A CLASS OF BINARY SIGNALING ALPHABETS 233 



seven of trace and seven of trace —2. The latter seven matrices cor- 

 respond to elements of even weight and together with / must represent 

 a subgroup of order 8. We associate them with the subgroup generated 

 by the elements 2, 3, and 4. We have therefore 



x(/) = 10, x(2) = x(3) = x(4) = x(23) 



= x(24) = x(34) = x(234) = -2. 



Examination of the symmetries involved shows that it doesn't matter 

 how the remaining Xi ai"e associated with the remaining group elements. 

 We take, for example 



x(l) = 4, x(12) = x(13) = x(14) = x(123) 



= x(124) = x(134) = x(1234) = 0. 



Now form the sum shown in equation (14) with /3 = 1234 (i.e., with the 

 character x^" obtained from column 1234 of the Table VI by means 

 of substitution (13). There results c?i234 = V-i which is impossible. There- 

 fore there does not exist a (10, 4) -alphabet with weights 35 6 . 



The weights 5 46 correspond to a representation of d with character 

 x(/) = 10, 0^, 2, ( — 2)^ We take the subgroup of elements of even weight 

 to be generated by 2, 3, and 4. Except for the identity, it is clearly im- 

 material to w^hich of these elements we assign the character 2. We make 

 the following assignment: x(/) = 10, x(2) = 2, x(3) = x(4) = x(23) = 

 x(24) = x(34) = x(234) = -2, x(l) = x(12) = x(13) = x(14) = 

 x(123) = x(124) = x(134) = x(1234) = 0. The use of equation (14) 

 shows that ^2 = \'2 which is impossible. 



It follows that of the 53,743,987 (10, 4)-alphabets, none is better than 

 the one listed on Table III. 



Not all the entries of Table III were established in the manner just 

 demonstrated for the (10, 4)-alphabet. In many cases the search for a 

 l)est alphabet was narrowed down to a few alphabets by simple argu- 

 ments. The standard arrays for the alphabets were constructed and the 

 best alphabet chosen. For large n the labor in making such a table can 

 be considerable and the operations involved are highly liable to error 

 when performed by hand. 



I am deeply indebted to V. M. Wolontis who programmed the IBM 

 CPC computer to determine the a's of a given alphabet and who pa- 

 tiently ran off many such alphabets in course of the construction of 

 Tables II and III. I am also indebted to Mrs. D. R. Fursdon who eval- 

 uated many of the smaller alphabets by hand. 



