224 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



tation operators. The symbols for these operators are identical with the 

 sequences of zeros and ones that form the coordinates of the 2 points. 

 It is readily seen that these rotation operators form a group which is 

 transitive on the letters of the alphabet and which leave the unit cube 

 invariant. Theorem 2 then follows. 



Theorem 2 also follows readily from consideration of the array (4). 

 For example, the maximum likelihood region associated with / is the 

 set of points I, So , S3 , • • • , Sy . The maximum likelihood region asso- 

 ciated with A; is the set of points Ai , AiS^ , AiSs , ■ • ■ , AiSy . The 

 rotation (successive reflections in symmetry planes of the cube) whose 

 symbol is the same as the coordinate sequence of Ai sends the maximum 

 likelihood region of / into the maximum likelihood region oi Ai , i = 

 1, 2, • • • , M. 



2.5 PROOF OF THEOREM 3 



That every systematic alphabet is a group alphabet follows trivially 

 from the fact that the sum mod 2 of two letters satisfying parity checks 

 is again a letter satisfying the parity checks. The totality of letters satis- 

 fying given parity checks thus constitutes a finite group. 



To prove that every group alphabet is a systematic code, consider 

 the letters of a given (w, /c) -alphabet listed in a column. One obtains in 

 this way a matrix with 2 rows and n columns whose entries are zeros 

 and ones. Because the rows are distinct and form a group isomorphic to 

 Ck , there are k linearly independent rows (mod 2) and no set of more 

 than h independent rows. The rank of the matrix is therefore h. The 

 matrix therefore possesses k linearly independent (mod 2) columns and 

 the remaining n — k columns are linear combinations of these A;. Main- 

 taining only these k linearly independent columns, we obtain a matrix of 

 k columns and 2*' rows with rank k. This matrix must, therefore, have k 

 linearly independent rows. The rows, however, form a group under mod 

 2 addition and hence, since k are linearly independent, all 2" rows must 

 be distinct. The matrix contains only zeros and ones as entries; it has 2 

 distinct rows of k entries each. The matrix must be a listing of the num- 

 bers from to 2^^ — 1 in binary notation. The other n — k columns of 

 the original matrix considered are linear combinations of the columns of 

 this matrix. This completes the proof of Theorem 3 and Proposition 4. 



2.6 PROOF OF THEOREM 4 



To prove Theorem 4 we first note that the parity check sequence of 

 the product of two elements of Bn is the mod 2 sum of their separate 



