## Residual Code

Every linear [*s*, *n*, *d*]-code C over **F**_{b} 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: 2015-09-03.
http://mint.sbg.ac.at/desc_CResidual.html