NON-BINARY ERROR CORRECTION CODES 



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checks will contain the same set of allowable messages as some systematic 

 code, was proved by Hamming. ^ In particular, such a theorem indicates 

 that a Reed-Muller code will contain the same set of allowable messages 

 as some systematic code. This was also proved by Slepian, who has given 

 a simple method of deriving a systematic code generating the same set 

 of messages as a Reed-Muller code. For convenience, such a code will be 

 called an SERM code (Systematic Equivalent Reed-Muller code). 



A Reed-Muller decoder serves to derive the information digits from a 

 message in Reed-Muller code which may have been mutilated by noise. 

 If a Reed-Muller decoder is followed by a Reed-Muller encoder, the com- 

 bination serves as a noise eliminator (provided the noise is within the 

 correction bounds of the code), since the output of the encoder is the 

 noiseless Reed-Muller code message that is equivalent to the noisy 

 message that entered the decoder. This property is useful since it means 

 that any message, drawn from the set of Reed-Muller code messages, 

 which has not been mutilated outside the bounds set up by a particular 

 Reed-Muller code, will be restored to its original form, by a Reed-Muller 

 decoder followed by a Reed-Muller encoder. Since an SERM code will 

 produce only messages included in the set of messages of the correspond- 

 ing Reed-Muller code, the SERM code can be used in conjunction with a 

 Reed-]\Iuller decoder and encoder to permit transmission over a noisy 

 channel in a systematic code. 



The two systems shown in Fig. 3 are therefore equivalent in their 

 error correction properties. In both cases, messages from the set of Reed- 

 Muller code messages are sent, and since the same decoder is used ini- 

 tially, both systems will correct errors in the received message in the 

 same manner. The Reed-Muller encoder in the second system is re- 

 (luired because a Reed-Muller decoder does not correct a message but 

 derives information digits from the received message directly. The 

 derived information digits, however, necessarily correspond to some 

 corrected form of the received message and, in effect, the decoder performs 

 the same correction as it would perform by deriving the corrected form of 

 the message first. 



Fig. 3 — Equivalent systems using SERJM and Reed-Muller codes. 



