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
Famous quotes containing the word proof:
“To cease to admire is a proof of deterioration.”
—Charles Horton Cooley (1864–1929)
“War is a beastly business, it is true, but one proof we are human is our ability to learn, even from it, how better to exist.”
—M.F.K. Fisher (1908–1992)
“There is no better proof of a man’s being truly good than his desiring to be constantly under the observation of good men.”
—François, Duc De La Rochefoucauld (1613–1680)