A CLASS OF BINARY SIGNALING ALPHABETS 



219 



Table V — Coset Leaders and Parity Check Sequences 



FOR (10, 5) -Alphabet 



tector should print 61(62 -j- l)b3lhh^ if the parity check sequence is 01111, 

 00111, 01011, 01101, or OHIO; the detector should print hMb-i + 1)6465 

 if the parity check sequence is 10111, 10011, 10101, or 10110; the de- 

 tector should print 616263(64 -j- 1)65 if the parity check sequence is 11011, 

 11001, 11010; and finally the detector should print 61626364(65 -j- 1) if the 

 parity check sequence is 11101 or 11100. 



Simpler rules of operation for the detector may possibly be obtained 

 by choice of a different set of S's in Table V. These quantities in general 

 are not unique. Also there may exist non-equivalent alphabets with 

 simpler detector rules that achieve the same probability of error as the 

 alphabet in question. 



I'vrt II — Additional Theory and Proofs of Theorems of Part I 



' 2.1 the abstract group Cn 



It will be helpful here to say a few more words about Br, , the group 



of n-place binary sequences under the operation of addition mod 2. This 



j group is simply isomorphic with the abstract group Cn generated by n 



\ commuting elements of order two, say ai, a-2 , ■ ■ ■ , a„ . Here a,:ay = 



<i,ai and a/ = /, i, j = 1, 2, • • • , n, where / is the identity for the 



group. The eight distinct elements of C3 are, for example, /, o-i , a-y , 



(h , (iici-, , aio-.i , a-itti , aia-ittz . The group C„ is easily seen to be isomorphic 



I with the Ai-fold direct product of the group Ci with itself. 



