Kohichi Sakaniwa

Quadratic optimization of fixed points of nonexpansive mappings in hubert space

Finding an optimal point in the intersection of the fixed point sets of a family of nonexpansive mappings is a frequent problem in various areas of mathematical science and engineering. Let be nonexpansive mappings on a Hilbert space H, and let be a quadratic function defined by for all , where is a strongly positive bounded self-adjoint linear operator. Then, for each sequence of scalar parameters (λn) satisfying certain conditions, we propose an algorithm that generates a sequence converting strongly to a unique minimizer u* of Θ over the intersection of the fixed point sets of all the Ti’s. This generalizes some results of Halpern (1967), Lions (1977), Wittmann (1992), and Bauschke (1996). In particular, the minimization of Θ over the intersection of closed convex sets Ci can be handled by taking Ti to the metric projection onto Ci without introducing any special inner products that depends on A. We also propose an algorithm that generates a sequence converging to a unique minimizer of Θ over , where K...

Quantum Error Correction Beyond the Bounded Distance Decoding Limit

In this paper, we consider quantum error correction over depolarizing channels with nonbinary low-density parity-check codes defined over Galois field of size 2p. The proposed quantum error correcting codes are based on the binary quasi-cyclic Calderbank, Shor, and Steane (CSS) codes. The resulting quantum codes outperform the best known quantum codes and surpass the performance limit of the bounded distance decoder. By increasing the size of the underlying Galois field, i.e., 2p, the error floors are considerably improved.

Multiplicatively Repeated Nonbinary LDPC Codes

We propose nonbinary LDPC codes concatenated with multiplicative repetition codes. By multiplicatively repeating the (2,3)-regular nonbinary LDPC mother code of rate 1/3, we construct rate-compatible codes of lower rates 1/6, 1/9, 1/12,.... Surprisingly, such simple low-rate nonbinary LDPC codes outperform the best low-rate binary LDPC codes so far. Moreover, we propose the decoding algorithm for the proposed codes, which can be decoded with almost the same computational complexity as that of the mother code.

Some Classes of Quasi-Cyclic LDPC Codes: Properties and Efficient Encoding Method*This paper was presented in part at the 26th Symposium on Information Theory and Its Applications, Higashiura, Hyogo, Japan, December 15--18, 2003 and at IEEE International Symposium on Information Theory, Chicago, IL USA, June 27--July 2, 2004.

Low-density parity-check (LDPC) codes are one of the most promising next-generation error-correcting codes. For practical use, efficient methods for encoding of LDPC codes are needed and have to be studied. However, it seems that no general encoding methods suitable for hardware implementation have been proposed so far and for randomly constructed LDPC codes there have been no other methods than the simple one using generator matrices. In this paper we show that some classes of quasi-cyclic LDPC codes based on circulant permutation matrices, specifically LDPC codes based on array codes and a special class of Sridhara-Fuja-Tanner codes and Fossorier codes can be encoded by division circuits as cyclic codes, which are very easy to implement. We also show some properties of these codes.

Spatially-coupled MacKay-Neal codes and Hsu-Anastasopoulos codes

Kudekar et al. recently proved that for transmission over the binary erasure channel (BEC), spatial coupling of LDPC codes increases the BP threshold of the coupled ensemble to the MAP threshold of the underlying LDPC codes. One major drawback of the capacity-achieving spatially coupled LDPC codes is that one needs to increase the column and row weight of parity-check matrices of the underlying LDPC codes.

Spatially coupled LDPC codes for decode-and-forward in erasure relay channel

We consider spatially-coupled LDPC codes for the three terminal erasure relay channel. It is observed that BP threshold value of spatially-coupled LDPC codes, in particular spatially-coupled MacKay-Neal code, is close to the theoretical limit for the relay channel. Empirical results suggest that spatially-coupled LDPC codes have great potential to achieve theoretical limit of a general relay channel.

Algebraic multidimensional phase unwrapping and zero distribution of complex polynomials-characterization of multivariate stable polynomials

We define the multidimensional unwrapped phase for any finite extent multidimensional signal that may have its zero on the distinguished boundary of the unit polydisc. By using this definition, we deduce that multivariate stable polynomials can be simply characterized in terms of the proposed unwrapped phase. A rigorous symbolic algebraic solution to the exact phase unwrapping problem for multidimensional finite extent signals is also proposed. This solution is based on a newly developed general Sturm sequence and does not need any numerical root finding or numerical integration technique. Furthermore, it is shown that the proposed algebraic phase unwrapping algorithm can be used to determine the exact zero distribution of any univariate complex polynomial without suffering the so-called singular case problem.

Fountain Coding via Multiplicatively Repeated Non-Binary LDPC Codes

We study fountain codes transmitted over the binary-input symmetric-output channel. For channels with small capacity, receivers in fountain coding systems needs to collects many channel outputs to recover information bits. Since a collected channel output yields a check node in the decoding Tanner graph, the channel with small capacity leads to large decoding complexity. In this paper, we introduce a novel fountain coding scheme with non-binary LDPC codes. The decoding complexity of the proposed fountain code does not depend on the channel. Numerical experiments show that the proposed codes exhibit better performance than conventional fountain codes, especially for moderate number of information bits.

Weight distributions of multi-edge type LDPC codes

For a (λ(x); ρ(x)) standard irregular LDPC code ensemble, the growth rate of the average weight distribution for small relative weight ω is given by log(λ′(0)ρ′(1))ω + O(ω2) in the limit of code length n. If λ′(0)ρ′(1) ≪ 1, there exist exponentially few code words of small linear weight, as n tends to infinity. It is known that the condition coincides with the stability condition of density evolution over the erasure channels with the erasure probability 1. In this paper, we show that this is also the case with multi-edge type LDPC (MET-LDPC) codes. MET-LDPC codes are generalized structured LDPC codes introduced by Richardson and Urbanke. The parameter corresponding λ′(0)ρ′(1) appearing in the conditions for MET-LDPC codes is given by the spectral radius of the matrix defined by extended degree distributions.

Weight and Stopping Set Distributions of Two-Edge Type LDPC Code Ensembles

In this paper, we explicitly formulate the average stopping set distributions and their asymptotic exponents of two instances of two-edge type LDPC code ensembles. Further we investigate the relation between the asymptotic exponents of those two code ensembles