  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.
