On sets of vectors of a finite vector space in which every subset of basis size is a basis II

This article contains a proof of the MDS conjecture for k ≤ 2p − 2. That is, that if S is a set of vectors of $${{\mathbb F}_q^k}$$ in which every subset of S of size k is a basis, where q = ph, p is prime and q is not and k ≤ 2p − 2, then |S| ≤ q + 1. It also contains a short proof of the same fact for k ≤ p, for all q.

[1]  Simeon Ball,et al.  On sets of vectors of a finite vector space in which every subset of basis size is a basis II , 2012, Designs, Codes and Cryptography.

[2]  David G. Glynn.  The non-classical 10-arc of PG(4, 9) , 1986, Discret. Math..

[3]  K. A. Bush.  Orthogonal Arrays of Index Unity , 1952 .

[4]  Tim Penttila,et al.  A unified construction of finite geometries associated with q-clans in characteristic 2 , 2003 .

[5]  José Felipe Voloch.  Arcs in projective planes over prime fields , 1990 .

[6]  James G. Oxley,et al.  Matroid theory , 1992 .

[7]  A. Blokhuis,et al.  Arcs in PG(n, q), MDS-codes and three fundamental problems of B. Segre — Some extensions , 1990 .

[8]  H. Kaneta,et al.  An elementary proof and an extension of Thas' theorem on k-arcs , 1989, Mathematical Proceedings of the Cambridge Philosophical Society.

[9]  B. Segre.  Curve razionali normali ek-archi negli spazi finiti , 1955 .

[10]  William Cherowitzo,et al.  α-Flocks and Hyperovals , 1998 .

[11]  David G. Glynn,et al.  Two new sequences of ovals in finite desarguesian planes of even order , 1983 .

[12]  Tim Penttila,et al.  Flocks and ovals , 1996 .

[13]  O. Antoine,et al.  Theory of Error-correcting Codes , 2022 .

[14]  Gábor Korchmáros,et al.  On the Embedding of an Arc into a Conic in a Finite Plane , 1996 .

[15]  J. Hirschfeld,et al.  The packing problem in statistics, coding theory and finite projective spaces : update 2001 , 2001 .

[16]  B. Segre.  Ovals In a Finite Projective Plane , 1955, Canadian Journal of Mathematics.

[17]  J. Thas,et al.  General Galois geometries , 1992 .