On the dimensions of certain LDPC codes based on q-regular bipartite graphs

TL;DR

This paper proves a conjecture about the dimensions of a family of binary LDPC codes, called LU(3,q), using symplectic geometry and group actions, advancing understanding of their structural properties.

Contribution

It provides a proof for the conjectured dimensions of LU(3,q) LDPC codes when q is odd, based on symplectic geometry and group theory.

Findings

Confirmed the conjecture for odd q

Connected code dimensions to symplectic vector space geometry

Utilized group actions to analyze code structure

Abstract

An explicit construction of a family of binary LDPC codes called LU(3,q), where q is a power of a prime, was recently given. A conjecture was made for the dimensions of these codes when q is odd. The conjecture is proved in this note. The proof involves the geometry of a 4-dimensional symplectic vector space and the action of the symplectic group and its subgroups.