## Construction for [*s*, *s*−6, 6]-Code from [*s*, 3, *s*−3]-Code

In [1, Theorem 13] it is shown how a linear [*s*, *s*−6, 6]-code can be constructed from a linear [*s*, 3, *s*−3]-code C, both over the same field **F**_{b}.

The columns of the generator matrix of C form a plane (*s*, 3)-arc in the projective plane PG(2, *b*). Applying the transformation *φ* : PG(2, *b*)→PG(5, *b*) defined by

*φ*(

*a*:

*b*:

*c*) := (

*a*

^{2}:

*ab*:

*ac*:

*b*

^{2}:

*bc*:

*c*

^{2})

to each of these points yields *s* points in PG(5, *b*), such that every 5 of them are in general position. Hence, the resulting matrix is the parity check matrix of an [*s*, *s*−6, 6]-code over **F**_{b}.

### References

[1] | Mario A. de Boer. Almost MDS codes. Designs, Codes and Cryptography, 9(2):143–155, October 1996.doi:10.1007/BF00124590 |

### 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. “Construction for [*s*, *s*−6, 6]-Code from [*s*, 3, *s*−3]-Code.”
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