= Solution
For a block code, the <minimum distance of a code> is the minimum <Hamming distance> between distinct codewords, where <Hamming distance> counts positions in which two equal-length binary strings differ. We use equal-length codewords, so the fact that the codomain allows arbitrary finite strings creates no ambiguity. Minimum distance $\delta$ means every two distinct codewords differ in at least $\delta$ positions and some pair differs in exactly $\delta$.
Changing at most $\delta-1$ bits cannot turn a codeword into another codeword. Testing membership therefore detects every such nonzero error. For $t=\lfloor(\delta-1)/2\rfloor$, suppose a received word lies within distance $t$ of two codewords. The <triangle inequality> would put their mutual distance at most $2t<\delta$, a contradiction. Hence the radius-$t$ <Hamming balls> are disjoint and nearest-codeword decoding corrects all errors of weight at most $t$.
For an alphabet of size $m\geq2$, start with $0$ and the $m-1$ standard unit vectors in $\{0,1\}^{m-1}$. Their <minimum distance of a code> is one. Repeat every coordinate $\delta$ times; every distance is multiplied by $\delta$, so the resulting code has \b[<minimum distance of a code> exactly $\delta$]. A one-letter alphabet has no pair of distinct codewords and therefore no finite <minimum distance of a code> under this definition; the construction concerns the usual nontrivial coding situation.
Back to article page
