1450 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 



both xi and X2 , and .T3 and Xi leave this matrix unchanged so that 

 ^21437" = T. The possibiHty of priming different combinations of the 

 columns which have an equal number of O's and I's must now be con- 

 sidered. Certain of the possible combinations can be excluded before- 

 hand. In Table III (a) the only possibility which must be considered is 

 that of priming both xi and X2 . If only xi or X2 is primed, there will be 

 no row which has all zeros. No permutation of the columns of this 

 matrix (with Xi or .1-2 primed) can produce a row with all zeros. Therefore 

 this matrix cannot possibly be made equal to the original matrix by re- 

 arranging rows and columns. Priming both Xi and X2 must be considered 

 since the 12-row will be converted into a row with all zeros. The opera- 

 tion of priming .Ti and X2 is written symbolical!}^ as A^ioo = A''i2 . In 

 general, if the matrix has a row consisting of all zeros, only those Ni 

 operations for which i is the number of some row in the matrix, need be 

 considered. If the row does not have an all-zero row, only those A'', for 

 which i is not the number of some row need be considered. Similarly, if 

 the matrix has a row consisting of all I's, only those A\- for which there 

 is some row of the matrix which will be convei'ted into an all-one row% 

 need be considered. This is equivalent to considering only those Ni for 

 which some row has a number /c = 2" — 1 — t* where n is the number 

 of columns. If the matrix does not have an all-one row, only those A^, for 

 which no row has a number A: = 2" — 1 — i should be considered. 



Each priming operation which is not excluded by these rules is applied 

 to the transmission matrix. The matrices so formed are then partitioned 

 as described previously. Any of these matrices that have the same par- 

 titioning as the original matrix are then inspected to see if any row and 

 column permutations will convert them to the original matrix. For the 

 matrix of Table III (a) the operation of priming both Xi and X2 was not 

 excluded. The matrix which results when these columns are primed is 

 shown in Table Ill(b). Inspection of this figure shows that interchange 

 of either Xs and .T4 or Xi and X2 will convert the matrix back to the 

 matrix of Table III (a). Therefore, for the transmission of this table 

 SuizNimT = T and S2i3iNnmT = T. 



2 TOTAL SYMMETRY 



There are certain transmissions whose value depends not on which 

 relays are operated but only on how many relays are operated. For 



* The number of the row which has all ones is 2" — 1 . If Ni operating on some 

 row, k, is to produce the all-one row, i must have I's wherever k has O's and vice 

 versa. This means that 



i + k = 2"" - 1 or A; = 2" - 1 - i. 



