Proof
First observe that there are many q-ary words of length, since each letter in such a word may take one of different values, independently of the remaining letters.
Now let be an arbitrary q-ary block code of minimum distance . Clearly, all codewords are distinct. If we delete the first letters of each codeword, then all resulting codewords must still be pairwise different, since all original codewords in have Hamming distance at least from each other. Thus the size of the code remains unchanged.
The newly obtained codewords each have length
and thus there can be at most
of them. Hence the original code shares the same bound on its size :
Read more about this topic: Singleton Bound
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