Minimizing using a Karnaugh map when given as subscripts F4,2655

Question

I have to minimize the expression using minterms and a Karnaugh map:

$F_{4,2655}$

How might I get this expression I am given into a form much like a typical boolean algebra minification question? I do not understand the given notation.

From a classmate they commented that the notation means the following:

2655 would be the sum of all the minterm values of function F, expressed in decimal (which when converted to binary gives the Boolean representation that corresponds to truth table).

So...

$2655_{10} = 0000 1010 0101 1111_2$

Then I need to do something with this from here, but I am not sure what. I included an extra 4 bits as I believe the $2^4$ in the question represents the amount of bits. Now having this somewhat converted I am not sure where to go from here.

Answer 1

I assume that F4,2655 is a shorthand notation for the Boolean expression with four inputs which has a truthtable with the binary equivalent of 2655 as output column.

The truthtable looks as follows:

A B C D F    
0 0 0 0 0
0 0 0 1 0
0 0 1 0 0
0 0 1 1 0
0 1 0 0 1
0 1 0 1 0
0 1 1 0 1
0 1 1 1 0
1 0 0 0 0
1 0 0 1 1
1 0 1 0 0
1 0 1 1 1
1 1 0 0 1
1 1 0 1 1 
1 1 1 0 1
1 1 1 1 1  

Translated into a Karnaugh map

             AB
       00  01  11  10
      +---+---+---+---+
   00 | 0 | 1 | 1 | 0 |
      +---+---+---+---+
   01 | 0 | 0 | 1 | 1 |
CD    +---+---+---+---+
   11 | 0 | 0 | 1 | 1 |
      +---+---+---+---+
   10 | 0 | 1 | 1 | 0 |
      +---+---+---+---+

Resulting minimized expression:

A & D  +  B & !D

---

Reversing the order of bits in the output column results in:

Truthtable:

A B C D  F
0 0 0 0  1
0 0 0 1  1
0 0 1 0  1
0 0 1 1  1
0 1 0 0  1
0 1 0 1  0
0 1 1 0  1
0 1 1 1  0
1 0 0 0  0
1 0 0 1  1
1 0 1 0  0
1 0 1 1  1
1 1 0 0  0
1 1 0 1  0 
1 1 1 0  0
1 1 1 1  0 

Karnaugh map:

             AB
       00  01  11  10
      +---+---+---+---+
   00 | 1 | 1 | 0 | 0 |
      +---+---+---+---+
   01 | 1 | 0 | 0 | 1 |
CD    +---+---+---+---+
   11 | 1 | 0 | 0 | 1 |
      +---+---+---+---+
   10 | 1 | 1 | 0 | 0 |
      +---+---+---+---+ 

Minimized expression:

!B & D + !A & !D

So, both cases yield a somewhat similar expression