## Bound for MDS Codes

The MDS-code-conjecture states that the only linear orthogonal arrays with index unity and linear MDS-codes (i.e., codes meeting the Singleton bound with equality) are (extended) Reed-Solomon codes, codes from the hyperoval and codes with dimension or codimension equal to 0 or 1, namely repetition codes, parity-check codes, codes without redundancy, and trivial codes.

In other words, the conjecture states that for every linear OA(*b*^{m}, *s*,**F**_{b}, *m*) and every linear [*s*, *s*−*m*, *m* + 1]-code over **F**_{b} we have

*s*≤

For *m* ∈ {0, 1} the bound is trivial. For *m* ∈ {2, 3} it follows e.g. from the Hamming bound and for *m* ≥ *b* it can be derived from the Plotkin bound. So the only interesting cases are 4 ≤ *m* ≤ *b*−1 and therefore *b* ≥ 5.

Linear [*s*, *s*−*m*, *m* + 1]-codes are equivalent to *s*-arcs in the projective space PG(*m*−1, *b*). Therefore, proving the MDS-code-conjecture is also a problem of finite geometry.

The MDS-code-conjecture is known to be true for all *b* ≤ 27, with the result for 11 ≤ *b* ≤ 19 due to [1] and the result for 23 ≤ *b* ≤ 27 due to [2]. For *b* = 32 it is known to hold at least for *m* ≤ 7 (and therefore for *m* ≥ 27 by considering the dual MDS code), see [3, Table 3.1].

More information on *s*-arcs can be found in [3, Section 3] and [4].

### References

[1] | Jin-Ming Chao and H. Kaneta. Classical arcs in PG( r, q) for 11 ≤ q ≤ 19.Discrete Mathematics, 174(1–3):87–94, September 1997.doi:10.1016/S0012-365X(96)00319-6 |

[2] | Jin-Ming Chao and H. Kaneta. Classical arcs in PG( r, q) for 23 ≤ q ≤ 29.Discrete Mathematics, 226(1–3):377–385, January 2001.doi:10.1016/S0012-365X(00)00169-2 |

[3] | James W. P. Hirschfeld and Leo Storme. The packing problem in statistics, coding theory and finite projective spaces: Update 2001. In Finite Geometries, volume 3 of Developments in Mathematics, pages 201–246. Kluwer Academic Publishers, 2001. |

[4] | James W. P. Hirschfeld. Complete arcs. Discrete Mathematics, 174(1–3):177–184, September 1997.doi:10.1016/S0012-365X(96)00330-5 |

### 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 MDS Codes.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2008-04-04.
http://mint.sbg.ac.at/desc_CBoundT0Lin.html