The structure of dual Schubert union codes

TL;DR

This paper demonstrates that Schubert union codes are Tanner codes derived from Grassmannian point-line incidence geometry, providing a lengthening algorithm, an encoding method, and the minimum distance of these codes.

Contribution

It introduces a lengthening algorithm and an iterative encoding method for Schubert union codes, establishing their Tanner code structure and determining their minimum distance.

Findings

Schubert union codes are Tanner codes from Grassmannian geometry.

A systematic encoding algorithm with linear complexity is developed.

The minimum distance of Schubert union codes is determined.

Abstract

In this article we prove that Schubert union codes are Tanner codes constructed with the point--line incidence geometry that Schubert varieties inherit from the Grassmannian. We do this by first finding an lengthening algorithm for Tanner codes. This algorithm finds the entries of a codeword of a Tanner code from the entries in a given subset of its positions. We find sufficient conditions on the initial set and the initial positions such that a codeword is determined from the component codes only. We find an iterative and systematic encoding algorithm for Schubert union codes with linear complexity. With this encoder we also determine the minimum distance of Schubert union codes.