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:
“The thing with Catholicism, the same as all religions, is that it teaches what should be, which seems rather incorrect. This is “what should be.” Now, if you’re taught to live up to a “what should be” that never existed—only an occult superstition, no proof of this “should be”Mthen you can sit on a jury and indict easily, you can cast the first stone, you can burn Adolf Eichmann, like that!”
—Lenny Bruce (1925–1966)
“The chief contribution of Protestantism to human thought is its massive proof that God is a bore.”
—H.L. (Henry Lewis)
“Talk shows are proof that conversation is dead.”
—Mason Cooley (b. 1927)