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 Fb.

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) := (a2 : ab : ac : b2 : bc : c2)

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 Fb.

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.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. http://mint.sbg.ac.at/desc_CAMDSM6.html

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