Upper bounds for partial spreads

A partial $t$-spread in $\mathbb{F}_q^n$ is a collection of $t$-dimensional subspaces with trivial intersection such that each non-zero vector is covered at most once. We present some improved upper bounds on the maximum sizes.Comment: 4 page

Partial Spreads in Random Network Coding

Following the approach by R. K\"otter and F. R. Kschischang, we study network codes as families of k-dimensional linear subspaces of a vector space F_q^n, q being a prime power and F_q the finite field with q elements. In particular, following an idea in finite projective geometry, we introduce a class of network codes which we call "partial spread codes". Partial spread codes naturally generalize spread codes. In this paper we provide an easy description of such codes in terms of matrices, discuss their maximality, and provide an efficient decoding algorithm

Improved upper bounds for partial spreads

A partial $(k-1)$-spread in $\operatorname{PG}(n-1,q)$ is a collection of $(k-1)$-dimensional subspaces with trivial intersection, i.e., each point is covered at most once. So far the maximum size of a partial $(k-1)$-spread in $\operatorname{PG}(n-1,q)$ was known for the cases $n\equiv 0\pmod k$, $n\equiv 1\pmod k$ and $n\equiv 2\pmod k$ with the additional requirements $q=2$ and $k=3$. We completely resolve the case $n\equiv 2\pmod k$ for the binary case $q=2$.Comment: 8 page

An Overview of the Different Kinds of Vector Space Partitions

In a finite vector space V (n,q), where V is n-dimensional over a finite field with q elements, a collection P of subspaces is called a vector space partition. The property of this set P is that any vector that is not zero may be found in exactly one element of P. Partitions of vector spaces have strong ties to design theory, error-correcting algorithms, and finite projective planes. The first portion of my talk will focus on the mathematical fields that share common ground with vector space partitions. The rest of the lecture will go over some of the most well-known results on vector space partition classification. Heden and Lehmann's result on vector space partitions and maximal partial spreads, as well as El-Zanati et al.'s recent findings on the types found in spaces V(n, 2) for n = 8 or less, the Beutelspacher and Heden theorem on T-partitions, and their newly established condition for the existence of a vector space partition will all be covered. Furthermore, I will demonstrate Heden's theorem about the tail length of a vector space split. Finally, I shall provide some historical notes