1430 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 



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that all possible combinations of trial rows are considered. The set of 

 fewest selected rows is the actual set of basis rows. '| \\ 



Table VII illustrates the process of determining basis rows for a 

 cyclic prime implicant table. After rows G and J have been selected u| |et( 

 cyclic table results, Table VII (b). Rows A and B are then chosen as al 

 pair of trial basis rows since column has crosses in only these two rows. , 

 The selection of row A leads to the selection of rows D and E as given in ; 

 Table VII (c). Row A is marked with three asterisks to indicate that it 

 is a trial basis row. Table VII (d) illustrates the fact that the selection! 

 of rows C and F is brought about by the selection of row B. Since bothi 

 sets of selected rows have the same number of rows (5) they are both 

 sets of basis rows. Each set of basis rows corresponds to a different min- 

 imum sum so that there are two minimum sums for this function. 



Sometimes it is not necessary to determine all minimum sums 

 for the transmission being considered. In such cases, it may be possible 

 to shorten the process of determining basis rows. Since each column 

 must have a cross in some basis row, the total number of crosses in all 

 of the basis rows is equal to or greater than the number of columns. 

 Therefore, the number of columns divided by the greatest number of 

 crosses in any row (or the next highest integer if this ratio is not an 

 integer) is equal to the fewest possible basis rows. For example, in Table 

 VII there are ten columns and two crosses in each row. Therefore, 

 there must be at least 10 divided by 2 or 5 rows in any set of basis rows. 

 The fact that there are only five rows selected in Table VII (c) guaran- 

 tees that the selected rows are basis rows and therefore Table VII (d) is 

 unnecessary if only one minimum sum is required. In general, the process 

 of trying different combinations of trial rows can be stopped as soon as 

 a set of selected rows which contains the fewest possible number of basis 

 rows has been found (providing that it is not necessary to discover all 

 minimum sums) . It should be pointed out that more than the minimum 

 number of basis rows may be required in some cases and in these 

 cases all combinations of trial rows must be considered. A more accurate 

 lower bound on the number of basis rows can be obtained by considering 

 the number of rows which have the most crosses. For example, in Table 

 VI there are 15 columns and 4 crosses, at most, in any row. A lower 

 bound of 4 {—- = 3f ) is a little too optimistic since there are only three 

 rows which contain four crosses. A more realistic lower bound of 5 is 

 obtained by noting that the rows which have 4 crosses can provide crosses 

 in at most 12 columns and that at least two additional rows containing 

 two crosses are necessary to provide crosses in the three remaining col- 

 umns. 



