## Improvement by Maruta on the Griesmer Bound

In a series of papers Maruta determines cases in which the Griesmer bound can be improved by one, i.e., parameters *b*, *n*, and *d* where a linear [*s*, *n*, *d*]-code over **F**_{b} can only exist if

*s*≥ 1 + .

In [1] this is shown for *n* = 4, *d* = *b*^{3} – *b*^{2} – *b* – ⌈⌉ − 1, and all *b* ≥ 4. In [2] for *n* = 5, *d* = *b*^{4} – *b*^{3} – *b* – ⌈⌉ − 1, and all *b* ≥ 4. In [3] for *d* = (*n* – 2)*b*^{n−1} – (*n* – 1)*b*^{n−2} for *n* and *b* with either *n* = 3, 4, 5 and *b* ≥ *n* or *n* ≥ 6 and *b* ≥ 2*n*−3.

### References

[1] | Tatsuya Maruta. On the non-existence of linear codes attaining the Griesmer bound. Geometriae Dedicata, 60(1):1–7, March 1996.doi:10.1007/BF00150863 |

[2] | Tatsuya Maruta. On the minimum length of q-ary linear codes of dimension five.Geometriae Dedicata, 65(3):299–304, March 1997.doi:10.1023/A:1004901203236 |

[3] | Tatsuya Maruta. On the achievement of the Griesmer bound. Designs, Codes and Cryptography, 12(1):83–87, September 1997.doi:10.1023/A:1008250010928 |

### 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. “Improvement by Maruta on the Griesmer Bound.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2008-04-04.
http://mint.sbg.ac.at/desc_CBoundMaruta.html