Different Types of Binary Codes | BCD (8421), 2421, Excess-3, Gray

In this tutorial, we will learn about one of the basic requirements of digital electronics i.e., the Binary Codes of the Binary Number System. Some of the popular Binary Codes are BCD (8421), 2421, 5211, Excess-3, Gray.

Introduction

The first successful system of electrical communication was the telegraph, which was invented by Samuel F.B. Morse in the year 1832. The telegraph operators used a code of clicks to send the messages. If the key pressed for a short time it is the Morse code called ‘dot’ and if the key pressed for a long-time it is ‘Dash’. A sample Morse code is shown below:

Morse Code Symbols

If it is observed logically, various innumerable combinations of dots and dashes of any kind of words (even sentences) can be written utilizing the above code. In the same way, binary digits also be utilized to make such various innumerable combinations. These may be considered as Binary Codes.

Other than the commonly used 8421 code or BCD code, other Binary Codes like 2421 code, 5211 code, reflective code, sequential code, non-weighted code, excess-3 code and Gray code are also popular.

Commonly used Binary Codes

Before going into the details of individual binary codes, let us quickly take a look at some of the commonly used Binary Codes. The following is the list:

  • 8421 Codes
  • 2421 Codes
  • 5211 Codes
  • Excess-3 Codes
  • Gray Codes

In the above list, the first three i.e., 8421, 2421 and 5211 are Weighted Binary Codes while the other two are Non-Weighted Binary Codes.

Weighted Binary Systems

The values assigned to consecutive places in the decimal system which is a place value system are 10⁴, 10³, 10², 10¹, 10⁰, 10⁻¹, 10⁻², 10⁻³… and so on from left to right. It is easily can be understood that the weight of digit of the decimal system is ‘10’.

For example:

 (3546.25)10 = 3 x 10³ + 5 x 10² + 4 x 10¹ + 6 x 10⁰ + 2 x 10⁻¹ + 5 x 10⁻² 

In the same way, the values assigned to consecutive places in the binary system, which is also a place value system, are called as weighted binary system.

The weights in a binary system are 2⁴, 2³, 2², 2¹, 2⁰, 2⁻¹, 2⁻², 2⁻³… from left to right. It is easily can be understood that the weight of digit of the binary system is ‘2’.

For example:

 (1110110)2 = 1 x 2⁶+ 1 x  2⁵ + 1 x  2⁴ + 0 x  2³ + 1 x  2² + 1 x 2¹ + 0 x 2⁰ 

 = 64 + 32 + 16 + 0 + 4 + 2 + 0 = (118)10 

Binary Weights

Whenever any binary number appears, its decimal equivalent can be found easily as follows.

  • When there is 1 in a digit position, weight of that position should be added.
  • When there is 0 in a digit position, weight of that position should be disregarded.

For example binary number 1100 has a decimal equivalent of 8 + 4 + 0 + 0 = 12.

8421 Code or BCD Code

The decimal numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 can be expressed in Binary numbers as shown in the following table. All these binary numbers again expressed in the last column by expanding into 4 bits. As per the weighted binary digits, the 4 Bit binary numbers can be expressed according to their place value from left to right as 8421 (2³ 2² 2¹ 2⁰ = 8421).

Decimal Number
Binary Number
4 BIT Expression(8421)
0
0
0000
1
1
0001
2
10
0010
3
11
0011
4
100
0100
5
101
0101
6
110
0110
7
111
0111
8
1000
1000
9
1001
1001

As per the above expression all the decimal numbers written in the 4 Bit binary code in the form of 8421 and this is called as 8421 Code and also as Binary coded decimal BCD.

As this is a straight code, any Decimal number can be expressed easily because the weights of the positions are straight for easy conversion into this 8421 code.

There are other forms of codes which are not so popular but rather confusing. They are 2421 code, 5211 code, reflective code, sequential code, non- weighted coded, excess-3 code and Grey code. They are having their own importance for some of the exclusive applications and may be useful for some of the special applications.

2421 Code

This code also a 4 bit application code where the binary weights carry 2, 4, 2, 1 from left to right.

