Six-Variable K-map

Six-Variable K-map

• A 6- variable K-map requires 26 = 64 cells. These cells are divided into four identical 16-cells map as shown in the Fig. 3.7.1.

• If the variables are A, B, C, D, E and F, 16-cell maps contain C, D, E and F and each 16-cell map represents four combinations of A and B. The adjacencies between entries in each cell (16-cell map) are visualized identical to that of a four-variable map.

Example for Understanding

Ex. 3.7.1 Simplify the Boolean function.

F (A, B, C, D, E, F) = ∑m( 0, 5, 7, 8, 9,12,13, 23, 24, 25, 28, 29, 37, 40, 42, 44, 46, 55, 56, 57, 60, 61)

Sol. : Group 1 and group 2 are two pairs of Is in the first 16-cell map. Group 3 is formed by two isolated Is from first 16-cell map and third 16-cell map.

Group 4 is a combination of two quads from first 16-cell and second 16-cell map. Similarly group 5 is a combination of two quads from second 16-cell map and fourth 16-cell map. Group 6 is again a combination of isolated Is from second and fourth 16-cell maps. Finally group 7 is a quad within the third 16-cell map. This gives simplified expression as

Example with Solution

Ex. 3.7.2 Find the minimal SOP form for the following 6 variable switching function.

f(x₁,x₂,x₃,x₄,x₅,x₆) = ∑ m (2, 3, 6, 7, 10,14,18,19, 22, 23, 27, 37, 41, 43, 45, 46, 58, 59)

Implement the reduced function using NAND gates only. 

Sol. :

Implementation : Ttie AND-OR function can be implemented by NAND-NAND logic as shown in the Fig. 3.7.5.

1. Summary of Rules for K-Map Simplification

Rules for Simplifying logic function using K-map are :

1. Group should not include any cell containing a zero.

2. The number of cells in a group must be a power of 2, such as 1, 2, 4, 8 or 16.

3. Group may be horizontal, vertical but not diagonal.

4. Cell containing 1 must be included in at least one group.

5. Groups may overlap.

6. Each group should be as large as possible to get maximum simplification.

7. Groups may be wrapped around the map. The leftmost cell in a row may be grouped with the rightmost cell and the top cell in a column may be grouped with the bottom cell.

8. A cell may be grouped more than once. The only condition is that every group must have at least one cell that does not belong to any other group. Otherwise, redundant terms will result.

9. We need not group all don't care cells, only those that actually contribute to a maximum simplification.

10. All above rules are stated considering the SOP simplification. In case of POS simplification all rules are same except 0 (zero) takes place of 1 (one).

Review Question

1. State the rules for K-map simplification.