To guarantee correction of upto t errors, the minimum Hamming distance dmin in…

Key result: To guarantee correction of up to t errors the minimum Hamming distance must be at least 2t+1.

Reasoning:

• If the minimum distance d_min ≥ 2t+1, then for any two distinct codewords c1 and c2 we have d(c1,c2) ≥ 2t+1.

• Assume a received word r is within Hamming distance t of c1, so d(r,c1) ≤ t. If r were also within t of c2, then by the triangle inequality d(c1,c2) ≤ d(c1,r)+d(r,c2) ≤ t+t = 2t, which contradicts d(c1,c2) ≥ 2t+1.

• Thus spheres of radius t around different codewords are disjoint, and any received word with up to t errors decodes uniquely to the original codeword.

• Conversely, if d_min ≤ 2t then there exist two codewords at distance ≤ 2t, and a received word at distance ≤ t from both codewords can arise, causing ambiguous decoding. Therefore d_min must be at least 2t+1.

Conclusion: The minimum Hamming distance required to guarantee correction of up to t errors is 2t+1.