K-Maps

About Karnaugh Maps:

Some addtional notes on Gray codes and 5-bit K-maps

Consider the 4-bit K-map matrix:

(w, x, y, z)		Final Bits (y, z) - binary and decimal
(w, x, y, z)		00	01	11	10
Initial Bits (w, x)	00	0000 = 0	0001 = 1	0011 = 3	0010 = 2
	01	0100 = 4	0101 = 5	0111 = 7	0110 = 6
	11	1100 = 12	1101 = 13	1111 = 15	1110 = 14
	10	1000 = 8	1001 = 9	1011 = 11	1010 = 10

w		Final Bits
w		00	01	11	10
Initial Bits	00	0000	0001	0011	0010
	01	0100	0101	0111	0110
	11	1100	1101	1111	1110
	10	1000	1001	1011	1010

w'		Final Bits
w'		00	01	11	10
Initial Bits	00	0000	0001	0011	0010
	01	0100	0101	0111	0110
	11	1100	1101	1111	1110
	10	1000	1001	1011	1010

x		Final Bits
x		00	01	11	10
Initial Bits	00	0000	0001	0011	0010
	01	0100	0101	0111	0110
	11	1100	1101	1111	1110
	10	1000	1001	1011	1010

x'		Final Bits
x'		00	01	11	10
Initial Bits	00	0000	0001	0011	0010
	01	0100	0101	0111	0110
	11	1100	1101	1111	1110
	10	1000	1001	1011	1010

y		Final Bits
y		00	01	11	10
Initial Bits	00	0000	0001	0011	0010
	01	0100	0101	0111	0110
	11	1100	1101	1111	1110
	10	1000	1001	1011	1010

y'		Final Bits
y'		00	01	11	10
Initial Bits	00	0000	0001	0011	0010
	01	0100	0101	0111	0110
	11	1100	1101	1111	1110
	10	1000	1001	1011	1010

z		Final Bits
z		00	01	11	10
Initial Bits	00	0000	0001	0011	0010
	01	0100	0101	0111	0110
	11	1100	1101	1111	1110
	10	1000	1001	1011	1010

z'		Final Bits
z'		00	01	11	10
Initial Bits	00	0000	0001	0011	0010
	01	0100	0101	0111	0110
	11	1100	1101	1111	1110
	10	1000	1001	1011	1010

xz		Final Bits
xz		00	01	11	10
Initial Bits	00	0000	0001	0011	0010
	01	0100	0101	0111	0110
	11	1100	1101	1111	1110
	10	1000	1001	1011	1010

x'z'		Final Bits
x'z'		00	01	11	10
Initial Bits	00	0000	0001	0011	0010
	01	0100	0101	0111	0110
	11	1100	1101	1111	1110
	10	1000	1001	1011	1010

xy		Final Bits
xy		00	01	11	10
Initial Bits	00	0000	0001	0011	0010
	01	0100	0101	0111	0110
	11	1100	1101	1111	1110
	10	1000	1001	1011	1010

wx		Final Bits
wx		00	01	11	10
Initial Bits	00	0000	0001	0011	0010
	01	0100	0101	0111	0110
	11	1100	1101	1111	1110
	10	1000	1001	1011	1010

wxy'		Final Bits
wxy'		00	01	11	10
Initial Bits	00	0000	0001	0011	0010
	01	0100	0101	0111	0110
	11	1100	1101	1111	1110
	10	1000	1001	1011	1010

w'yz		Final Bits
w'yz		00	01	11	10
Initial Bits	00	0000	0001	0011	0010
	01	0100	0101	0111	0110
	11	1100	1101	1111	1110
	10	1000	1001	1011	1010

w'yz		Final Bits
w'yz		00	01	11	10
Initial Bits	00	0000	0001	0011	0010
	01	0100	0101	0111	0110
	11	1100	1101	1111	1110
	10	1000	1001	1011	1010

wx'yz'		Final Bits
wx'yz'		00	01	11	10
Initial Bits	00	0000	0001	0011	0010
	01	0100	0101	0111	0110
	11	1100	1101	1111	1110
	10	1000	1001	1011	1010

w'x'yz		Final Bits
w'x'yz		00	01	11	10
Initial Bits	00	0000	0001	0011	0010
	01	0100	0101	0111	0110
	11	1100	1101	1111	1110
	10	1000	1001	1011	1010

Gray Codes - From Carpinelli, page 9

New Gray codes are produced by taking the previous, appening the list reversed, and putting 0's before the first half and 1's before the second half, thus:

Bits: 
   0    1    2    3     4       5
   0    0   00   000   0000   00000
        1   01   001   0001   00001
            11   011   0011   00011
            10   010   0010   00010
                 110   0110   00110
                 111   0111   00111
                 101   0101   00101
                 100   0100   00100
                       1100   01100
                       1101   01101
                       1111   01111
                       1110   01110
                       1010   01010
                       1011   01011
                       1001   01001
                       1000   01000
                              11000
                              11001
                              11011
                              11010
                              11110
                              11111
                              11101
                              11100
                              10100
                              10101
                              10111
                              10110
                              10010
                              10011
                              10001
                              10000

5-Bit K-Maps - not quite so useful

`(v, w, x, y, z)`		`Final Bits (y, z) - binary and decimal`
`(v, w, x, y, z)`		`00`	`01`	`11`	`10`
`Initial Bits (v, w, x)`	`000`	`00000 = 0`	`00001 = 1`	`00011 = 3`	`00010 = 2`
	`001`	`00100 = 4`	`00101 = 5`	`00111 = 7`	`00110 = 6`
	`011`	`01100 = 12`	`01101 = 13`	`01111 = 15`	`01110 = 14`
	`010`	`01000 = 8`	`01001 = 9`	`01011 = 11`	`01010 = 10`
	`110`	`11000 = 24`	`11001 = 25`	`11011 = 26`	`11010 = 26`
	`111`	`11100 = 28`	`11101 = 29`	`11111 = 31`	`11110 = 30`
	`101`	`10100 = 20`	`10101 = 21`	`10111 = 23`	`10110 = 22`
	`100`	`10000 =16`	`10001 = 17`	`10011 = 19`	`10010 = 18`

This representation is nice in v, w, y and z, but expression involving x are not so nice, see the following example:

x'z'		`Final Bits (y, z) - binary and decimal`
x'z'		`00`	`01`	`11`	`10`
`Initial Bits (v, w, x)`	`000`	`00000 = 0`	`00001 = 1`	`00011 = 3`	`00010 = 2`
	`001`	`00100 = 4`	`00101 = 5`	`00111 = 7`	`00110 = 6`
	`011`	`01100 = 12`	`01101 = 13`	`01111 = 15`	`01110 = 14`
	`010`	`01000 = 8`	`01001 = 9`	`01011 = 11`	`01010 = 10`
	`110`	`11000 = 24`	`11001 = 25`	`11011 = 26`	`11010 = 26`
	`111`	`11100 = 28`	`11101 = 29`	`11111 = 31`	`11110 = 30`
	`101`	`10100 = 20`	`10101 = 21`	`10111 = 23`	`10110 = 22`
	`100`	`10000 =16`	`10001 = 17`	`10011 = 19`	`10010 = 18`

w

w'

x

x'

y

y'

z

z'

xz

x'z'

xy

wx

wxy'

w'yz

w'yz

wx'yz'

w'x'yz

Gray Codes - From Carpinelli, page 9

5-Bit K-Maps - not quite so useful

x'z'