Skip to main content

Capability of a code with dmin = 4

Short answer

Hamming distance is the number of differing bit positions between two codewords; d_min is the minimum such distance among all codeword pairs. For d_min = 4, it detects up to 3 errors, corrects 1 error, and can simultaneously correct 1 error and detect up to 2 errors (SEC-DED).

Define Hamming Distance and minimum Hamming distance. Suppose a code has a Hamming distance dmin = 4. What is the error detection and correction capability of this scheme?

Hamming distance

The Hamming distance d(x, y) between two equal-length binary words is the number of positions in which they differ (the weight of x ⊕ y). Example: d(1011, 1101) = 2.

Minimum Hamming distance

d_min is the smallest Hamming distance between any two distinct codewords of a code. Example: for {000, 011, 101, 110}, d_min = 2.

Capability for d_min = 4

  • Error detection: up to d_min − 1 = 3 errors can be detected.
  • Error correction: up to t = ⌊(d_min − 1)/2⌋ = ⌊3/2⌋ = 1 error can be corrected.
  • Simultaneous: with d_min ≥ t + s + 1 (t corrected, s detected, s ≥ t), it can correct 1 error and also detect up to 2 errors (4 ≥ 1 + 2 + 1). This is the single-error-correcting, double-error-detecting (SEC-DED) mode, as in an extended Hamming code.

Rules used: detect s errors needs d_min ≥ s + 1; correct t errors needs d_min ≥ 2t + 1.

✓ Answer checked by Claude Sonnet 5.5

Related questions

All Data Communication chapter 8 questions →