Excess-3 (XS-3) BCD — which code represents each decimal digit by its 4-bit binary value after adding 3 to that digit (i.e., code = binary(digit + 3))?

Introduction / Context:Several BCD variants exist to simplify arithmetic and error detection. Excess-3 (also written XS-3) is a self-complementing code that shifts each decimal digit by 3 before encoding it in 4-bit binary. This property eases 9’s complement and subtraction operations in certain hardware designs.

Given Data / Assumptions:

• Per-digit BCD with 4-bit fields.
• Mapping formula: codeword = binary(digit + 3).
• Valid for digits 0–9 (so 3–12 in binary before mapping; 10–12 codes are still 4-bit patterns).

Concept / Approach:By adding 3 to the decimal digit, Excess-3 ensures that the 9’s complement of a number corresponds to bitwise complement of its code, simplifying subtraction via complements. For example, digit 0 maps to 0011, 1→0100, …, 9→1100. This is not a weighted code and differs from 8421 BCD, which maps 0→0000 through 9→1001 directly without the +3 offset.

Step-by-Step Solution:

Verification / Alternative check:Take d = 5: 5 + 3 = 8 → 1000; d = 9: 9 + 3 = 12 → 1100. The mapping matches standard XS-3 tables.

Why Other Options Are Wrong:

Common Pitfalls:Confusing Excess-3 with 8421 due to both using 4 bits; overlooking its self-complementing advantage for subtraction operations.

Final Answer:Excess-3 code