228 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



We henceforth exclude consideration of the column / of a modular 

 representation table. Its inckision in an (n, /v)-alphabet is clearly a waste 

 of 1 binary digit. 



It is easy to show that every column of a modular representation table 

 for Ch contains exactly 2 " ones. Since an (n, /v)-alphabet is made from 

 n such columns the alphabet contains a total of n2 '~ ones and we have 



Proposition 6. The weights of an (n, /c)-alphabet form a partition of 

 n2''~^ into 2* — 1 non-zero parts, each part being an integer from the set 

 1,2, ■■■ ,n. 

 The identity element always has weight zero, of course. 



It is readily established that the product of two elements of even 

 weight is again an element of even weight as is the product of two ele- 

 ments of odd weight. The product of an element of even weight with an 

 element of odd weight yields an element of odd weight. 



The elements of even weight of an (n, A;) -alphabet form a subgroup 

 and the preceding argument shows that this subgroup must be of order 

 2*" or 2*""^ If the group of even elements is of order 2''~\ then the collec- 

 tion of even elements is a possible (n, k — l)-alphabet. This (n, k — 1) 

 alphabet may, however, contain the column / of the modular represen- 

 tation table of Ck-i ■ We therefore have 



Proposition 7. The partition of Proposition 6 must be either into 

 2^ — 1 even parts or else into 2 " odd parts and 2^—1 even parts. 

 In the latter case, the even parts form a partition of a2 "" where a is 

 some integer of the set k — I, k, ■ • • , n and each of the parts is an in- 

 teger from the set 1, 2, • • • , n. 



2.9 THE CHARACTERS OF Ck 



Let us replace the elements of Bn (each of which is a sequence of zeros 

 and ones) by sequences of 4-1 's and — I's by means of the following 

 substitution 



The multiplicative properties of elements of Bn can be preserved iti this 

 new notation if we define the product of two 4-1,-1 symbols to be the 

 symbol whose tth component is the ordinary product of the ?'th compo- 

 nents of the two factors. For example, 1011 and 01 10 become respectively 

 -11 -1 -1 and 1 -1 -11. We have 



(-11 -1 -1)(1 -1 -11) = (-1 -11 -1) 



