Construction Y1

Shortening allows the construction of a linear [su, nu, d]-code from a linear [s, n, d]-code for u = 1,…, n. However, in some situations it is possible to obtain a code-dimension of nu + 1 instead of nu.

Given a linear [s, n, d]-code C, then a new linear [su, nu + 1, d]-code Cʹ over the same field exists, given that the dual distance d of C (the distance of C, the dual code of C) is less or equal to u.

The new code is obtained by removing u columns from the parity check matrix of C such that all d non-zero coordinates of the minimum weight code word of C are removed.

Since MinT has no knowledge about the minimum distance of C, it has to assume the worst and use the upper bound on the minimum distance of any [s, sn]-code.

Parameters of the Involved Codes

This propagation rule is used by MinT in three different circumstances:

See Also

References

[1]F. Jessie MacWilliams and Neil J. A. Sloane.
The Theory of Error-Correcting Codes.
North-Holland, Amsterdam, 1977.

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 Y1.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2008-04-04. http://mint.sbg.ac.at/desc_CConsB.html

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