id: 06026375
dt: j
an: 06026375
au: Ball, Simeon
ti: On sets of vectors of a finite vector space in which every subset of basis
    size is a basis.
so: J. Eur. Math. Soc. (JEMS) 14, No. 3, 733-748 (2012).
py: 2012
pu: European Mathematical Society Publishing House, Zürich
la: EN
cc: 
ut: arcs; maximum distance separable codes; uniform matroids; vector space;
    basis; rank; Reed-Solomon codes
ci: 
li: doi:10.4171/JEMS/316
ab: Summary: It is shown that the maximum size of a set ${ S}$ of vectors of a
    $k$-dimensional vector space over ${\mathbb F}_q$, with the property
    that every subset of size $k$ is a basis, is at most $q+1$, if $k \leq
    p$, and at most $q+k-p$, if $q \geq k \geq p+1 \geq 4$, where $q=p^h$
    and $p$ is prime. Moreover, for $k\leq p$, the sets $S$ of maximum size
    are classified, generalising Beniamino Segre’s “arc is a conic”
    theorem. These results have various implications. One such implication
    is that a $k\times (p+2)$ matrix, with $k \leq p$ and entries from
    ${\mathbb F}_p$, has $k$ columns which are linearly dependent. Another
    is that the uniform matroid of rank $r$ that has a base set of size $n
    \geq r+2$ is representable over ${\mathbb F}_p$ if and only if $n \leq
    p+1$. It also implies that the main conjecture for maximum distance
    separable codes is true for prime fields; that there are no maximum
    distance separable linear codes over ${\mathbb F}_p$, of dimension at
    most $p$, longer than the longest Reed-Solomon codes. The
    classification implies that the longest maximum distance separable
    linear codes, whose dimension is bounded above by the characteristic of
    the field, are Reed-Solomon codes.
rv: