Error-Correcting Codes for Multiple-Level Transmission

A groat deal of work has been done on error-correcting codes for the binary channel. In this paper we consider codes for a channel that can transmit more than two levels. Multiple-level transmission is practical if the channel is sufficiently quiet, as, for example, the submarine voice cable. It results in a substantial increase in bit rate and in added flexibility in choosing a code. One now has four parameters to adjust -- the number of levels of transmission, the number of information symbols, 2S1 282 T H E B E L L SYSTEM TECHNICAL J O U R N A L , JANUARY 1 9 6 1 the number of redundant symbols, and the number of errors it is desirable to detect and/or correct. Of course it cannot be decided without detailed analysis whether these advantages will more than compensate for the added complexity of the terminal equipment. In the binary case, systematic error-correcting codes have certain advantages; 1 in particular, they are amenable to known mathematical techniques. It has been shown by Slepian" that the words of a systematic code form a group under place-by-place addition mod 2. The natural generalization of a group code over the field (0,1) appears to be a vector space over a finite field of q elements. We call such vector spaces alphabets, and their individual elements are called letters. In the general case, a "code" becomes an "alphabet" and a word (unfortunately!) becomes a "letter." Each letter is a row of n symbols picked from the ground field; the alphabet is a space of row vectors of length n.