### Abstract

A set of n + k points (k > 0) in projective space of dimension n is said to be an (n + k)-arc if there is no hyperplane containing any n + 1 points of the set. It is well-known that for the projective space PG(n, q), this is equivalent to a maximum distance separable linear code with symbols in the finite field GF(q), of length n + k, dimension n + 1, and distance d = k that satisfies the Singleton bound d ≤ k. We give an algebraic condition for such a code, or set of points, and this is associated with an identity involving determinants.

Original language | English |
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Pages (from-to) | 215-218 |

Number of pages | 4 |

Journal | Designs Codes and Cryptography |

Volume | 58 |

Issue number | 2 |

DOIs | |

Publication status | Published - Feb 2011 |

### Keywords

- Arc
- Determinant
- General position
- Grassmannian
- MDS code
- Projective space

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## Cite this

Glynn, D. (2011). A condition for arcs and MDS codes.

*Designs Codes and Cryptography*,*58*(2), 215-218. https://doi.org/10.1007/s10623-010-9404-x