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.
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Hirschfeld J.W.P.: Projective Geometries over Finite Fields, 2nd edn. Oxford University Press, Oxford (1998)
Hirschfeld J.W.P., Thas J.A.: General Galois Geometries. Oxford University Press, Oxford (1991)
Pless V.S., Hoffman W.C., Brualdi R.A.: Handbook of Coding Theory. Elsevier, Amsterdam (1998)
Tonchev V.D.: A note on MDS codes, n-arcs, and complete designs. Des. Codes Cryptogr. 29, 247–250 (2003)
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Communicated by Simeon Ball.
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Glynn, D.G. A condition for arcs and MDS codes. Des. Codes Cryptogr. 58, 215–218 (2011). https://doi.org/10.1007/s10623-010-9404-x
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DOI: https://doi.org/10.1007/s10623-010-9404-x