Digital Circuits – Codes

Binary codes or digital codes– In the digital logic circuits, each decimal or piece of information is represented through an equivalent combination of binary digits.

Binary codes can be classified into two types.

Weighted codes – Positional Weights (positively weighted codes and negatively weighted codes)
Unweighted codes

Example:

8421 code
2421 code
84-2-1 code
Excess 3 Code
Gray Code

Binary Codes for Decimal digits

The following table shows the various binary codes for decimal digits 0 to 9.

Decimal Digit	8421 Code	2421 Code (0-4) Normal Binary Code (5-9) 2421 codes	84-2-1 Code	Excess 3 Code
0	0000	0000	0000	0011
1	0001	0001	0111	0100
2	0010	0010	0110	0101
3	0011	0011	0101	0110
4	0100	0100	0100	0111
5	0101	1011	1011	1000
6	0110	1100	1010	1001
7	0111	1101	1001	1010
8	1000	1110	1000	1011
9	1001	1111	1111	1100

We have 10 digits in decimal number system. To represent these 10 digits in binary, we require minimum of 4 bits. But, with 4 bits there will be 16 unique combinations of zeros and ones. Since, we have only 10 decimal digits, the other 6 combinations of zeros and ones are not required.

1) 8421 Code

Its weights are 8, 4, 2 and 1.
It has all +ve weights. So, it is a positively weighted code.
Also called as natural BCD BinaryCodedDecimal code.

Example

Let us find the BCD equivalent of the decimal number 372. This number has 3 decimal digits 3, 7 and 2. From the table, we can write the BCD 8421 codes of 3,7 and 2 are 0011, 0111 and 0010 respectively.

∴ 372₁₀ = (0011 0111 0010)_BCD

There are 12 bits in BCD representation, since each BCD code of decimal digit has 4 bits.

2) 2421 Code

Its weights are 2, 4, 2 and 1.
It has all positive weights. So, it is a positively weighted code.
It is an unnatural BCD code. Sum of weights of unnatural BCD codes is equal to 9.
It is a self-complementing code or Reflective Code. Self-complementing codes provide the 9’s complement of a decimal number, just by interchanging 1’s and 0’s in its equivalent 2421 representation.

Example 1: Convert Binary of 5 into 2421 code

Solution:

2421 or (2+4+2+1) have to sequence
Value is 5 so we take (2+2+1) only & ignore 4 as it
2+2+1=5 became and Wherever these 3 positions, write 1, the rest is 0.
So 2421 = 1011

Example 2: Convert 7 of 8421 into 2421 code

Solution:

8421 values same as binary values

Convert in 2+4+2+1 & make its sum as 7
So we can take 2+4+1 = 7
So 2421 = 1101

Example 3: Convert 372 into 2421 code

Solution:

3 -> 0011

7 -> 1101 (2+4+2+1 => (2+4+1) =7 so replace the value as 1 (1101))

2 -> 0010

So equivalent of the decimal number 372 = 0011 1101 0010

3) 8 4 -2 -1 Code

Its weights are 8, 4, -2 and -1.
It has negative weights along with positive weights. So, it is a negatively weighted code.
It is an unnatural BCD code.
It is a self-complementing code.

Example 1: Convert 6 into 8 4 -2 -1 code

Solution:

Same as previous rule

Value needed = 6 = ( 8-2)

So 8 -2 is taken as ‘1’ & others ‘0’

ie. 8 4 -2 -1 = 1010

Example 2: Convert 862 into 8 4 -2 -1 code

Solution

8 => 1000 (8 4 -2 -1 => replace 8 position as ‘1’ others ‘0’)

6 => 1010 (8 4 -2 -1 => (8-2 = 6) => replace 8 and -2 position as ‘1’ and others ‘0’

2 => 0110 (8 4 -2 -1 => (4-2=2) => replace 4 and -2 position as ‘1’ and others ‘0’

Value of 862 = 1000 1010 0110

4) Excess 3 Code

It doesn’t have any weights. So, it is an un-weighted code.
Binary value of 3 – 0011
Add value of 3 with all
It is a self-complementing code.

Example 1: Find access-3 value of 5?

Solution:

5 = 0101

3 = 0011

So, 5+3 => 0101+0011 => 1000

Example 1: Find access-3 value of 9?

Solution:

9 = 1001

3 = 0011

So 9+3 => 1001 + 0011 = 1100

5) Gray Code

It doesn’t have any weights. So, it is an un-weighted code.
In the below table, the successive Gray codes are differed in one bit position only. Hence, this code is called as unit distance code.
A Gray code is an encoding of numbers so that adjacent numbers have a single digit differing by 1. The term Gray code is often used to refer to a “reflected” code, or more specifically still, the binary reflected Gray code.

The following table shows the 4-bit Gray codes corresponding to each 4-bit binary code.

Decimal Number	Binary Code	Gray Code
0	0000	0000
1	0001	0001
2	0010	0011
3	0011	0010
4	0100	0110
5	0101	0111
6	0110	0101
7	0111	0100
8	1000	1100
9	1001	1101
10	1010	1111
11	1011	1110
12	1100	1010
13	1101	1011
14	1110	1001
15	1111	1000

Binary code to Gray Code Conversion

Follow these steps for converting a binary code into its equivalent Gray code.

Consider the given binary code and place a zero to the left of MSB.
Compare the successive two bits starting from zero. If the 2 bits are same, then the output is zero. Otherwise, output is one.
Repeat the above step till the LSB of Gray code is obtained.

Example

From the table, we know that the Gray code corresponding to binary code 1000 is 1100. Now, let us verify it by using the above procedure.

Given, binary code is 1000.

Step 1 − By placing zero to the left of MSB, the binary code will be 01000.

Step 2 − By comparing successive two bits of new binary code, we will get the gray code as 1100.

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