## ConstructionÂ Y1

Shortening allows the construction of a linear [*s*âˆ’*u*, *n*âˆ’*u*, *d*]-code from a linear [*s*, *n*, *d*]-code for *u* = 1,â€¦, *n*. However, in some situations it is possible to obtain a code-dimension of *n*âˆ’*u* + 1 instead of *n*âˆ’*u*.

Given a linear [*s*, *n*, *d*]-code C, then a new linear [*s*âˆ’*u*, *n*âˆ’*u* + 1, *d*]-code CÊ¹ over the same field exists, given that the dual distance *d*^{âŠ¥} of C (the distance of C^{âŠ¥}, 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*^{âŠ¥} non-zero coordinates of the minimum weight code word of C^{âŠ¥} are removed.

Since MinT has no knowledge about the minimum distance of C^{âŠ¥}, it has to assume the worst and use the upper bound on the minimum distance of any [*s*, *s*âˆ’*n*]-code.

### Parameters of the Involved Codes

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

The straightforward way is to obtain CÊ¹ based on C. This method is listed as non-constructive because there is no efficient method for determining the minimum-weight codeword in C

^{âŠ¥}. CÊ¹ is listed as parent of the construction.This method can also be used for showing that the minimum distance of C

^{âŠ¥}is greater than*u*, provided that CÊ¹ 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 CÊ¹ 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 CÊ¹, and the second base code is the (non existent) dual C

^{âŠ¥}of C.

### See Also

[1, Ch.Â 18, SectionÂ 9.1]

### References

[1] | F.Â Jessie MacWilliams and Neil J.Â 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. “ConstructionÂ Y1.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2024-09-05.
http://mint.sbg.ac.at/desc_CConsB.html