iV-TERMINAL SWITCHING CIRCUITS 685 



(14) log (2^ + 1) + 2 . 

 ^+ log log (/,0V) ^ 2' 



Then 



(15) FW < 2"(<^(iV))-'^''-' 



which approaches zero uniformly as M increases. 



For most of the switching functions of practical interest H is much 

 bigger than log log 0(iV). In these cases the number 



E -1 log // 



is larger than (13) by a factor of about NP/\og N. In the case of two-ter- 

 minal relay circuits the corresponding factor found by Shannon was only 8. 

 It is not clear whether this difference indicates that there is a wider range 

 of complexity for iV-terminal networks than for two-terminal networks or 

 that our methods for obtaining upper and lower bounds lose some of their 

 effectiveness as N increases. Nevertheless, (13) is surprisingly large, as we 

 shall see in the example which follows. 



Example. Consider a telephone central office with 10,000 lines. If the 

 office must be able to connect the lines together in pairs in any arrangement 

 and to remember which line of a pair originated the call, a count of the 

 number of different states which must be produced reveals that the office 

 needs a memory of at least H = 64,000 bits, which can be supplied using 

 19,200 switches with 10 positions each. The number of other switching func- 

 tions that one might ask these 19,200 switches to perform is 



.^(10,000)^"-°" = (io»'»«»)'«""° = 10'»"'"° approx. 



To apply theorem VIII to these other functions we first note that (14) will 

 be satisfied as long as we pick e greater than .006. Then, substituting in 

 (13) and (15) we discover that the chance that one of these switching func- 

 tions chosen at random can be synthesized with less than about 



^ rvl9 ,000 , , 



10 contacts 

 is less than some number of the order of 



1019,200 



If the same calculation is repeated for a 10,000-line office which is capable 



