1382 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1957 



Table XVI — Multiple Small Error Correction Code 

 Using SERM Codes with Decimal Channel 



This means that a Reed-Muller code can be adapted to the system 

 shown m Fig. 2. The Systematic Binary Error Correction Code Encoder 

 is simply an SERM encoder; this is permissible since the SERM codes 

 are .systematic. The Binary Decoder and Corrector is simply a Reed- 

 Muller decoder followed b}^ a Reed-Muller encoder. Everything else 

 remains unchanged. 



This scheme offers flexibility for the correction of large numbers of 

 small errors. Proper initial error correction encoding of the original in- 

 formation digits will permit correction of a small number of large errors. 



Table X^T shows some tj^pical cases of the correction of many small 

 errors in a decimal message as a function of the number of information 

 and check digits in a message of constant length. For convenience, every- 

 thing is sho^^^l in equivalent decimal digits, even though in the actual 

 code, binary and quinary information digits are used. Only the first few 

 entries are considered, since the message composed exclusively of the 

 digits 0, 3, 6, 9 in which any number of small errors in a decimal channel 

 may be corrected (this code is described by the first entry of Table Y) is 

 more efficient than the codes corresponding to subsequent entries on 

 Table XXl. This code, which is very easy to instrument, will transmit 

 the equivalent of 77 decimal digits in a 128 decimal digit message. 



One problem not efficiently solved by these techniques is the multiple- 

 error correction ternary channel problem. A technique which can be 

 used is a code identical to the regular binary Reed-Muller Code, except 

 that all equations will be modulo 3 instead of modulo 2. In decoding, this 

 will sometimes require subtraction instead of addition; in modulo 2 

 ecjuations there is no difference between these operations, but in modulo 3 

 equations, the two operations are distinct. The same procedure can be 

 used for correcting multiple unrestricted errors in any base. 



VII. iterative codes 



All the codes described above have one disadvantage; occasional ex- 

 cessive noise will jdeld a non-correctable message. In order to approach 



