[Paper Review] Upper bounds for partial spreads

Sascha Kurz|arXiv (Cornell University)|Jun 28, 2016

Cooperative Communication and Network Coding9 references3 citations

TL;DR

This paper presents improved upper bounds for the maximum size of partial $t$-spreads in $ F_q^n$, using a generalized averaging technique on the number of holes (1-dimensional subspaces not covered by the spread). By analyzing the distribution of holes in lower-dimensional subspaces and applying modular arithmetic constraints, the authors derive tighter bounds for $A_q(n,2t;t)$, significantly improving prior results for several cases, including $A_2(15,12;6) \leq 515$ and $A_3(17,12;6) \leq 177280$.

ABSTRACT

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.

Motivation & Objective

• To improve known upper bounds on the maximum size of partial $t$-spreads in $\mathbb{F}_q^n$, denoted $A_q(n,2t;t)$, especially for cases where exact values are unknown.
• To generalize and streamline the hole-averaging method introduced by N"astase and Sissokho to derive tighter constraints on the number of uncovered 1-dimensional subspaces (holes) in lower-dimensional subspaces.
• To establish new theoretical bounds for $A_q(n,2t;t)$ using recursive modular constraints on hole counts across nested subspaces, particularly for $n = kt + r$ with $r < t$.
• To provide explicit, numerically tight upper bounds for various $q$, $n$, and $t$, including cases with $q=2,3,4,5,7,8,9$ and dimensions up to $n=18$, improving over prior results.
• To unify and extend existing approaches from finite geometry and constant-dimension codes by leveraging vector space partition theory and combinatorial averaging arguments.

Proposed method

• Utilizes vector space partitions of type $t^{m_t}1^{m_1}$, where $m_t$ is the number of $t$-dimensional subspaces and $m_1$ the number of uncovered 1-dimensional subspaces (holes), to model partial $t$-spreads.
• Applies Lemma 2.2 to analyze the number of holes in a hyperplane $H$, showing that the number of holes in $H \cap \mathcal{P}$ satisfies $\widehat{m}_1 \equiv \frac{m_1 + x - 1}{q} \pmod{q^{s-1}}$, where $x$ is the remainder in the decomposition of the total number of $s$-dimensional subspaces.
• Employs Lemma 2.3 to show that there exists a hyperplane with strictly fewer holes than the average, leading to a descent argument on hole counts in lower-dimensional subspaces.
• Generalizes the hole-counting descent to $(n-j)$-dimensional subspaces via Corollary 2.4, deriving bounds on hole counts modulo $q^{s-j}$, with explicit expressions involving $c = m_1 \mod q^s$ and $x$.
• Uses Theorem 2.10 to derive a contradiction when assuming too many holes exist in a subspace, based on quadratic Diophantine constraints and the non-negativity of certain expressions involving $\theta(i)$ and $\lambda$.
• Combines these modular constraints with known bounds from prior work (e.g., [3], [4]) to refine upper bounds for $A_q(n,2t;t)$, especially for $q=2,3,4,5,7,8,9$ and various $n,t$.

Experimental results

Research questions

• RQ1 What are the tightest possible upper bounds for $A_q(n,2t;t)$, the maximum size of a partial $t$-spread in $\mathbb{F}_q^n$, for $n > 2t$ and $n \not\equiv 0 \pmod{t}$?
• RQ2 How can the distribution of holes (uncovered 1-dimensional subspaces) in lower-dimensional subspaces be used to derive stronger theoretical constraints on the size of partial spreads?
• RQ3 Can the hole-averaging method of N"astase and Sissokho be generalized and strengthened to yield improved bounds for $A_q(n,2t;t)$ across a wide range of $q$, $n$, and $t$?
• RQ4 What are the exact or tight upper bounds for $A_q(n,2t;t)$ in cases where $n = kt + r$ with $r < t$, especially when $t > \genfrac{[}{]}{0.0pt}{}{r}{1}_q$?
• RQ5 How do the new bounds compare to existing ones, particularly for $q=2$, $q=3$, and higher $q$, in dimensions up to $n=18$?

Key findings

• The paper improves the upper bound for $A_2(15,12;6)$ from 516 to 515, refining a previously known result.
• For $A_2(17,14;7)$, the bound is tightened from 1028 to 1026, demonstrating significant improvement in the binary case.
• For $A_9(18,16;8)$, the bound is reduced from 3,486,784,442 to 3,486,784,420, showing the method's effectiveness for larger $q$.
• The paper establishes tight bounds of the form $q^d l + 1 \leq A_q(n,2t;t) \leq q^d l + c$ for various $q$, $d$, and $n = dk + r$, with explicit values for $l$ and $c$.
• For $A_3(17,12;6)$, the upper bound is $177,280$, derived from the general form $3^6 l + 133$ with $l = \frac{3^{6k-1} - 3^5}{3^6 - 1}$ and $k=3$.
• The method yields improved bounds for $A_8(14,12;6) \leq 16,777,237$ and $A_9(13,10;5) \leq 43,047,086$, showing broad applicability across different parameters.