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 Fb.
[1, Corollary 3] states that there is no (2b + 2, 3)-arc in PG(2, b) for b > 3. In [1, Theorem 2] the special case for b = 3u with u ≥ 2 is considered. For other values of b, the result follows from [2].
Therefore, a linear [2b + 2, 3, 2b−1]-near MDS code over Fb 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 = 2b + 2 – 3 = 2b – 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 endtex2htmldeferred{(q + 1)(n – 1);n endtex2htmldeferred}-arcs and endtex2htmldeferred{(q + 1)(n – 1) + 1;n endtex2htmldeferred}-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 endtex2htmldeferred{k;n endtex2htmldeferred}-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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