GATE EC 2022 Set 1 Q52 — Digital Communications

Consider communication over a memoryless binary symmetric channel using a $(7, 4)$ Hamming code. Each transmitted bit is received correctly with probability $(1 – \in),$ and flipped with probability $\in.$ For each codeword transmission, the receiver performs minimum Hamming distance decoding, and correctly decodes the message bits if and only if the channel introduces at most one bit error.

For $\in = 0.1,$ the probability that a transmitted codeword is decoded correctly is ___________(rounded off to two decimal places).