212 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



with the use of an (n, fc) -alphabet. If the original message is presented as 

 a long sequence of zeros and ones, the sequence is broken into blocks of 

 length k places. Each block is used as the first k places of a letter of 

 the signaling alphabet. The last n-k places of the letter are determined 

 by fixed parity checks over the first k places. 



Theorem 4 demonstrates the relative ease of instrumenting the maxi- 

 mum hkelihood detector (8) for use with an (n. A:) -alphabet. When an 

 element T of Bn is received at the channel output, it is subjected to the 

 n-k parity checks of the alphabet being used. This results in a parity 

 check sequence R{T). R(T) serves to identify a unique coset leader, say 

 Si . The product SiT is then formed and produced as the detector out- 

 put. The probability that this be the correct letter of the alphabet is Qi . 



1.10 BEST GROUP ALPHABETS 



Two important questions regarding (n, fc)-alphabets naturally arise. 

 What is the maximum value of Qi possible for a given n and k and which 

 of the N(n, k) different subgroups give rise to this maximum Qi? The 

 answers to these questions for general n and k are not known. For many 

 special values of n and k the answers are known. They are presented in 

 Tables II, III and IV, which are explained below. 



The probability Qi that a transmitted letter be produced correctly by 

 the detector is the sum, Qi = ^i f{Si) of the probabilities of the coset 

 leaders. This sum can be rewritten as Qi = 2Zi=o «« P^Q^~^ where a, is 

 the number of coset leaders of weight i. One has, of course, ^a, = v = 



/ y) \ T? ' 



2^"'' for an (n, /(;)-alphabet. Also «> ^ ( . ) = -7-7 — '■ — n- ! since this is the 



\t / tlin — t) 



number of elements of Bn of weight i. 



The (Xi have a special physical significance. Due to the noise on the 

 channel, a transmitted letter, A, , of an (n, /c)-alphabet will in general be 

 received at the channel output as some element T of Bn different from 

 Ai .li T differs from Ai in s places, i.e., if w{AiT) = s, we say that an 

 s-tuple error has occurred. For a given (n, fc)-alphabet, ai is the number 

 of i-tuple errors which can be corrected by the alphabet in question, 

 i = 0, 1,2, ■ • • , n. 



Table II gives the a{ corresponding to the largest possible value of Qi 

 for a given k and ?i for k = 2,3, •••w— l,n = 4--- ,10 along with a 

 few other scattered values of n and k. For reference the binomial coeffi- 

 cients ( . ) are also listed. For example, we find from Table II that the 

 best group alphabet with 2 =16 letters that uses n = 10 places has a 



