Plotkin border

In channel coding , block codes are used to detect and correct errors in data streams. A block code of the length above a -nary alphabet with a minimum distance fulfills the Plotkin limit , also known as the Plotkin limit , ${\displaystyle C}$ ${\displaystyle n}$ ${\displaystyle q}$ ${\displaystyle d}$

${\displaystyle |C|\leq {\frac {d}{d-({\frac {q-1}{q}})\cdot n}}}$

then when the denominator is positive. Thus, the Plotkin limit only provides a result if it is sufficiently close to . ${\displaystyle d}$ ${\displaystyle n}$

If a code adopts the Plotkin limit, it is particularly true that the distance between any two code words is exact . ${\displaystyle C}$ ${\displaystyle d}$

Is and with , even the closer relationship applies: ${\displaystyle q\geq 3}$ ${\displaystyle |C|=a\cdot q+b}$ ${\displaystyle b<q}$

${\displaystyle d{|C| \choose 2}\leq n\left({|C| \choose 2}-b{a+1 \choose 2}-(q-b){a \choose 2}\right)}$

For example, the Plotkin limit for , and only , provides the tightening , since there is a contradiction for and . ${\displaystyle q=3}$ ${\displaystyle n=9}$ ${\displaystyle d=7}$ ${\displaystyle |C|\leq 7}$ ${\displaystyle |C|\leq 6}$ ${\displaystyle a=2}$ ${\displaystyle b=1}$

It was published by Morris Plotkin in 1960 .

See also

• Optimal code

Individual evidence

1. ↑ M. Plotkin: Binary codes with specified minimum distance, IRE Transactions on Information Theory, 6: 445-450, 1960 (Engl.).
2. ^ WC Huffman, V. Pless: Fundamentals of Error-Correcting Codes, Cambridge University Press, 2003.
3. ↑ The Plotkin limit and its tightening .