210 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



We prefer to use the word "alphabet" in place of "code" since the 

 latter has many meanings. In a systematic alphabet, the places in any 

 letter can be divided into two classes : the information places — A; in 

 number for an (n, /c)-alphabet — and the check positions. All letters 

 have the same information places and the same check places. If there 

 are k information places, these may be occupied by any of the 2 /v-place 

 binary sequences. The entries in the n — k check positions are fixed 

 linear (mod 2) combinations of the entries in the information positions. 

 The rules by which the entries in the check places are determined are 

 called parity checks. Examples: for the (4, 2)-alphabet of (6), namely 

 0000, 1100, 0011, nil, positions 2 and 3 can be regarded as the informa- 

 tion positions. If a letter of the alphabet is the sequence aia^a^ai , then 

 ai = a2 , tti = az are the parity checks determining the check places 1 

 and 4. For the (5, 3)-alphabet 00000, 10001, 01011, 00111, 11010, 10110, 

 01100, 11101 places 1, 2, and 3 (numbered from the left) can be taken 

 as the information places. If a general letter of the alphabet is aiazazaiai , 

 then a4 = a2 -j- as , Ob = ai -j- a2 -|- ^3 . 



Two group alphabets are called equivalent if one can be obtained from 

 the other by a permutation of places. Example: the 7 distinct (3, 2)- 

 alphabets given in (3) separate into three equivalence classes. Alpha- 

 bets (i), (ii), and (iv) are equivalent; alphabets (iii), (v), (vi), are equiva- 

 lent; (vii) is in a class by itself. 



Proposition S. Equivalent (n, fc) -alphabets have the same probability 

 Qi of correct transmission for each letter. 



Proposition 4- Every (n, /c) -alphabet is equivalent to an (n, k)- 

 alphabet whose first k places are information places and whose last n — k 

 places are determined by parity checks over the first k places. 



Henceforth we shall be concerned only with (n. A;) -alphabets w^hose 

 first k places are information places. The parity check rules can then 

 be written 



k 

 ai = S Tij-ay , t = /b -j- 1, • • • , n (9) 



where the sums are of course mod 2. Here, as before, a typical letter of 

 the alphabet is the sequence aia^ • ■ - ttn . The jn are k(n — k) quantities, 

 zero or one, that serve to define the particular (n, A;)-alphabet in question. 



1.8 MAXIMUM LIKELIHOOD DETECTION BY PARITY CHECKS 



For any element, J\ of Bn we can form the sum given on the right of 

 (9). This sum maj^ or may not agree with the symbol in the ?'th place of 



