Which of the following binary codes for decimal digits are self complementing?…

Correct answer: 2421 code and excess-3 code.

Definition: A code is self-complementing if the bitwise complement of the codeword for digit d equals the codeword for 9−d.

• 2421 code: This is a weighted code with weights 2, 4, 2, 1. The sum of weights is 2+4+2+1 = 9, so when every bit is complemented the represented value becomes 9 minus the original value. Therefore the bitwise complement of the code for d is the code for 9−d, and 2421 is self-complementing.

• Excess-3 code: Each digit d is represented by the 4-bit binary of (d+3). The bitwise complement of binary(d+3) equals binary(15−(d+3)) = binary(12−d) = binary((9−d)+3), which is exactly the excess-3 code for 9−d. Example: excess-3 for 2 is binary(5)=0101 and for 7 is binary(10)=1010, which are complements.

Why the other codes fail:

• 8421 code is not self-complementing because its weights sum to 8+4+2+1 = 15, not 9; bitwise complementing a 8421 codeword does not produce the 9's complement codeword.

• Excess-3 Gray code is not self-complementing because applying Gray coding to excess-3 codewords breaks the direct bitwise-complement relation; the Gray transform does not map complements to complements in general.