GATE ECE 2025 | Question: 23

Question

The generator matrix of a $(6,3)$ binary linear block code is given by

$$G=\left[\begin{array}{llllll} 1 & 0 & 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 & 1 & 0
\end{array}\right]$$

The minimum Hamming distance $d_{\text {min }}$ between codewords equals _________ (answer in integer).

Answer 1

• For a linear block code, the minimum Hamming distance ($d_{\min}$) is equal to the minimum Hamming weight (number of 1s) of the non-zero codewords.

• The codewords are generated by all possible linear combinations (sums modulo-2) of the rows of $G$.

• Since there are 3 rows ($k=3$), there are $2^3 - 1 = 7$ non-zero codewords to check.

 

Combination

Calculation (Modulo-2 Sum)

Codeword

Weight

Row 1(Direct)1 0 0 1 0 1 3

Row 2(Direct)0 1 0 0 1 1 3

Row 3(Direct)0 0 1 1 1 0 3

R1 + R2 100101 + 0100111  1 0 1 1 0 4

R1 + R3 100101 + 0011101  0 1 0 1 1 4

R2 + R3 010011 + 0011100  1 1 1 0 1 4

R1 + R2 + R3 110110 + 0011101  1 1 0 0 0 3

 

Looking at the weights calculated above ($3, 3, 3, 4, 4, 4, 3$), the smallest non-zero weight is 3.

Answer:

The minimum Hamming distance $d_{\min}$ equals 3.

Answer 2

4