In order that a code is ‘\(t\)’ error correcting, the minimum Hamming distance…

Concept: For a code to correct up to t bit errors, the minimum Hamming distance between any two valid codewords must satisfy d_min ≥ 2t + 1. This follows from the nearest-codeword decoding rule: every valid codeword defines a sphere of radius t (all words reachable by at most t bit-flips from it), and decoding is guaranteed to recover the correct codeword only when no two such spheres overlap.

Application: Applying this rule to the item:

1. Let C1 and C2 be any two valid codewords, decoded by nearest-codeword matching.

2. First check that a distance of only 2t is NOT enough: if d(C1, C2) = 2t exactly, construct a received word W that agrees with C1 in t of the differing positions and with C2 in the other t differing positions. Then d(C1, W) = t = d(C2, W), so a single t-bit error pattern produces a W equidistant from both codewords — decoding is ambiguous, so 2t cannot guarantee correction.

3. Now check that d(C1, C2) ≥ 2t + 1 IS enough: if C1 is transmitted with at most t errors producing W, then d(C1, W) ≤ t.

4. By the triangle inequality, d(C1, C2) ≤ d(C1, W) + d(W, C2), which rearranges to d(W, C2) ≥ d(C1, C2) − d(C1, W) ≥ (2t + 1) − t = t + 1.

5. Since d(W, C2) ≥ t + 1 > t ≥ d(C1, W), the received word W is always strictly closer to C1 than to C2, so nearest-codeword decoding always recovers C1 correctly. So 2t + 1 is both necessary (step 2) and sufficient (steps 3–4), i.e. the minimum Hamming distance of the whole code must satisfy d_min ≥ 2t + 1.

Cross-check: The well-known (7, 4) Hamming code has d_min = 3 and is documented to correct exactly 1-bit errors (t = 1). Substituting into the derived relation: 2t + 1 = 2(1) + 1 = 3, which matches d_min = 3 exactly — confirming the relation independently.

Result: The minimum Hamming distance required for a code to be t-error-correcting is 2t + 1.