JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech III Year ERROR CORRECTING CODES Previous Question Paper (August - 2022 - R18)

For a (15, 11) Hamming code, the block length is n = 15 and the number of information bits is k = 11. The code has a minimum distance d_min = 3, meaning it can correct all single-bit errors and detect all double-bit errors. An undetected error occurs when the channel noise transforms a transmitted codeword into another valid codeword.

The probability of an undetected error P_ue is given by the sum of the probabilities of receiving a weight-i error pattern that matches one of the A_i codewords of weight i (excluding the all-zero codeword):

For a Hamming code, the weight distribution A_i is determined by the MacWilliams identity. For a (15, 11) code, the non-zero weights are A₃ = 35, A₄ = 105, A₅ = 168, A₆ = 280, A₇ = 435, A₈ = 435, A₉ = 280, A₁₀ = 168, A₁₁ = 105, A₁₂ = 35, A₁₅ = 1.

Given p = 0.01, the probability is dominated by the first few terms:

Substituting the values: P_ue ≈ 35(0.01)³(0.99)¹² + 105(0.01)⁴(0.99)¹¹ ≈ 3.5 × 10^-5(0.886) + 1.05 × 10^-6(0.895) ≈ 3.1 × 10^-5.

Error control strategies are techniques used to ensure reliable data transmission over noisy channels. The primary strategies are:

• Automatic Repeat Request (ARQ): This is a closed-loop strategy where the receiver detects errors using error-detecting codes (like CRC) and requests a retransmission of the corrupted data. It includes Stop-and-Wait, Go-Back-N, and Selective Repeat protocols. It is highly reliable but introduces variable latency.
• Forward Error Correction (FEC): This is an open-loop strategy where the transmitter adds redundant bits to the data using error-correcting codes (like Hamming, Reed-Solomon, or Convolutional codes). The receiver uses these bits to detect and correct errors without requesting retransmission. It is suitable for real-time applications where retransmission is not feasible.
• Hybrid ARQ (HARQ): This combines FEC and ARQ. The receiver attempts to correct errors using FEC; if the error is uncorrectable, it requests a retransmission via ARQ. This provides the efficiency of FEC with the high reliability of ARQ.