230 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



(h = ^. E x{A)AA) (14) 



2^- 



A C Ck 



Every (n, A)-alphabet furnishes iis with a matrix representation of Ck 

 by means of (13) and the procedure outUned below (13). The trace xi^.) 

 of the matrix representing the element A of C\ is related to the weight 

 of the letter by 



x(A) = n - 2w(A) (15) 



Equations (14) and (15) permit us to compute from the weights of an 

 (u, /,)-alphabet what irreducible representations are present in the alpha- 

 bet and how many times each is contained. It is assumed here that the 

 given alphabet has been made isomorphic to Ck and that the weights are 

 labelled by elements of Ck ■ 



Consider the converse problem. Given a set of mmibers ivi , Wn , • ■ ■ , 

 W'lk that satisfy Propositions 6 and 7. From these we can compute 

 cjuantities %/ = n — 2wi as in (15). It is clear that the given ty's will 

 constitute the weights of an (/t, A)-alphabet if and only if the 2^ x» can 

 be labelled with elements of (\ so that the 2 sums (14) {fi ranges over 

 all elements of Ck) are non-negative integers. The integers d^ tell what 

 representations to choose to construct an in, A)-alphabet with the given 

 weights Wi . 



2.10 CONSTRUCTION OF BEST ALPHABETS 



A great many different techniques were used to construct the group 

 alphabets listed in Tables II and III and to show that for each n and k 

 there are no group alphabets with smaller probability of error. Space 

 prohibits the exhibition of proofs for all the alphabets listed. We content 

 ourseh'es here with a sample argument and treat the case n = 10, k = 

 4 in detail. 



According to (2) there are A^(10, 4) = 53,743,987 different (10, 4)- 

 alphabets. We now show that none is better than the one given in Table 

 III. The letters of this alphabet and weights of the letters are 



1 



167 8 10 5 



2 6 7 9 10 5 



3 5 6 8 9 10 6 



4 5 7 8 9 10 6 

 1289 4 

 13579 5 



