Algebraic constructions of LDPC codes with no short cycles

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.