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A matrix-theoretic approach for studying quasi-cyclic codes based on matrix transformations via Fourier transforms and row and column permutations is developed. These transformations put a parity-check matrix in the form of an array of circulant matrices into a diagonal array of matrices of the same size over an extension field. The approach is amicable to(More)
This paper is concerned with construction and structural analysis of both cyclic and quasi-cyclic codes, particularly low-density parity-check (LDPC) codes. It consists of three parts. The first part shows that a cyclic code given by a parity-check matrix in circulant form can be decomposed into descendant cyclic and quasi-cyclic codes of various lengths(More)
This paper is concerned with construction and structural analysis of both cyclic and quasi-cyclic codes, particularly LDPC codes. It consists of three parts. The first part shows that a cyclic code given by a parity-check matrix in circulant form can be decomposed into descendant cyclic and quasi-cyclic codes of various lengths and rates. Some fundamental(More)
This paper presents a technique to decompose a cyclic code given by a parity-check matrix in circulant form into descendant cyclic and quasi-cyclic codes of various length and rates. Based on this technique, cyclic finite geometry (FG) LDPC codes are decomposed into a large class of cyclic FG-LDPC codes and a large class of quasi-cyclic FG-LDPC codes.
Several classes of quasi-cyclic LDPC codes have been proposed in the literature and shown to have excellent performance over noisy channels when decoded with iterative message-passing algorithms. However, by and large, important properties of the codes, including their dimensions, are only given for specific codes based on computer programming. Using(More)
This paper shows that a cyclic code can be put into quasi-cyclic form by decomposing a circular parity-check matrix through column and row permutations. Such a decomposition of a circular parity-check matrix of a cyclic code produces a group of shorter cyclic or quasi-cyclic codes and leads to a new method for constructing long cyclic codes from short(More)
The trapping set structure of LDPC codes constructed using finite geometries is analyzed. A trapping set is modeled as a sub-geometry of the geometry used to construct an LDPC code. The variable nodes of a trapping set are viewed as points of the geometry and the check nodes adjacent to the variable nodes are viewed as the lines passing through any of these(More)
An approach for studying quasi-cyclic codes based on matrix transformations via Fourier transforms and row and column permutations is presented. These transformations put a parity-check matrix in the form of an array of circulant matrices into a diagonal array of matrices of the same size over an extension field. The approach is used to characterize certain(More)
THIS PAPER IS ELIGIBLE FOR THE STUDENT PAPER AWARD. This paper analyzes trapping set structure of binary regular LDPC codes whose parity-check matrices satisfy the constraint that no two rows (or two columns) have more than one place where they both have non-zero components, which is called row-column (RC) constraint. For a (γ,ρ)-regular LDPC(More)
The structure of certain subgraphs of the Tanner graph of an LDPC code, the trapping sets, has been identified as important for the error floor performance of iterative decoding algorithms. To investigate such sets requires the parity check matrix of the code to be generated with sufficient structure that allows useful information to be obtained while(More)