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 VarÅ¡amov Bound.”
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