Residual Code

Every linear [s, n, d]-code C over Fb yields a linear [sd, n – 1,⌈d /b⌉]-code Cʹ over the same field ([1] for binary codes, [2] for the general case).

Construction

Without loss of generality let the first row of the generator matrix of C be a code word of the form (1,…, 1, 0,…, 0) with weight d. Then a generator matrix of Cʹ is given by discarding this first row as well as the first d columns from the generator matrix of C.

See Also

References

[1]James H. Griesmer.
A bound for error-correcting codes.
IBM Journal of Research and Development, 4:532–542, 1960.
[2]Graham Solomon and Jack J. Stiffler.
Algebraically punctured cyclic codes.
Information and Control, 8(2):170–179, April 1965.
doi:10.1016/S0019-9958(65)90080-X
[3]F. Jessie MacWilliams and Neil J. A. Sloane.
The Theory of Error-Correcting Codes.
North-Holland, Amsterdam, 1977.
[4]Jürgen Bierbrauer.
Introduction to Coding Theory.
Discrete Mathematics and its Applications. Chapman & Hall/CRC, Boca Raton, London, New York, Washington D.C., 2004.
MR2079734 (2005f:94001)

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

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