148 BELL SYSTEM TECHNICAL JOURNAL 



Computers/ the 3 out of 7 code used for radio telegraphy ,2 and the word 

 count sent at the end of telegrams. 



In some situations self checking is not enough. For example, in the Model 

 5 Relay Computers built by Bell Telephone Laboratories for the Aberdeen 

 Proving Grounds/ observations in the early period indicated about two 

 or three relay failures per day in the 8900 relays of the two computers, repre- 

 senting about one failure per two to three million relay operations. The self- 

 checking feature meant that these failures did not introduce undetected 

 errors. Since the machines were run on an unattended basis over nights and 

 week-ends, however, the errors meant that frequently the computations 

 came to a halt although often the machines took up new problems. The 

 present trend is toward electronic speeds in digital computers where the 

 basic elements are somewhat more reliable per operation than relays. How- 

 ever, the incidence of isolated failures, even when detected, may seriously 

 interfere with the normal use of such machines. Thus it appears desirable 

 to examine the next step beyond error detection, namely error correction. 



We shall assume that the transmitting equipment handles information 

 in the binary form of a sequence of O's and I's. This assumption is made 

 both for mathematical convenience and because the binary system is the 

 natural form for representing the open and closed relays, flip-flop circuits, 

 dots and dashes, and perforated tapes that are used in many forms of com- 

 munication. Thus each code symbol will be represented by a sequence of 

 O's and I's. 



The codes used in this paper are called systematic codes. Systematic codes 

 may be defined^ as codes in which each code symbol has exactly n binary 

 digits, where m digits are associated with the information while the other 

 k = n — m digits are used for error detection and correction. This produces 

 a redundancy R defined as the ratio of the number of binary digits used to 

 the minimum number necessary to convey the same information, that is, 



R = n/m. 



This serves to measure the efficiency of the code as far as the transmission 

 of information is concerned, and is the only aspect of the problem discussed 

 in any detail here. The redundancy may be said to lower the effective channel 

 capacity for sending information. 



The need for error correction having assumed importance only recently, 

 very little is known about the economics of the matter. It is clear that in 



' Franz Alt, "A Bell Telephone Laboratories' Computing Machine" — I, II. Mathe- 

 matical Tables and Other Aids to Computation, Vol. 3, pp. 1-13 and 60-84, Jan. and 

 Apr. 1948. 



* S. Sjjarks, and R. G. Kreer, "Tape Relay System for Radio Telegraph Operation," 

 /2.C./1. /^m'ew, Vol. 8, pp. 393-426, (especially p. 417), 1947. 



' In Section 7 this is shown to be equivalent to a much weaker appearing definition. 



