136-i THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1957 



d, would contain only the digits and d, thus wasting most of the pos- 

 sible corrector values.* 



It is possible to encode and decode using the prime factors of the 

 number base, performing separate and independent corrections on each 

 factor. This is also inefficient, since for many cases, information as to 

 which digit is in error is found independently in two or more ways, while 

 for certain values of the error, it can be found in only one way. Working 

 with mixed digits and check bases, a lower than 6, is not satisfactory 

 since certain values of the error {a in particular) will never show up in a 

 particular check. The technique used for primes will not work since 

 multiples of two different characteristics may be identical; for example, 

 base 10, characteristics 11 and 13, error 5, will both yield correctors of 55. 



Another technique that is relatively efficient is, however, available. 

 It involves performing all check, encoding and decoding operations in a 

 number base p, where p is some prime number (usually, the lowest) 

 that is equal to or greater than 6. (In case 6 is a prime, we use the pro- 

 cedure outlined above, which is a special case of the procedure to be 

 described below.) 



The obvious difficulty in such a procedure is that while the informa- 

 tion channel can only handle h levels, the check digits may assume p 

 levels, corresponding to the required p check states. This dilemma can 

 be resolved by adding an adjustment digit. The object of this digit is to 

 permit check hiformation to be transmitted in a base greater than h, 

 the channel base. The idea of an adjustment digit can best be illus- 

 trated by an example. Suppose for a decimal channel, checks are performed 

 in a unodecimal (base 11) code. Let 7 represent the value corresponding to 

 ten. (The consecutive integers in a unodecimal system are then 0, 1, 2, 

 3, • • • , 9, 7, 10, 11, • • • , 19, I7, 20, etc.) Suppose in a particular mes- 

 sage, four check digits, Zi , Zo , zs , Zi , calculated modulo 1 1 from decimal 

 information digits are used, whose values are 1, 0, 7, 8. A fifth digit, 

 Zo is added such that the sums modulo 11 of 21 + 20 , 22 + zo , Zs + ^o , 

 24 + Zq are kept constant at 1, 0, 7, 8 respectively. There are eleven dif- 

 ferent words satisfying the condition: [1, 0, 7, 8] = [(^i + Zo), (zo + Zo), 

 (23 + Zo), (24 + Zo)]. These are shown in Table VI. Of these words, six do 

 not contain the digit 7, and so may be transmitted over a decimal chan- 

 nel. Thus, an adjustment digit permits check digits which are calculated 

 in a number system of a higher base than h, to be transmitted over a base 

 h channel. When an adjustment digit is used in base p for adjusting m 

 digits so that transmission over a channel in base h is possible, a mini- 



* A waste of corrector values is equivalent to an excessive number of check 

 states for a message, which in turn implies an excessive number of check digits. 



