22G THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



>r% n—k 

 .. — 2 - 



this alphabet is as good as any other alphabet of 2 letters and n places. 



In the first place, we observe that every letter of this (n, A-)-alphabet 

 (except /) has unprimed numbers in its symbols. It follows that each of 

 the 2' letters /, 1', 2', • ■ • , (', V2', ■■■ , V2' ■■■ (' occurs in a different 

 coset of the given (n, A-)-alphabet. For, if two of these letters appeared 

 in the same coset, their product (which contains only primed numbers) 

 would have to be a letter of the (n, k) alphabet. This is impossible since 

 every letter of the (/i, A) alphabet has unprimed numbers in its symbol. 

 Since there are precisely 2 cosets we can designate a coset by the single 

 element of the list Bi , Bi , ■ • ■ , B-ii = I which appears in the coset. 



We next observe that the condition 



71 ^ 2 — 



guarantees that J5a+i is of weight 3 or less. For, the given condition is 

 equivalent to 



'-■-©-o-o-e 



We treat several cases depending on the weight of Bu+i . 



If Bk+\ is of weight 3, we note that for i = 1,2, • • • , A-, the coset con- 

 taining Bi also contains an element of weight one, namely the element 

 i obtained as the product of Bi with the letter iBi of the given (n, A;)- 

 alphabet. Of the remaining (2 — A') 5's, one is of weight zero, C are of 



weight one, f j are of weight 2 and the remaining are of weight 3. We 



have, then an = 1, ai = f + A- = n. Now every B of weight 4 occurs in' 

 the list of generators \Bi , 2B-2 , • • • , kBk . It follows that on multi- 

 plying this list of generators by any B of weight 3, at least one element 

 of weight two will result. (E.g., (l'2'3')(il'2'3'40 = j4') Thus every 

 coset with a B of weight 2 or 3 contains an element of weight 2 and 

 a2 = 2 — ao — cn] . 



The argument in case Bk+i is of weight two or one is similar. 



2.8 MODULAR REPRESENTATIONS OF C„ 



In order to explain one of the methods used to obtain the best (//, A)- 

 alphabets listed in Tal)les II and III, it is necessary to digress here lo 

 present additional theory. 



I 



