## Dual Code (with Bound on *d* by Construction Y1)

Shortening allows the construction of a linear [*s*−*u*, *n*−*u*, *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 *n*−*u* + 1 instead of *n*−*u*.

Given a linear [*s*, *n*, *d*]-code C, then a new linear [*s*−*u*, *n*−*u* + 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*, *s*−*n*]-code.

### Parameters of the Involved Codes

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

The straightforward way is to obtain Cʹ based on C. This method is listed as non-constructive because there is no efficient method for determining the minimum-weight codeword in C

^{⊥}. Cʹ is listed as parent of the construction.This method can also be used for showing that the minimum distance of C

^{⊥}is greater than*u*, provided that Cʹ does not exist. Applied in this way this rule is constructive, because the dual code can always be constructed efficiently. MinT lists C is listed as parent, the non-existence of Cʹ as the second parent of this construction.When MinT uses this method for establishing bounds for codes, the non-existence is proven for the code C, whereas the first base code listed is the (non existent) resulting code Cʹ, and the second base code is the (non existent) dual C

^{⊥}of C.

### See Also

[1, Ch. 18, Section 9.1]

### 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. “Dual Code (with Bound on *d* by Construction Y1).”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2015-09-03.
http://mint.sbg.ac.at/desc_CConsB-dual.html