GATE CS 1995 Q1.12 — Data Link Layer: Framing, Error Detection, MAC, Ethernet Bridging

The distance of a code (also known as the minimum Hamming distance, $d_{\min}$) is defined as the minimum Hamming distance between any pair of distinct codewords in the code.

The Hamming distance between two codewords is the number of positions at which the corresponding bits differ.

Let the given codewords be:

• $c_1 = 000000$
• $c_2 = 010101$
• $c_3 = 000111$
• $c_4 = 011001$
• $c_5 = 111111$

Let us calculate the Hamming distance $d(c_i, c_j)$ for some pairs:

• $d(c_1, c_2) = \text{weight}(000000 \oplus 010101) = \text{weight}(010101) = 3$
• $d(c_2, c_3) = \text{weight}(010101 \oplus 000111) = \text{weight}(010010) = 2$
• $d(c_2, c_4) = \text{weight}(010101 \oplus 011001) = \text{weight}(001100) = 2$

Since the minimum possible Hamming distance between any two distinct codewords in this set is $2$, the distance of the code is $2$.

Therefore, the correct option is A.