Decimal Number
Binary Number
2421 Code
0
0
0000
1
1
0001
2
10
0010
3
11
0011
4
100
0100
5
101
1011
6
110
1100
7
111
1101
8
1000
1110
9
1001
1111

5211 Code

This code is also a 4 bit application code where the binary weights carry 5, 2, 1, 1 from left to right.

Decimal Number
Binary Number
5211 Code
0
0
0000
1
1
0001
2
10
0011
3
11
0101
4
100
0111
5
101
1000
6
110
1010
7
111
1100
8
1000
1110
9
1001
1111

Reflective Code

It can be observed that in the 2421 and 5211 codes, the code for decimal 9 is the complement of the code for decimal 0, the code for decimal 8 is the complement of the code for decimal 1, the code for decimal 7 is the complement of the code for decimal 2, the code for decimal 6 is the complement of the code for decimal 3, the code for decimal 5 is the complement of the code for decimal 4. These codes are called as Reflective Codes. The same can be observed in the following table:

Decimal Number
Binary Number
2421 Code
5211 Code
0
0
0000
0000
1
1
0001
0001
2
10
0010
0011
3
11
0011
0101
4
100
0100
0111
5
101
1011
1000
6
110
1100
1010
7
111
1101
1100
8
1000
1110
1110
9
1001
1111
1111

NOTE: 8421 code is not a reflective code.

Sequential Codes

Sequential codes are the codes in which 2 subsequent numbers in binary representation differ by only one digit. The 8421 and Excess-3 codes are examples of sequential codes. 2421 and 5211 codes do not come under sequential codes.

Decimal Number
Binary Number
8421 Code
Excess-3
0
0
0000
0011
1
1
0001
0100
2
10
0010
0101
3
11
0011
0110
4
100
0100
0111
5
101
0101
1000
6
110
0110
1001
7
111
0111
1010
8
1000
1000
1011
9
1001
1001
1100

Non-Weighted Codes

Some of the codes will not follow the weights of the sequence binary numbers these are called as non-weighted codes. ASCII code and Grey code are some of the examples where they are coded for some special purpose applications and they do not follow the weighted binary number calculations.

Excess-3 Code

As mentioned above, some of the codes will not follow the binary weights, Excess-3 code is an example of it and it is an important 4 bit code. The excess – 3 code of a decimal number is achieved by adding the number 3 to the 8421 code.

For example to convert 15 to an excess-3 code, first 3 to be added to each digit as shown below.

Excess-3 Code Explanation

Excess – 3 Code Examples

  1. Find the excess-3 code of (237.75)10
  2. Find the decimal number of excess-3 number 110010100011.01110101.

Sol:

1) The excess-3 code for (237)10 is obtained by adding 3 to all the digits individually, that is 2, 3 and 7 will become 5, 6 and 10 respectively. These 5, 6 and 10 decimals have to be converted into binary form and the result is 010101101010.

The excess-3 code for (.75)10 is obtained by replacing 7 and 5 with 10 and 8 respectively by adding 3 to each digit. That is, the excess-3 code for (.75)10 is .10101000.

Combining the results of the integral and fractional parts, the excess-3 code for (237.75)10 is 010101101010.10101000.

2) The excess-3 code is 110010100011.01110101

By separating 4 bits as group the equivalent excess-3 code is given as 1100 1010 0011.0111 0101.

Subtracting 0011 from each four-bit group, we obtain the new number as: 1001 0111 0000.0100 0010.

Therefore, the decimal equivalent is (970.42)10.

Gray Code

The gray code is the code where one bit will be differed to the preceding number. For example, decimal numbers 13 and 14 are represented by gray code numbers 1011 and 1001, these numbers differ only in single position that is the second position from the right. In the same way first position on the left changes for 7 and 8 which are 0100 and 1100 and this is also called Unit-distance code. The gray code has very special place in digital electronics.

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

Conclusion

An introductory tutorial on Binary Codes. You learned different binary codes like BCD (8421), 2421, 5211, Excess-3 and Gray Codes.

4 Responses

  1. Good article, just one minor correction. You listed the weights as 5, 4, 2 and 1 for 5211 code, when they are really 5, 2, 1 and 1.

Leave a Reply

Your email address will not be published. Required fields are marked *