Code Converters - BCD(8421) to/from Excess-3

Last Updated : 23 Oct, 2026

Code converters are combinational circuits that convert data from one binary code to another. BCD (8421) and Excess-3 are two commonly used decimal codes in digital electronics, each designed for different representation and arithmetic requirements.

  • BCD (8421) represents each decimal digit directly using a unique 4-bit weighted binary code and is commonly used in calculators, digital clocks, and display systems.
  • Excess-3 is a non-weighted, self-complementing code obtained by adding 3 (0011) to the corresponding BCD value, making it useful for complement-based arithmetic operations.
bcd_to_excess_3

Converting BCD(8421) to Excess-3

A BCD (8421) digit is converted to its corresponding Excess-3 code by adding 3 (0011) to the BCD value. Since only decimal digits 0–9 are valid, the remaining input combinations are treated as don't-care (X) conditions.

Let A, B, C, and D represent the BCD (8421) input bits, where A is the MSB and D is the LSB. Similarly, let w, x, y, and z represent the Excess-3 output bits, where w is the MSB and z is the LSB.

The truth table for the conversion is given below. The X's mark is don't care condition. 

To obtain the corresponding digital circuit, K-Maps are used to minimize the Boolean expression for each Excess-3 output bit using the BCD (8421) input bits.

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Minimized Boolean expressions (partial):

w = A + BC + BD
x = B′C + B′D + BC′D′

The corresponding digital circuit:

Converting Excess-3 to BCD(8421)

An Excess-3 code is converted back to BCD (8421) using the reverse conversion process.

  • Let w, x, y, and z represent the Excess-3 input bits, where w is the MSB and z is the LSB.
  • Similarly, let A, B, C, and D represent the BCD (8421) output bits, where A is the MSB and D is the LSB.
  • Let w, x, y, and z represent the Excess-3 code bits

The truth table for the conversion is given below. The X's mark is don't care condition. 

K-Map for D

K-Map for C

K-Map for B

K-Map for A

 
Corresponding Minimized Boolean Expressions for BCD (8421) Output Bits:

A = wx + wyz
B = x′y′ + x′z′ + xyz
C = y′z + yz′
D = z′

The corresponding digital circuit:

Here, E3, E2, E1, and E0 represent the Excess-3 input bits (w, x, y, z), while B3, B2, B1, and B0 represent the BCD (8421) output bits (A, B, C, D).
 

Excess-3 to BCD

BCD vs Excess-3

BCDExcess-3
Represents each decimal digit directly using its 4-bit binary equivalent.Represents each decimal digit by adding 3 (0011) to its BCD value.
Requires additional correction logic during arithmetic operations.Simplifies complement and subtraction operations due to its self-complementing property.
Does not provide built-in error detection.Invalid code combinations are easier to detect.
Uses a 4-bit binary code for each decimal digit.Uses a 4-bit code obtained by adding 3 to the corresponding BCD value.
Directly represents decimal digits.Conversion requires adding or subtracting 3 between BCD and Excess-3.
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