Algebraic constructions of LDPC codes with no short cycles

TL;DR

This paper introduces an algebraic group ring method for constructing LDPC codes that avoid short cycles in their check matrices, enhancing code performance and reliability.

Contribution

It presents a novel algebraic approach using group rings to systematically construct LDPC codes with no short cycles, improving upon existing methods.

Findings

Constructed LDPC codes with no short cycles from group ring elements

Simulated and compared these codes with standard LDPC codes used in wireless

Demonstrated improved performance and structural properties of the new codes

Abstract

An algebraic group ring method for constructing codes with no short cycles in the check matrix is derived. It is shown that the matrix of a group ring element has no short cycles if and only if the collection of group differences of this element has no repeats. When applied to elements in the group ring with small support this gives a general method for constructing and analysing low density parity check (LDPC) codes with no short cycles from group rings. Examples of LDPC codes with no short cycles are constructed from group ring elements and these are simulated and compared with known LDPC codes, including those adopted for wireless standards.