Dual of MDS Code Is Again an MDS Code

The dual code of a linear MDS code is again a linear MDS code. In other words, the dual of every linear [s, n, s−n + 1]-code is a linear [s, s−n, n + 1]-code over the same field. A code is an MDS code if it meets the Singleton bound [1] with equality, i.e., d = s−n + 1.

This follows from a result by Delsarte [2], who shows that an orthogonal array has index unity if and only if it is an MDS code when interpreted as a code. If the OA is linear, its dual can be constructed, which is obviously also an MDS code.

References

[1]	Richard C. Singleton. Maximum distance q-nary codes. IEEE Transactions on Information Theory, 10(2):116–118, April 1964.
[2]	Philippe Delsarte. An algebraic approach to the association schemes of coding theory. Philips Research Reports Supplement, 10, 1973.

Copyright

Copyright © 2004, 2005, 2006, 2007, 2008, 2009, 2010 by Rudolf Schürer and Wolfgang Ch. Schmid.
Cite this as: Rudolf Schürer and Wolfgang Ch. Schmid. “Dual of MDS Code Is Again an MDS Code.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2024-09-05. http://mint.sbg.ac.at/desc_CDualT0IsT0.html