A CLASS OF BINARY SIGNALING ALPHABETS 213 



1 A Q C 'J ** Q 



probability of correct transmission Qi = q + lOg p + 39g p" + l-Ag'p . 

 The alphabet corrects all 10 possible single errors. It corrects 39 of the 



possible f .^ j = 45 double errors (second column of Table II) and in 



addition corrects 14 of the 120 possible triple errors. By adding an addi- 

 tional place to the alphabet one obtains with the best (11, 4)-alphabet 

 an alphabet with 16 letters that corrects all 11 possible single errors and 

 all 55 possible double errors as well as 61 triple errors. Such an alphabet 

 might be useful in a computer representing decimal numbers in binary 

 form. 



For each set of a's listed in Table II, there is in Table III a set of 

 parity check rules which determines an {n, A)-alphabet having the given 

 a's. The notation used in Table III is best explained by an example. A 

 (10, 4)-alphabet which realizes the a's discussed in the preceding para- 

 graph can be obtained as follows. Places 1, 2, 3, 4 carrj- the information. 

 Place 5 is determined to make the mod 2 sum of the entries in places 

 3, 4, and 5 ecjual to zero. Place 6 is determined by a similar parity check 

 on places 1, 2, 3, and 6; place 7 by a check on places 1, 2, 4, and 7, etc. 



It is a surprising fact that for all cases investigated thus far an {n, k)- 

 alphabet best for a given value of p is uniformly best for all values of 

 p, ^ p ^ 1 2. It is of course conjectured that this is true for all n and /,-. 



It is a further (perhaps) surprising fact that the best {n, fc) -alphabets 

 are not necessarily those with greatest nearest neighbor distance be- 

 tween letters when the alphabets are regarded as point configurations on 

 the n-cube. For example, in the best (7, 3)-alphabet as listed in Table 

 III, each letter has two nearest neighbors distant 3 edges away. On the 

 other hand, in the (7, 3)-alphabet given by the parity check rules 413, 

 512, 623, 7123 each letter has its nearest neighbors 4 edges away. This 

 latter alphabet does not have as large a value of Qi , however, as does 

 the (7, 3)-alphabet listed on Table III. 



The cases /.; = 0, 1, /? — 1, n have not been listed in Tables II and III. 

 The cases k = and k = n are completely trivial. For k = 1, all n > 1 

 the best alphabet is obtained using the parity rule a> = 03= • • • = 

 a„ = oi . If n = '2j, 



If n = 2j + 1, Qi = i: (^') pY-\ 



For k = n — 1, /; > 1. the maximum Qi is Qi = g"~ and a parity rule 

 for an alphabet realizing this Qi is o„ = oi . 



If the a's of an (/<, A)-alphabet are of the form a, = ( . j , i = 0, 1, 



