On sets of subspaces with two intersection dimensions and a geometrical junta bound

TL;DR

This paper investigates constant dimension subspace codes with two fixed intersection dimensions, establishing bounds on their structure and classifying extremal cases, especially when the intersection values are consecutive.

Contribution

It introduces an upper bound on the dimension of non-junta codes with two intersection values and classifies extremal codes when these values are consecutive.

Findings

Established an upper bound for the span of non-junta codes with two intersection dimensions.

Proved the bound is tight when the two intersection values are consecutive.

Classified the extremal codes into four infinite families.

Abstract

In this article, constant dimension subspace codes whose codewords have subspace distance in a prescribed set of integers, are considered. The easiest example of such an object is a {\it junta}; i.e. a subspace code in which all codewords go through a common subspace. We focus on the case when only two intersection values for the codewords, are assigned. In such a case we determine an upper bound for the dimension of the vector space spanned by the elements of a non-junta code. In addition, if the two intersection values are consecutive, we prove that such a bound is tight, and classify the examples attaining the largest possible dimension as one of four infinite families.