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. A code is considered "binary" if the codewords use symbols from the binary alphabet
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
over the finite field
be the minimum distance of
is the Hamming distance between
The expression
represents the maximum number of possible codewords in a binary code of length
and minimum distance
The Plotkin bound places a limit on this expression.
Theorem (Plotkin bound): i) If
, then ii) If
is even, then iv) If
denotes the floor function.
be the Hamming distance of
be the number of elements in
is equal to
The bound is proved by bounding the quantity
in two different ways.
On the one hand, there are
), it follows that On the other hand, let
matrix whose rows are the elements of
be the number of zeros contained in the
'th column contains
Each choice of a zero and a one in the same column contributes exactly
) to the sum
and therefore The quantity on the right is maximized if and only if
holds for all
(at this point of the proof we ignore the fact, that the
are integers), then Combining the upper and lower bounds for
is even, it follows that This completes the proof of the bound.