1442 THE BELL SYSTEM TECHXICAL JOURNAL, NOVEMBER 1956 



Table XV — Determination of the Prime Implicants for the 

 Transmission of Table XV Specified as a 

 Sum of Product Terms 



ter has a zero (one). In Table X\'a the (0, 1) character has a zero in the 

 .r2-position while the (3, 7) character has a one in the .ro-position. A new 

 character is fornied (1, 3) which has a dash in ihe .<-2-p()sition. 



This rule for constructing new characters is actually a generalization 

 of the rule used in Section 3 and corresponds to the theorem. 



.ri.r2 + .r/.rii = XiX-s + .ri'.r;5 + .r2.r3 . 



Repeated application of this rule will lead to the complete set of prime 

 implicants. As described in Section 3, any character which has all of the 

 numbers of its decimal label appearing in the label of another character 

 should be checked. The unchecked characters then represent the prime 

 implicants. The process described in this section was discussed fi'om a 

 slightly different point of view by Quine.^ 



12 summary and conclusions 



In this paper a method has been presented for writing any transmis- 

 sion as a minimum sum. This method is similar to that of Quine; how- 

 ever, several significant improvements have been made. The notation 

 has been simplified by using the symbols 0, 1 and - instead of primed 

 and unprimed variables. While it is not completeh^ new in itself, this 

 notation is especially appropriate for the arrangement of terms used in 

 determining the prime implicants. Listing the terms in a column which 

 is partitioned so as to place terms containing the same number of 1 's in 

 the same partition reduces materially the labor involved in determining 

 the prime implicants. Such a list retains some of the advantage of the 

 arrangement of squares in the Karnaugh Chart without reciuiring a 

 geometrical representation of an n-dimensional hj^percube. Since the 



