Plotkin bound
In the mathematics of coding theory, the Plotkin bound, named after Morris Plotkin, is a limit (or bound) on the maximum possible number of codewords in binary codes of given length "n" and given minimum distance "d". Statement of the bound. A code is considered "binary" if the codewords use symbols from the binary alphabet formula_1. In particular, if all codewords have a fixed length "n", then the binary code has length "n". Equivalently, in this case the codewords can be considered elements of vector space formula_2 over the finite field formula_3. Let formula_4 be the minimum distance  of formula_5, i.e. where formula_7 is the Hamming distance between formula_8 and formula_9. The expression formula_10 represents the maximum number of possible codewords in a binary code of length formula_11 and minimum distance formula_4. The Plotkin bound places a limit on this expression. Theorem (Plotkin bound): i) If formula_4 is even and formula_14, then ii) If formula_4 is odd and formula_17, then iii) If formula_4 is even, then iv) If formula_4 is odd, then where formula_23 denotes the floor function. Proof of case i. Let formula_7 be the Hamming distance of formula_8 and formula_9, and formula_27 be the number of  elements in formula_5 (thus, formula_27 is equal to formula_10). The bound is proved by bounding the quantity formula_31 in two different ways. On the one hand, there are formula_27 choices for formula_8 and for each such choice, there are formula_34 choices for formula_9. Since by definition formula_36 for all formula_8 and formula_9 (formula_39), it follows that On the other hand, let formula_41 be an formula_42 matrix whose rows are the elements of formula_5. Let formula_44 be the number of zeros contained in the formula_45'th column of formula_41. This means that the formula_45'th column contains formula_48 ones. Each choice of a  zero and a one in the same column contributes exactly formula_49 (because formula_50) to the sum formula_51 and therefore The quantity on the right is maximized if and only if formula_53 holds for all formula_45 (at this point of the proof we ignore the fact, that the formula_44 are integers), then Combining the upper and lower bounds for formula_57 that we have just derived, which given that formula_59 is equivalent to Since formula_27 is even, it follows that This completes the proof of the bound.