New Combinatorial Construction Techniques for Low-Density Parity-Check Codes and Systematic Repeat-Accumulate Codes

This paper presents several new construction techniques for low-density parity-check (LDPC) and systematic repeat-accumulate (RA) codes. Based on specific classes of combinatorial designs, the improved code design focuses on high-rate structured codes with constant column weights 3 and higher. The proposed codes are efficiently encodable and exhibit good structural properties. Experimental results on decoding performance with the sum-product algorithm show that the novel codes offer substantial practical application potential, for instance, in high-speed applications in magnetic recording and optical communications channels.Comment: 10 pages; to appear in "IEEE Transactions on Communications

Quantum Error Correction via Codes over GF(4)

The problem of finding quantum error-correcting codes is transformed into the problem of finding additive codes over the field GF(4) which are self-orthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on such codes of length up to 30 qubits.Comment: Latex, 46 pages. To appear in IEEE Transactions on Information Theory. Replaced Sept. 24, 1996, to correct a number of minor errors. Replaced Sept. 10, 1997. The second section has been completely rewritten, and should hopefully be much clearer. We have also added a new section discussing the developments of the past year. Finally, we again corrected a number of minor error

Sparse Graph Codes for Quantum Error-Correction

We present sparse graph codes appropriate for use in quantum error-correction. Quantum error-correcting codes based on sparse graphs are of interest for three reasons. First, the best codes currently known for classical channels are based on sparse graphs. Second, sparse graph codes keep the number of quantum interactions associated with the quantum error correction process small: a constant number per quantum bit, independent of the blocklength. Third, sparse graph codes often offer great flexibility with respect to blocklength and rate. We believe some of the codes we present are unsurpassed by previously published quantum error-correcting codes.Comment: Version 7.3e: 42 pages. Extended version, Feb 2004. A shortened version was resubmitted to IEEE Transactions on Information Theory Jan 20, 200

An Adaptive Entanglement Distillation Scheme Using Quantum Low Density Parity Check Codes

Quantum low density parity check (QLDPC) codes are useful primitives for quantum information processing because they can be encoded and decoded efficiently. Besides, the error correcting capability of a few QLDPC codes exceeds the quantum Gilbert-Varshamov bound. Here, we report a numerical performance analysis of an adaptive entanglement distillation scheme using QLDPC codes. In particular, we find that the expected yield of our adaptive distillation scheme to combat depolarization errors exceed that of Leung and Shor whenever the error probability is less than about 0.07 or greater than about 0.28. This finding illustrates the effectiveness of using QLDPC codes in entanglement distillation.Comment: 12 pages, 6 figure

Binary Cyclic Codes from Explicit Polynomials over \gf(2^m)

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, monomials and trinomials over finite fields with even characteristic are employed to construct a number of families of binary cyclic codes. Lower bounds on the minimum weight of some families of the cyclic codes are developed. The minimum weights of other families of the codes constructed in this paper are determined. The dimensions of the codes are flexible. Some of the codes presented in this paper are optimal or almost optimal in the sense that they meet some bounds on linear codes. Open problems regarding binary cyclic codes from monomials and trinomials are also presented.Comment: arXiv admin note: substantial text overlap with arXiv:1206.4687, arXiv:1206.437