Array Convolutional Low-Density Parity-Check Codes

This paper presents a design technique for obtaining regular time-invariant low-density parity-check convolutional (RTI-LDPCC) codes with low complexity and good performance. We start from previous approaches which unwrap a low-density parity-check (LDPC) block code into an RTI-LDPCC code, and we obtain a new method to design RTI-LDPCC codes with better performance and shorter constraint length. Differently from previous techniques, we start the design from an array LDPC block code. We show that, for codes with high rate, a performance gain and a reduction in the constraint length are achieved with respect to previous proposals. Additionally, an increase in the minimum distance is observed.Comment: 4 pages, 2 figures, accepted for publication in IEEE Communications Letter

On the Minimum/Stopping Distance of Array Low-Density Parity-Check Codes

In this work, we study the minimum/stopping distance of array low-density parity-check (LDPC) codes. An array LDPC code is a quasi-cyclic LDPC code specified by two integers q and m, where q is an odd prime and m <= q. In the literature, the minimum/stopping distance of these codes (denoted by d(q,m) and h(q,m), respectively) has been thoroughly studied for m <= 5. Both exact results, for small values of q and m, and general (i.e., independent of q) bounds have been established. For m=6, the best known minimum distance upper bound, derived by Mittelholzer (IEEE Int. Symp. Inf. Theory, Jun./Jul. 2002), is d(q,6) <= 32. In this work, we derive an improved upper bound of d(q,6) <= 20 and a new upper bound d(q,7) <= 24 by using the concept of a template support matrix of a codeword/stopping set. The bounds are tight with high probability in the sense that we have not been able to find codewords of strictly lower weight for several values of q using a minimum distance probabilistic algorithm. Finally, we provide new specific minimum/stopping distance results for m <= 7 and low-to-moderate values of q <= 79.Comment: To appear in IEEE Trans. Inf. Theory. The material in this paper was presented in part at the 2014 IEEE International Symposium on Information Theory, Honolulu, HI, June/July 201

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

Invertible Bloom Lookup Tables with Listing Guarantees

The Invertible Bloom Lookup Table (IBLT) is a probabilistic concise data structure for set representation that supports a listing operation as the recovery of the elements in the represented set. Its applications can be found in network synchronization and traffic monitoring as well as in error-correction codes. IBLT can list its elements with probability affected by the size of the allocated memory and the size of the represented set, such that it can fail with small probability even for relatively small sets. While previous works only studied the failure probability of IBLT, this work initiates the worst case analysis of IBLT that guarantees successful listing for all sets of a certain size. The worst case study is important since the failure of IBLT imposes high overhead. We describe a novel approach that guarantees successful listing when the set satisfies a tunable upper bound on its size. To allow that, we develop multiple constructions that are based on various coding techniques such as stopping sets and the stopping redundancy of error-correcting codes, Steiner systems, and covering arrays as well as new methodologies we develop. We analyze the sizes of IBLTs with listing guarantees obtained by the various methods as well as their mapping memory consumption. Lastly, we study lower bounds on the achievable sizes of IBLT with listing guarantees and verify the results in the paper by simulations