TL;DR
This paper determines the exact dimensions of LU(3,q) LDPC codes for all q, using geometric and algebraic methods, extending previous bounds to exact values.
Contribution
It proves the previously known lower bound for LU(3,q) code dimensions when q is a power of 2, establishing it as exact, via symplectic geometry and representation theory.
Findings
Exact dimension formula for LU(3,q) codes for all q.
Extension of previous bounds to precise dimensions.
Application of symplectic geometry and polynomial rings.
Abstract
A family of LDPC codes, called LU(3,q) codes, has been constructed from q-regular bipartite graphs. Recently, P. Sin and Q. Xiang determined the dimensions of these codes in the case that q is a power of an odd prime. They also obtained a lower bound for the dimension of an LU(3,q) code when q is a power of 2. In this paper we prove that this lower bound is the exact dimension of the LU(3,q) code. The proof involves the geometry of symplectic generalized quadrangles, the representation theory of Sp(4,q), and the ring of polynomials.
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Taxonomy
TopicsCoding theory and cryptography · Cooperative Communication and Network Coding · Finite Group Theory Research