## Bound for Almost-MDS-Codes with Dimension *n* = 3

(*s*, *r*)-caps are sets of *s* points in some space such that not more than *r* points are collinear. In this terminology normal *s*-caps are (*s*, 2)-caps. (*s*, *r*)-caps in the projective plane PG(2, *b*) are also known as plane (*s*, *r*)-arcs. (*s*, 3)-caps in PG(2, *b*) are equivalent to linear [*s*, 3, *s*−3]-near-MDS-codes over **F**_{b}.

[1, Corollary 3] states that there is no (2*b* + 2, 3)-arc in PG(2, *b*) for *b* > 3. In [1, Theorem 2] the special case for *b* = 3^{u} with *u* ≥ 2 is considered. For other values of *b*, the result follows from [2].

Therefore, a linear [2*b* + 2, 3, 2*b*−1]-near MDS code over **F**_{b} with *b* > 3 cannot exist. Furthermore, it is shown in [3] that every [*s*, *s*−*m*, *m*]-almost-MDS code (i.e., a code with Singleton defect *t* = 1) with *m* > *b* is also a near-MDS code. The code mentioned above has *m* = 2*b* + 2 – 3 = 2*b* – 1 > *b*. Therefore, every code with these parameters would be a near-MDS code therefore cannot exist (see also [4, Theorem 8]).

### References

[1] | Joseph A. Thas. Some results concerning endtex2html_{d}eferred{(q + 1)(n – 1);nendtex2html_{d}eferred}-arcs and endtex2html_{d}eferred{(q + 1)(n – 1) + 1;nendtex2html_{d}eferred}-arcs in finite projective planes of order q.Journal of Combinatorial Theory, Series A, 19(2):228–232, September 1975.doi:10.1016/S0097-3165(75)80012-4 |

[2] | Adriano Barlotti. Sui endtex2html_{d}eferred{k;nendtex2html_{d}eferred}-archi di un piano lineare finito.Bollettino della Unione Matematica Italiana., 11:553–556, 1956.MR0083141 (18,666e) |

[3] | Stefan M. Dodunekov and Ivan N. Landjev. On near-MDS codes. Journal of Geometry, 54(1–2):30–43, November 1995.doi:10.1007/BF01222850 |

[4] | 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. “Bound for Almost-MDS-Codes with Dimension *n* = 3.”
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Version: 2015-09-03.
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