A CLASS OF BINARY SIGNALING ALPHABETS 



231 



14569 

 23578 

 24568 

 3 4 6 7 

 12 3 5 7 9 

 12 4 5 7 10 



1 3 4 8 10 



2 3 4 9 10 



12 3 4 6 7 8 9 



5 

 5 

 5 

 4 

 6 

 6 

 5 

 5 

 8 



The notation is that of Section 2.1. By actually forming the standard 

 array of this alphabet, it is verified that 



ao =1, Oil = 10, 



«2 



39, 



a:i 



14. 



Table II shows ( .-> ) = ^5, whereas a-z = 39, so the given alphabet 



does not correct all possible double errors. In the standard array for the 

 alphabet, 39 coset leaders are of weight 2. Of these 39 cosets, 33 have 

 only one element of weight 2; the remaining 6 cosets each contain two 

 elements of weight 2. This is due to the two elements of weight 4 in the 

 given group, namely 1289 and 3467. A portion of the standard array 

 that demonstrates these points is 



1289 



3467 



In order to have a smaller probability of error than the exhibited 

 alphabet, it is necessary that a (10, 4)-alphabet have an a^ > 39. We 

 proceed to show that this is impossible by consideration of the weights 

 of the letters of possible (10, 4)-alphabets. 



We first show that every (10, 4)-alphabet must have at least one ele- 

 ment (other than the identity, /) of weight less than 5. By Propositions 

 • ') and 7, Section 2.8, the weights must form a partition of 10-8 = 80 into 

 1 5 positive parts. If the weights are all even, at least two must be less 

 than 6 since 14-6 = 84 > 80. If eight of the weights are odd, we see from 

 8-5 + 7-() = 82 > 80 that at least one weight must be less than 5. 



