Vector Space Partitions
Start Date
August 2025
End Date
August 2025
Location
ALT 306
Abstract
A vector space is a set of elements that can be added together and multiplied by numbers and that satisfy certain algebraic axioms. A subspace is a non-empty subset of a vector space that is closed under vector addition and scalar multiplication. The dimension of a vector space is the number of elements that completely determine it.
A finite field of order q is a finite set of numbers dependent on a prime power q for which the operations of multiplication, addition, subtraction, and division are defined, and that satisfy certain algebraic rules.
Let V (n, q) denote the vector space of dimension n over a finite field of order q and let n>= 2. A 2-spread is a collection of subspaces of V:=V (n, q), all of dimension 2, such that each nonzero vector of V appears in exactly one of the subspaces. Creating 2-spreads gets increasingly more difficult as the dimension of a vector space and q grow. We present an efficient method that we built for every q to inductively create 2-spreads of a vector space of dimension 2k+2 by using a 2-spread from a 2k-dimensional vector space.
A partial t-spread of V(n, q) is a collection of t-dimensional subspaces of V (n, q) that intersect only in the zero vector. In this talk, we will also discuss our attempts to create a partial 3-spread of size 245 of the V(8,3) vector space; this would have been larger than the known lower bound.
Vector Space Partitions
ALT 306
A vector space is a set of elements that can be added together and multiplied by numbers and that satisfy certain algebraic axioms. A subspace is a non-empty subset of a vector space that is closed under vector addition and scalar multiplication. The dimension of a vector space is the number of elements that completely determine it.
A finite field of order q is a finite set of numbers dependent on a prime power q for which the operations of multiplication, addition, subtraction, and division are defined, and that satisfy certain algebraic rules.
Let V (n, q) denote the vector space of dimension n over a finite field of order q and let n>= 2. A 2-spread is a collection of subspaces of V:=V (n, q), all of dimension 2, such that each nonzero vector of V appears in exactly one of the subspaces. Creating 2-spreads gets increasingly more difficult as the dimension of a vector space and q grow. We present an efficient method that we built for every q to inductively create 2-spreads of a vector space of dimension 2k+2 by using a 2-spread from a 2k-dimensional vector space.
A partial t-spread of V(n, q) is a collection of t-dimensional subspaces of V (n, q) that intersect only in the zero vector. In this talk, we will also discuss our attempts to create a partial 3-spread of size 245 of the V(8,3) vector space; this would have been larger than the known lower bound.
