Spanoids - an abstraction of spanning structures, and a barrier for LCCs

by   Zeev Dvir, et al.

We introduce a simple logical inference structure we call a spanoid (generalizing the notion of a matroid), which captures well-studied problems in several areas. These include combinatorial geometry, algebra (arrangements of hypersurfaces and ideals), statistical physics (bootstrap percolation) and coding theory. We initiate a thorough investigation of spanoids, from computational and structural viewpoints, focusing on parameters relevant to the applications areas above and, in particular, to questions regarding Locally Correctable Codes (LCCs). One central parameter we study is the rank of a spanoid, extending the rank of a matroid and related to the dimension of codes. This leads to one main application of our work, establishing the first known barrier to improving the nearly 20-year old bound of Katz-Trevisan (KT) on the dimension of LCCs. On the one hand, we prove that the KT bound (and its more recent refinements) holds for the much more general setting of spanoid rank. On the other hand we show that there exist (random) spanoids whose rank matches these bounds. Thus, to significantly improve the known bounds one must step out of the spanoid framework. Another parameter we explore is the functional rank of a spanoid, which captures the possibility of turning a given spanoid into an actual code. The question of the relationship between rank and functional rank is one of the main open problem we raise as it may reveal new avenues for constructing new LCCs (perhaps even matching the KT bound). As a modest first step, we develop an entropy relaxation of functional rank and use it to demonstrate a small constant gap between the two notions. To facilitate the above results we also develop some basic structural results on spanoids including an equivalent formulation of spanoids as set systems and properties of spanoid products.


page 1

page 2

page 3

page 4


Rank-Metric Codes and q-Polymatroids

We study some algebraic and combinatorial invariants of rank-metric code...

Rank and pairs of Rank and Dimension of Kernel of ℤ_pℤ_p^2-linear codes

A code C is called ℤ_pℤ_p^2-linear if it is the Gray image of a ℤ_pℤ_p^2...

ℤ_pℤ_p^2-linear codes: rank and kernel

A code C is called _p_p^2-linear if it is the Gray image of a _p_p^2-add...

On the lower bound for the length of minimal codes

In recent years, many connections have been made between minimal codes, ...

Rank-Metric Codes, Semifields, and the Average Critical Problem

We investigate two fundamental questions intersecting coding theory and ...

On the number of resolvable Steiner triple systems of small 3-rank

In a recent work, Jungnickel, Magliveras, Tonchev, and Wassermann derive...

Please sign up or login with your details

Forgot password? Click here to reset