Construction X with Varšamov Bound

Construction X allows the construction of a new code based on three linear codes C₁, C₂, and C_e, where C₁ is a subcode of C₂, and C_e 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: 2015-09-03. http://mint.sbg.ac.at/desc_CConsXGV.html