Masazumi Kurihara

A Fast Parallel Implementation of the Berlekamp-Massey Algorithm with a One-D Systolic Array Architecture

In this paper we present a fast parallel version of the BM algorithm based on a one-dimensional (1D) or linear systolic array architecture which is composed of a series of m cells (processing units), where m is the size of the given data, i.e., the length of the input sequence. The 1D systolic array has only local communication links between each two neighboring cells without any global or nonlocal links between distant cells. Each cell executes a small fixed number of operations at every time unit. Our implementation with the 1D systolic array architecture attains time complexity \(\mathcal{O}\left( m \right)\) so that we can have the optimal total complexity \(\mathcal{O}\left( {m^2 } \right)\), which means that both requirements of (1) maximum throughput rate and of (2) local communication are satisfied, as is the case with some fast parallel implementations of the extended Euclidean algorithm. Our method gives not only another proof of equivalence between the Berlekamp-Massey algorithm and the extended Euclidean algorithm, in particular in the realm of parallel processing, but also alternatives of practical and efficient architectures for R.S. decoders.

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.

A systolic array architecture for implementing a fast parallel decoding algorithm of one-point AG codes

Previously we proposed a fast parallel decoding algorithm for general one-point algebraic geometric (AG) codes with a systolic array architecture. But, designing the detailed structure of the systolic array and scheduling the relevant procedures remained to be given explicitly, because the proper discontinuities (i.e., gaps) accompanied with a pole-order (nongap) sequence make the problem very complicated. Based on a version of the fast decoding algorithm expressed in terms of matrices with univariate polynomials as entities, we present a method of complete scheduling on a revised architecture of systolic array for implementing this algorithm in parallel. For the size m of input data, i.e. the syndrome, and the first nongap /spl rho/, the time complexity is O(m) and the space complexity is O(/spl rho//sup 2/ m), where the time complexity O(/spl rho/m/sup 2/) was required by the serial algorithm.

A Systolic Array Architecture for Fast Decoding of One-Point AG Codes and Scheduling of Parallel Processing on It

Since before we have proposed a systolic array architecture for implementing fast decoding algorithm of one point AG codes. In this paper we propose a revised architecture which is as its main framework a one-dimensional systolic array, in details, composed of a three-dimensional arrangement of processing units called cells, and present a method of complete scheduling on it, where not only our scheme has linear time complexity but also it satisfies restriction to local communication between nearest cells so that transmission delay is drastically reduced.

ON GENERALIZED HYPERBOLIC CASCADED REED-SOLOMON CODES

In this paper, a generalized version of the Hyperbolic Cascaded Reed–Solomon (HCRS) code proposed by K. Saints and C. Heegard is considered. The lower bound of the minimum distance (the design distance) is investigated. This is a linear code that can be expressed as (n, k, d min = d ) on GF(q). Here, n, k, dmin, and d are the code length, dimension, minimum distance, and design distance, respectively. The proposed code has more degrees of freedom in selection of the code length than the HCRS code. Codes with various code length can be constructed easily. However, in the m-dimensional space on GF(q), the selectable code length is less than qm. When the proposed code with a code length less than qm is compared with the shortened code with the same code length obtained by shortening the m-dimensional HCRS code with a code length of qm [or (q−1)m], the design distances are identical, whereas the proposed code has an equal or greater dimension compared with the HCRS code. Also, from the proposed code, it is relatively easy to construct a code of the same code length as the algebraic geometric code (geometric Goppa code) and the improved geometric Goppa code proposed by Feng and Rao. Further, when the performance (parameters) is compared at a high coding rate (k/n), the proposed code sometimes has equal or better performance.