Hajime Matsui

On generator and parity-check polynomial matrices of generalized quasi-cyclic codes

Generalized quasi-cyclic (GQC) codes have been investigated as well as quasi-cyclic (QC) codes, e.g., on the construction of efficient low-density parity-check codes. While QC codes have the same length of cyclic intervals, GQC codes have different lengths of cyclic intervals. Similarly to QC codes, each GQC code can be described by an upper triangular generator polynomial matrix, from which the systematic encoder is constructed. In this paper, a complete theory of generator polynomial matrices of GQC codes, including a relation formula between generator polynomial matrices and parity-check polynomial matrices through their equations, is provided. This relation generalizes those of cyclic codes and QC codes. While the previous researches on GQC codes are mainly concerned with 1-generator case or linear algebraic approach, our argument covers the general case and shows the complete analogy of QC case. We do not use Grobner basis theory explicitly in order that all arguments of this paper are self-contained. Numerical examples are attached to the dual procedure that extracts one from each other. Finally, we provide an efficient algorithm which calculates all generator polynomial matrices with given cyclic intervals.

Systolic array architecture implementing Berlekamp-Massey-Sakata algorithm for decoding codes on a class of algebraic curves

We construct a two-dimensional systolic array implementing the Berlekamp-Massey-Sakata (BMS) algorithm to provide error-locator polynomials for codes on selected algebraic curves. This array is constructed by introducing some new polynomials in order to increase the parallelism of the algorithm. The introduced polynomials are used in the majority logic scheme by Sakata et al. to correct errors up to the designed minimum distance without affecting its high speed. The arrangement of the nearest local connection of processing units in the systolic array is obtained for the general case. Furthermore, shortened systolic arrays that reduce the circuit scale and have the same function are constructed with only a slight modification of the connections and controls; this enables the adjustment of the circuit scale for different types of systems.

Encoding via Gröbner bases and discrete Fourier transforms for several types of algebraic codes

We propose a novel encoding scheme for algebraic codes such as codes on algebraic curves, multidimensional cyclic codes, and hyperbolic cascaded Reed-Solomon codes and present numerical examples. We employ the recurrence from the Grobner basis of the locator ideal for a set of rational points and the two- dimensional inverse discrete Fourier transform. We generalize the functioning of the generator polynomial for Reed-Solomon codes and develop systematic encoding for various algebraic codes.

Practical iterative decoding scheme using Reed-Solomon codes for magnetic recording channels

The short-term iterative decoding implementation proposed in this paper not only uses conventional long-distance Reed-Solomon codes (RS codes), but also uses short-distance RS codes consisting of redundant symbols P and Q periodically inserted into the data in 512-byte sectors. A single parity matrix composed of redundant symbol P is decoded by using a belief propagation algorithm (BPA) such as low density parity check (LDPC) decoding. The Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm is used for EEPR4 channel decoding. Serial iterative decoding is done by using log likelihood ratios produced by both algorithms. Simulations of the use of 28 redundant symbols of the short-distance RS codes and 30 symbols of the long-distance RS codes have confirmed that at a block error rate of about 10/sup -1/ (bit-error rate) /spl ap/10/sup -3/ the proposed system can reduce the block error rate more than tenfold. Consequently, one block erasure correction including 30 symbols per sector can be achieved at the same error rate.

On generator matrices and parity check matrices of generalized integer codes

Generalized integer codes are defined as codes over rings of integers modulo

A Class of Generalized Quasi-Cyclic LDPC Codes: High-Rate and Low-Complexity Encoder for Data Storage Devices

n

Low Complexity Encoder for Generalized Quasi-Cyclic Codes Coming from Finite Geometries

n in which individual code symbols generally have different moduli. In this paper, we use a certain type of matrix identities to derive a necessary and sufficient condition for integer matrices to be equal to the generator matrices of generalized integer codes. Moreover, it is shown that the parity check matrix is generated from this matrix identity of the generator matrix. We also show the close connection between the listing of a certain type of integer codes and Hecke rings. Finally, an efficient algorithm that enumerates theoretically all of the generator matrices of generalized integer codes is provided.

Inverse-Free Implementation of Berlekamp-Massey-Sakata Algorithm for Decoding Codes on Algebraic Curves

In this paper, we establish a lemma in algebraic coding theory that frequently appears in the encoding and decoding of, e.g., Reed-Solomon codes, algebraic geometry codes, and affine variety codes. Our lemma corresponds to the nonsystematic encoding of affine variety codes, and can be stated by giving a canonical linear map as the composition of an extension through linear feedback shift registers from a Gröbner basis and a generalized inverse discrete Fourier transform. We clarify that our lemma yields the error-value estimation in the fast erasure-and-error decoding of a class of dual affine variety codes. Moreover, we show that systematic encoding corresponds to a special case of erasure-only decoding. The lemma enables us to reduce the computational complexity of error-evaluation from O(n3) using Gaussian elimination to O(qn2) with some mild conditions on n and q, where n is the code length and q is the finite-field size.