1448 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 



be true since rearranging the rows of a matrix does not change the total 

 jiumber of I's in each column. Similarly, if priming some columns of a 

 matrix leaves the rows unchanged, either each column must have an 

 equal number of I's and O's or else for each primed column which has an 

 unequal number of O's and I's there must be a second primed column 

 which has as many I's as the first primed column has O's and vice versa. 

 Such pairs of columns must also be interchanged to keep the total num- 

 ber of I's in each column invariant. For the matrix of Table 11(a) the 

 only operations that need be considered are either interchanging xi and 

 X2 or Xz and Xt or priming and interchanging x^ and .re . 



For the present it will be assumed that no columns of the matrix have 

 an equal number of O's and I's. It is possible to determine all permuting 

 and priming operations which leave such a matrix unchanged by con- 

 sidering only permutation operations on a related matrix. This related 

 matrix, called the standard matrix, is formed by priming all the columns 

 of the original matrix which have more I's than O's, the Xq column in the 

 matrix of Table 11(a). Each column of a standard matrix must contain 

 more O's than I's, Table 11(b). The NiSj operations which leave the 

 original matrix unchanged can be determined directly from the oper- 

 ations that leave the corresponding standard matrix unchanged. It is 

 therefore only necessary to consider standard matrices. 



Since no columns of a standard matrix have an equal number of I's 

 and O's and no columns have more I's than O's it is only necessary to 

 consider permuting operations. The number of I's in a column (or row) 

 will be called the weight of the column (or row). Only columns or rows 

 which have the same weights can be interchanged. The matrix should 

 be partitioned so that all columns (or rows) in the same partition have 

 the same weight. Table 11(b). It is now possible to interchange columns 

 in the same column partition and check whether pairs of rows from the 

 same row partition can then be interchanged to regain the original 

 matrix. This can usually be done by inspection. For example, in Table 

 11(b) if columns .r4 and .r3 are interchanged, then interchanging rows 4 

 and 8, 5 and 9, and 6 and 10 will regain the original matrix. 



The process of inspection can be simplified by carrying the partition- 

 ing further. In the matrix of Table 11(b), row 32 cannot be interchanged 

 with either row 8 or row 4. This is because it is not possible to make 

 row 32 identical with either row 8 or row 4 by interchanging columns .ti 

 and X2 . Row 32 has weight 1 in these columns while rows 8 and 14 both 

 have weight 0. In general, only rows which have the same weight in each 

 submatrix can be interchanged. Permuting columns of the same partition 

 does not change the weight of the rows in the corresponding submatrices. 



