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Proof. Clearly dH (x, y) = dH (x − z, y − z) for all z. In particular dH (x, y) = dH (x − y, y − y) = dH (x − y, 0) . 2 A consequence of the lemma is that minimum distance for linear codes is much easier to calculate than for arbitrary codes. One need only survey |C| codewords for weight rather than roughly |C|2 pairs for distance. Examples. Of course the minimum weight of the length n repetition code is n. Also the minimum weight of the parity check code is clearly 2. The minimum weight of the length 27 real Reed-Solomon code is equal to its minimum distance which we found to be 21.

With coset leader decoding, the error patterns that are corrected are exactly those that are the chosen coset leaders. In particular, the code will be e-error-correcting if and only if every vector of weight at most e is the unique leader of its coset. Coset leader decoding is an MDD algorithm for linear codes over fields of size m. Therefore knowledge of the coset leaders for C makes it easy to calculate PC on an mSC(p). Indeed, an error pattern will be corrected if and only if it is a chosen coset leader.

Correcting code in An , then |C| · |Se (∗)| ≤ |An | . 6) Corollary. ) If C is a m-ary e-error-correcting code of length n, then R3 e w W n (m − 1)i . |C| ≤ mn i i=0 perfect e-error-correcting code 2 A code C that meets the sphere packing bound with equality is called a perfect e-error-correcting code. Equivalently, C is a perfect e-error-correcting code if and only if SSe is a MDD algorithm. As examples we have the binary repetition codes of odd length. 1. 7) Theorem. ) There exists an m-ary e-error-correcting code C of length n such that R3 2e w W n (m − 1)i .

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