68 BELL SYSTEM TECHNICAL JOURNAL 



"random noise" to emphasize that (2-5) does not represent a random noise 

 current unless N and T approach infinity. 



The input to the receiver is /o(/) + /(/). Let the process of reception 

 consist of computing the K -\- \ integrals 



Xk = 2T~' f [h(!) - /o(/) - /(/)]' dl, k = 0,1, ■■■ ,K (2-6) 



J-T/2 



and selecting the smallest one (all of the A' + 1 encodings have been carried 

 to the receiver beforehand). If the value of k corresponding to the smallest 

 integral happens to be 0, as it will be if the noise /(/) is small, no error is 

 made. In any other case the receiver picks out the wrong message. 



When the representations (2-2), (2-3), and (2-5) are put in (2-6) and the 

 integrations performed, it is found that 



x,= i; (A['' - Ai'' - B.f, Xo= t Bl (2-7) 



n=—N n=— .V 



which have already appeared in equations (1-2) and (1-3). If, as in Section 

 1, Pk is interpreted as a point in 2xV + 1 — dimensional Euclidean space with 

 coordinates .1-a-, • • • , Ao''\ • • • , A^-' and Q is the point A-^ + B_x, • • • , 

 Ao ^ + Bg, . . . ,A]^' + Bx, then Xk is the square of thedistance between points 

 Pk and Q. Point Po corresponds to the signal actually sent, points Pi, • • • , 

 Pk to the remaining signals, and point Q to the signal plus noise at the 

 receiver. The expected distance between the origin and Pa- is (t(2X + 1)^'- 

 = (2ir.s)^'-, that between P„ and (J is vi2X + D^'- = (2W.s-yi-\ and that 

 between the origin and Q is 



(a- + i.2)i/2(2^r ^ 1)1/2 ^ (2ir.,. + IWsY" 



No error is made when Xo is less than every one of .vi, .vo, • • • , .Va-, i.e., 

 when none of the points Pi, • • • , Pk lies within the sphere S of radius .vi'" 

 centered on Q and passing through Pn. Therefore the probability of obtain- 

 ing no error when the first encoding scheme is used is equal to the probability 

 denoted by Prob. (PiQ, • • • , PkQ > PoQ) in the mathematical problem of 

 Section 1. 



One might wonder why probability theory has played such a prominent 

 part in the encoding scheme just described. It is used because we do not 

 know the best method of encoding. In fact, it would not be used if we knew 

 how to solve the following problem:* Arrange A' + 1 points Pq, • • • Pk on 

 the hyj)er-surface of the 2M + 1 — dimensional sphere of radius (2irs)^'^ 



* C. E. Shannon has commented that although the solution of this problem leads to a 

 good code, it may not be the best possiljle, i.e., it is not obvious that the code obtained 

 in this way is the same as the one obtained by choosing a set of points so as to minimize 

 the probability of error (calculated from the given set of points and some given W\) 

 averaged over ail A' -|- 1 points. 



