# The admissibility theorem for the spatial X-ray transform over the two element field

## TL;DR

This paper proves an admissibility theorem for the Radon transform over the two-element field, classifying minimal line collections ensuring injectivity, contrasting with more uniform transform cases.

## Contribution

It provides a classification of minimal line sets for injectivity of the Radon transform over the two-element field, addressing Gelfand's admissibility problem.

## Key findings

- Classified minimal line collections for injectivity
- Contrasted with affine hyperplane and projective line transforms
- Provided accessible presentation of the admissibility theorem

## Abstract

We consider the Radon transform along lines in an $n$ dimensional vector space over the two element field. It is well known that this transform is injective and highly overdetermined. We classify the minimal collections of lines for which the restricted Radon transform is also injective. This is an instance of I.M.~Gelfand's {\it admissibility problem}. The solution is in stark contrast to the more uniform cases of the affine hyperplane transform and the projective line transform, which are addressed in other papers, \cite{Feld-G,Gr1}. The presentation here is intended to be widely accessible, requiring minimum background.