantage of such a property, 

 other than codebook codes, except for isolated instances of short message 

 codes having symmetrical properties. For example, the single digit, 

 single small error correction decimal code having 0, 3, G, 9 as the allow- 

 able messages takes full advantage of this property, and is, at the same 

 time, a true semi-systematic code. 



It is extremely difficult to find the ultimate limits of efficiency of code- 

 book codes. The exhaustive procedures are totally impractical except for 

 very short messages. If an analysis is restricted to codes which do not 

 take advantage of the property that certain values of digits may be 

 corrected in only one direction, and it is assumed that each possible 

 message is mutilated to the same number of incorrect messages, one 



