LDPC AND CONVOLUTIONAL CODES FROM MATRIX AND GROUP RINGS

In this chapter, low density parity check (LDPC) codes and convolutional codes are constructed and analyzed using matrix and group rings. It is shown that LDPC codes may be constructed using units or zero-divisors of small support in group rings. From the algebra it is possible to identify where short cycles occur in the matrix of a group ring element thereby allowing the construction, directly and algebraically, of LDPC codes with no short cycles. It is then also possible to construction units of small support in group rings with no short cycles at all in their matrices, thus allowing a huge choice of LDPC codes with no short cycles which may be produced from a single unit element. A general method is given for constructing codes from units in abstract systems. Applying the general method to the system of group rings with their rich algebraic structure allows the construction and analysis of series of convolutional codes. Convolutional codes are constructed and analyzed within group rings in the the infinite cyclic group over rings which are themselves group rings.