Construction X with Varšamov Bound

Construction X allows the construction of a new code based on three linear codes C1, C2, and Ce, where C1 is a subcode of C2, and Ce is an appropriately chosen auxiliary code.

Obtaining Code and Subcode Using the Varšamov Bound

In [1] it is shown that every linear [s, n, d]-code C1 can be embedded in an [s + u, n + u, dʹ]-code C2, provided that the existence of C2 is guaranteed by the Varšamov bound. Since C2 is longer than C1, the code words of C1 have to be padded with u zeros to obtain a proper subcode of C2.

Since C2 is not available explicitly, this method is considered non-constructive.

References

[1]Yves Edel and Jürgen Bierbrauer.
Lengthening and the Gilbert-Varshamov bound.
IEEE Transactions on Information Theory, 43(3):991–992, May 1997.
doi:10.1109/18.568708

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. “Construction X with Varšamov Bound.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2008-04-04. http://mint.sbg.ac.at/desc_CConsXGV.html

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