Based On The Complete Cyclic Difference Sets Of Ldpc Codes Constructed

Low-density parity-check(LDPC) codes, which were introduced by Gallager in 1962. Several Low-density parity-check(LDPC) codes have been designed achieving performance results very close to the Shannon limit when iteratively decoded using the sum-product algorithm with linear complexity. During the past few years, intense research has been focused on pseudo-random LDPC codes with the aim of closing the gap between the Shannon limit and error correction performances of LDPC codes. Despite the excellent error-correcting properties of some known pseudo-random LDPC codes, complexity issues trend to dominate system architecture and design consideration. The high complexity of pseudo-random LDPC codes is a direct consequence of the fact that for these codes a large amount of information is necessary to specify positions of the non-zero elements in the parity-check matrix. The complexity drawbacks of pseudo-random LDPC codes can be overcome by using structured LDPC codes. Several classes of structured LDPC codes are known, including those based on combinatorial design, finite geometries and mutually orthogonal Latin squares. Some of the structured LDPC codes have the property of being quasi-cyclic. Quasi-cyclic LDPC codes (QC-LDPC) have encoding advantages over pseudo-random LDPC codes as they can be encoded using simple shift-registers with complexity linearly proportional their code length.The performance of an LDPC codes with iterative decoding depends on a number of code structure properties. One such structural property is the girth of the code that is defined as the length of the shortest cycle in the code's Tanner graph. For a code to perform well with iterative decoding, its Tanner graph must not contain too many short cycles of length 4. Therefore in code construction, cycles of length 4 must be prevented. Many experimental results show that the error-floor of an LDPC codes with iterative decoding very much depends on the minimum distance of an LDPC code and hence it is necessary to keep the minimum distance of an LDPC code reasonably large. For an irregular code with iterative decoding, its error performance also depends on the degree distributions of the variable and check nodes of its Tanner graphThis paper presents a new approach to construct low-density parity-check (LDPC) codes. Two classes of LDPC codes are constructed based on prefect cyclic difference sets. The first class is constructed by decomposition of incidence matrices of the prefect cyclic difference sets. The decomposition technique reduces the density of the parity check matrix and hence reduces the number of short cycles, which usually implies better performance. The parity-check matrix of the second class of LDPC codes are matrices composed of some small circulant permutation matrices and cover a large set of codes with different rates and column weights, as the class of codes which is known as array codes. So the LDPC codes constructed are quite suitable for parallel VLSL decoder implementations. Since there are no cycles of length 4 in the Tanner graphs, dependence among extrinsic information is greatly reduced during iteration and decoding performance is also improved. Experimental simulations in terms of Bit Error Rate and Frame Error Rate on the Additive While Gaussian Noise (AWGN) Channels show that our method has remarkable coding gains. Furthermore, they are quasi-cyclic and their encoding can be achieved in linear time and implemented with simple feedback shift registers .We note that this advantage is not shared by random LDPC codes in general.