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

Ball, Simeon. "On sets of vectors of a finite vector space in which every subset of basis size is a basis." Journal of the European Mathematical Society 014.3 (2012): 733-748. <http://eudml.org/doc/277697>.

@article{Ball2012,
abstract = {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 $k\le p$, and at most $q+k−p$, if $q\ge k \ge p+1\ge 4$, where $q=p^k$ and $p$ is prime. Moreover, for $k\le p$, 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 $k\le p$ 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 $n\ge r+2$ is representable over $\mathbb \{F\}_p$ if and only if $n\le p+1$. 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.},
author = {Ball, Simeon},
journal = {Journal of the European Mathematical Society},
keywords = {arcs; Maximum Distance Separable Codes (MDS codes); uniform matroids; vector space; basis; rank; Reed-Solomon codes; arcs; maximum distance separable codes; uniform matroids; vector space; basis; rank; Reed-Solomon codes},
language = {eng},
number = {3},
pages = {733-748},
publisher = {European Mathematical Society Publishing House},
title = {On sets of vectors of a finite vector space in which every subset of basis size is a basis},
url = {http://eudml.org/doc/277697},
volume = {014},
year = {2012},
}

TY - JOUR
AU - Ball, Simeon
TI - On sets of vectors of a finite vector space in which every subset of basis size is a basis
JO - Journal of the European Mathematical Society
PY - 2012
PB - European Mathematical Society Publishing House
VL - 014
IS - 3
SP - 733
EP - 748
AB - 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 $k\le p$, and at most $q+k−p$, if $q\ge k \ge p+1\ge 4$, where $q=p^k$ and $p$ is prime. Moreover, for $k\le p$, 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 $k\le p$ 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 $n\ge r+2$ is representable over $\mathbb {F}_p$ if and only if $n\le p+1$. 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.
LA - eng
KW - arcs; Maximum Distance Separable Codes (MDS codes); uniform matroids; vector space; basis; rank; Reed-Solomon codes; arcs; maximum distance separable codes; uniform matroids; vector space; basis; rank; Reed-Solomon codes
UR - http://eudml.org/doc/277697
ER -