Design of Finite-Length Irregular Protograph Codes with Low Error Floors over the Binary-Input AWGN Channel Using Cyclic Liftings

We propose a technique to design finite-length irregular low-density parity-check (LDPC) codes over the binary-input additive white Gaussian noise (AWGN) channel with good performance in both the waterfall and the error floor region. The design process starts from a protograph which embodies a desirable degree distribution. This protograph is then lifted cyclically to a certain block length of interest. The lift is designed carefully to satisfy a certain approximate cycle extrinsic message degree (ACE) spectrum. The target ACE spectrum is one with extremal properties, implying a good error floor performance for the designed code. The proposed construction results in quasi-cyclic codes which are attractive in practice due to simple encoder and decoder implementation. Simulation results are provided to demonstrate the effectiveness of the proposed construction in comparison with similar existing constructions.Comment: Submitted to IEEE Trans. Communication

The Trapping Redundancy of Linear Block Codes

We generalize the notion of the stopping redundancy in order to study the smallest size of a trapping set in Tanner graphs of linear block codes. In this context, we introduce the notion of the trapping redundancy of a code, which quantifies the relationship between the number of redundant rows in any parity-check matrix of a given code and the size of its smallest trapping set. Trapping sets with certain parameter sizes are known to cause error-floors in the performance curves of iterative belief propagation decoders, and it is therefore important to identify decoding matrices that avoid such sets. Bounds on the trapping redundancy are obtained using probabilistic and constructive methods, and the analysis covers both general and elementary trapping sets. Numerical values for these bounds are computed for the [2640,1320] Margulis code and the class of projective geometry codes, and compared with some new code-specific trapping set size estimates.Comment: 12 pages, 4 tables, 1 figure, accepted for publication in IEEE Transactions on Information Theor

Stopping Sets of Algebraic Geometry Codes

Abstract — Stopping sets and stopping set distribution of a linear code play an important role in the performance analysis of iterative decoding for this linear code. Let C be an [n, k] linear code over Fq with parity-check matrix H, wheretherowsof H may be dependent. Let [n] ={1, 2,...,n} denote the set of column indices of H. Astopping set S of C with parity-check matrix H is a subset of [n] such that the restriction of H to S does not contain a row of weight 1. The stopping set distribution {Ti (H)} n i=0 enumerates the number of stopping sets with size i of C with parity-check matrix H. Denote H ∗ , the paritycheck matrix, consisting of all the nonzero codewords in the dual code C ⊥. In this paper, we study stopping sets and stopping set distributions of some residue algebraic geometry (AG) codes with parity-check matrix H ∗. First, we give two descriptions of stopping sets of residue AG codes. For the simplest AG codes, i.e., the generalized Reed–Solomon codes, it is easy to determine all the stopping sets. Then, we consider the AG codes from elliptic curves. We use the group structure of rational points of elliptic curves to present a complete characterization of stopping sets. Then, the stopping sets, the stopping set distribution, and the stopping distance of the AG code from an elliptic curve are reduced to the search, counting, and decision versions of the subset sum problem in the group of rational points of the elliptic curve, respectively. Finally, for some special cases, we determine the stopping set distributions of the AG codes from elliptic curves. Index Terms — Algebraic geometry codes, elliptic curves, stopping distance, stopping sets, stopping set distribution, subset sum problem. I