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 Fb can only exist if

s ≥ 1 + $\displaystyle \sum_{{i=0}}^{{n−1}}$$\displaystyle \left\lceil\vphantom{ \frac{d}{b^{i}}}\right.$$\displaystyle {\frac{{d}}{{b^{i}}}}$$\displaystyle \left.\vphantom{ \frac{d}{b^{i}}}\right\rceil$.

In [1] this is shown for n = 4, d = b3 – b2 – b – ⌈$ \sqrt{{b}}$⌉ − 1, and all b ≥ 4. In [2] for n = 5, d = b4 – b3 – b – ⌈$ \sqrt{{b}}$⌉ − 1, and all b ≥ 4. In [3] for d = (n – 2)bn−1 – (n – 1)bn−2 for n and b with either n = 3, 4, 5 and b ≥ n or n ≥ 6 and b ≥ 2n−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: 2015-09-03. http://mint.sbg.ac.at/desc_CBoundMaruta.html

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