Shojiro Sakata

Finding a minimal set of linear recurring relations capable of generating a given finite two-dimensional array

We present an algorithm for finding a minimal set of two-dimensional linear recurring relations capable of generating a prescribed finite two-dimensional array. This is a two-dimensional extension of the Berlekamp-Massey algorithm for synthesizing a shortest linear feedback shift-register capable of generating a given finite sequence. The complexity of computation for an array of size n is 0(n^2) under some reasonable assumptions. Furthermore, we make clear some relationship between our algorithm and Grobner bases of bivariate polynomial ideals, where polynomials correspond one-to-one to linear recurring relations.

Reconstruction Of Surfaces Of 3-D Objects By M-array Pattern Projection Method

A common problem of Pattern projection methods to measure surfaces of 3-D objects is that an observed pattern possibly i ncludes disorders such as deficiency, d isplacement, and permutation of subpatterns. These disorders make it difficult to match observed patterns with its position on the projected one and cause wrong m easurements as a result. This paper proposes a new technique to correct pattern disorders by using a pattern made from an M-array which is a two-dimensional extension of a well-known M-sequence.

Decoding binary 2-D cyclic codes by the 2-D Berlekamp-Massey algorithm

A method of decoding two-dimensional (2-D) cyclic codes by applying the 2-D Berlekamp-Massey algorithm is proposed. To explain this decoding method, the author introduces a subclass of 2-D cyclic codes, which are called 2-D BCH codes due to their similarity with BCH codes. It is shown that there are some short 2-D cyclic codes with a better cost parameter value. The merit of the approach is verified by showing several simple examples of 2-D cyclic codes. >

N-Dimensional Berlekamp-Massey Algorithm for Multiple Arrays and Construction of Multivariate Polynomials with Preassigned Zeros

In this paper we propose an algorithm of finding a minimal set of linear recurring relations for a given finite set of n-dimensional arrays. This algorithm is an n-dimensional extension of the Berlekamp-Massey algorithm for multisequences as well as an extension of the n-dimensional Berlekamp-Massey algorithm for a single array. Our algorithm is used to obtain Groebner bases of ideals defined by preassigned zeros. The latter problem is an extension of that treated by Moeller and Buchberger in the sense that the zeros can be over any finite extension (tilde K) of the base field K. Our approach gives an efficient method of obtaining Groebner bases of ideals defined by zeros to construct n-dimensional cyclic codes (i.e. Abelian codes). In case that the dimension n is small, the computational complexity is of order O((ILd)2), where I, L and d are the degree of the extension of (tilde K) over K, the number of the zeros and the size of the independent point set for the Groebner basis, respectively.

Finding a Minimal Polynomial Vector Set of a Vector of nD Arrays

We propose an algorithm for finding efficiently a minimal set of correlated linear recurrences capable of generating a given vector of finite n-dimensional (nD) arrays. The output of the algorithm is a Grobner basis of a module over the multivariate polynomial ring, provided that the size of the given arrays is sufficiently large in comparison with the degrees of the characteristic polynomials of the correlated linear recurrences found by the method. This algorithm is also an extension of the Berlekamp-Massey algorithm for finding a minimal polynomial set of an nD array. Although the algorithm has a close connection with the nD Berlekamp-Massey algorithm for multiple nD arrays, the former will find a minimal set of compound linear recurrences which relate all the nD arrays of the given vector while the latter finds a minimal set of linear recurrences which are in common to all the given nD arrays.

Two-dimensional shift register synthesis and Gro¨bner bases for polynomial ideals over an integer residue ring

Abstract In this paper we present an algorithm for finding a simplest n -dimensional linear feedback shift register which generates a given n -dimensional array over the integer residue ring Z m , where the term simplest means that the degree of every connection polynomial is minimal. This problem is an extension of the (one-dimensional) shift register synthesis over a field not only to n dimensions but also over the ring Z m . The result is useful for implementing encoders and decoders of Abelian codes over Z m .

Synthesis of two-dimensional linear feedback shift registers and Groebner bases

In this paper we discuss about how to design a two-dimensional linear feedback shift register. It is a switching circuit capable of generating a prescribed doubly periodic array. In particular, it can be used as an encoder of a two-dimensional cyclic code. Our method is based on a two-dimensional extension of the Berlekamp-Massey algorithm for synthesis of a (one-dimensional) linear feedback shift register. In the sequel, we make clear that our problem is equivalent to constructing a Groebner basis of the ideal which is defined by the given array.

A Gröbner Basis and a Minimal Polynominal Set of a Finite nD Array

In this paper, the relationship between a Grobner basis and a minimal polynomial set of a finite nD array is discussed. A minimal polynomial set of a finite nD array is determined by the nD Berlekamp-Massey algorithm. It is shown that a minimal polynomial set is not always a Grobner basis even if the uniqueness condition is satisfied, and a stronger sufficient condition for a minimal polynomial set to be a Grobner basis is presented. Furthermore, a simple test whether a given set of polynomials is a Grobner basis is proposed based on the theory of nD linear recurring arrays. The observations will be important in applying the nD Berlekamp-Massey algorithm to decode some kinds of nD cyclic codes and algebraic geometry codes.

Cycle representatives of quasi-irreducible two-dimensional cyclic codes

The author presents a method of finding the cycle representatives of any quasi-irreducible (QIR) 2-D cyclic code by extending A.P. Kurdjukovs (Probl. Peredach. Inform., vol.12, no.4, p.107-8, 1976) result on quasi-irreducible (i.e. nonsquare-free) 10D cyclic codes. The algorithm is not strictly deterministic in the sense that it is necessary to obtain a set of representative arrays for the code by a trial-and-error method. The result is useful for finding the cycle representatives of any 2D cyclic code by combining QIR components with the aid of G. Sequins (1974) method to the case where the symbol field is the binary Galois field GF(2). In particular, the result is useful for determining the weight distribution of any two-dimensional cyclic code. >

Linear Recurrences on 2d Convex Lattices and Decoding of Some Codes from Algebraic Curves

We present a theory of linear recurrences defined on convex lattices in the 2D plane and propose a generalization of the 2D Berlekamp-Massey algorithm which finds a minimal set of linear recurrences capable of generating a 2D array on a 2D covex lattice. Furthermore we show that this algorithm is applicable to decoding efficiently some kinds of algebraic geometry codes, in particular codes introduced by S. Miura and N. Kamiya.