Residual Code
Every linear [s, n, d]-code C over Fb yields a linear [s – d, 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
Applying this propagation rule n−1 times leads to the Griesmer bound. Therefore, performing this propagation rule is also called a Griesmer step.
References
| [1] | James H. Griesmer. A bound for error-correcting codes. IBM Journal of Research and Development, 4:532–542, 1960. |
| [2] | Gustave 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: 2024-09-05.
http://mint.sbg.ac.at/desc_CResidual.html