GATE 2026 EC Q.63: solution and answer

Official answer: 0.10 to 0.20Source: Official answer key, GATE 2026 EC (IIT Guwahati)

1. A $(7,4,1)$ Hamming code has block length $n=7$, $k=4$ message bits and error-correcting capability $t=1$ (minimum distance $d_{min}=3$). It decodes a received word correctly whenever the word has at most one bit in error.

2. Take the bit errors as independent with probability $p=0.10$ at the decoder input, so the number of errors in a 7-bit word is binomial with $n=7$, $p=0.1$.

3. The decoder fails when the word has 2 or more errors, so $P_{\text{fail}}=1-P(0)-P(1)$.

4. $$P(0)=(0.9{)}^7\approx 0.4783,P(1)=7(0.1)(0.9{)}^6=7(0.1)(0.5314)\approx 0.3720$$
5. $$P_{\text{fail}}=1-(0.4783+0.3720)=1-0.8503\approx 0.1497$$

6. Rounded to two decimal places, $P_{\text{fail}}\approx 0.15$.