COMMUNICATION IN PRESENCE OF NOISE 69 



in such a way that the smallest of the A' (A' + l)/2 distances Pa -P^, k,( = 0, 1, 

 • • • ,K,k 9^ /, has the largest possible value. This would maximize the dif- 

 ference (as measured by the distance between their representative points) 

 between the two (or more) most similar encoding signals.! 



In this paper we have been forced to rely on the randomness of probability 

 theory to secure a more or less uniform scattering of the points Po, • • • , Pr. 

 In our work they do not lie exactly on a sphere of radius (2Wsy'' but this 

 causes us no trouble. 



3. The Second Encoding Scheme 



The second of the two encoding schemes is suggested by one of Shannon's 

 (2) proofs of the fundamental result (1-1). In this scheme the A -+- 1 mes- 

 sages are to be sent over a transmission system having a frequency band ex- 

 tending from zero to F cycles per second, and are to be sent during a time 

 interval of nominal length T. 



The first few steps in the encoding process are just the same as in the first 

 scheme. N is still given by (2-1) and a by (N + l/2)(r- = Ws- After drawing 

 A -f 1 sets of ^'s, with 2N + 1 in each set, the A -|- 1 messages are 

 encoded so that the signal corresponding to the ^th message, ^ = 0, 1, • • •, 

 A, is 



/.(/) = (FT-)"' ± A': "'" ;'jl^' - f (3-1) 



„=_Ar TT^lPt — n) 



From (3-1), the value of lk{0 at / = n/{2F) is zero if the integer n exceeds 

 A^ in absolute value. If the integer n is such that | w | < N, the corresponding 

 value of Ik{t) is (FTY'^An''. The energy in the ^th signal is obtained by 

 squaring both sides of (3-1) and integrating with respect to /. Thus 



r lliOdl = 2-'T i: A'!:'' (3-2) 



J-aa n=-N 



which has the expected value {N -f- l/2)<r-r. The average power developed 

 when this amount of energy is expended during the nominal signal length 

 r is (iV -h 1/2)(T- which is equal to W s, as it should be. 

 The noise introduced by the transmission system is taken to be 



J(t) = (fr)'« t B. '^f^^'-f (3-3) 



„ — N t{2FI — n) 



t Possibly if A' + 1 discrete unit charges of electricity were allowed to move freely 

 on the sphere, their mutual repulsion would separate them in the required manner. In 

 2N -\- 1 dimensions this leads to the problem of minimizing the mutual potential energy 



■where N >\ and the summation extends over k, I = 0,\, . . . K with k 9^ (. However, 

 this problem also appears to be difficult. 



