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### Page 3042 of Mathematical Reviews Vol. , Issue 94e [page]

1994 Mathematical Reviews
94e:94010 94e:94010 94A55 68Q40 Sakata, Shojiro Finding a minimal polynomial vector set of a vector of nD arrays.  ...  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.”  ...

### Solving Homogeneous Linear Equations Over GF(2) via Block Wiedemann Algorithm

Don Coppersmith
1994 Mathematics of Computation
Once one has the minimal polynomial f of B , one can set f-(3,) = f(3,)/Ak (dividing by the highest possible power of A) , compute f -( B ) z for any vector z , and apply powers of B successively until  ...  < 1 + 1 For any univariate polynomial r ( 1 ) over G F ( 2 ) , define and set dm= min, dm(i). Clearly, any polynomial r achieving this minimum is a divisor of f , the minimal polynomial of B .  ...

### Solving homogeneous linear equations over ${\rm GF}(2)$ via block Wiedemann algorithm

Don Coppersmith
1994 Mathematics of Computation
A block version of the algorithm allows us to perform 32 matrix-vector operations for the cost of one.  ...  We propose a method of solving large sparse systems of homogeneous linear equations over GF(2), the field with two elements. We modify an algorithm due to Wiedemann.  ...  In particular, Odlyzko suggested that the Wiedemann algorithm might be converted to block form, after we had discussed the present author's similar work on a  ...

### Extension of the Berlekamp-Massey algorithm to N dimensions

Shojiro Sakata
1990 Information and Computation
By it we can find a minimal set of linear recurring (LR) relations which are valid for a given two-dimensional (2D) array over any finite field.  ...  We present an algorithm for finding a minimal set of linear recurring relations which are valid for a given n-dimensional array over any field, where the "minimality" is defined with respect to the partial  ...  of any minimal polynomial set having order between q and p must have degree equal to some s E S.  ...

### Computing in GF(q)

Jacob T. B. Beard
1974 Mathematics of Computation
The third type of primitive polynomial examined permits the given representation of GF(<jr) to display a primitive normal basis over GF(p).  ...  This paper gives an elementary deterministic algorithm for completely factoring any polynomial over GF(q), q = p , criteria for the identification of three types of primitive polynomials, an exponential  ...  The set of all scalar matrices aln of order n, a € F, is and that g{x) is the minimal polynomial of C(g{x)) over F.  ...

### Sparse FGLM algorithms

Jean-Charles Faugère, Chenqi Mou
2017 Journal of symbolic computation
,xn] of degree D, the transformation of the ordering of its Groebner basis from DRL to LEX is a key step in polynomial system solving and turns out to be the bottleneck of the whole solving process.  ...  Furthermore, for generic polynomial systems, we present an explicit formula for the estimation of the sparsity of one main multiplication matrix, and prove its construction is free.  ...  This work is supported by the EXACTA grant of the French National Research Agency (ANR-09-BLAN-0371-01) and the National Science Foundation of China (NSFC 60911130369), the HPAC grant of the French National  ...

### Fast algorithm for change of ordering of zero-dimensional Gröbner bases with sparse multiplication matrices

Jean-Charles Faugère, Chenqi Mou
2011 Proceedings of the 36th international symposium on Symbolic and algebraic computation - ISSAC '11
This almost matches the complexity of computing the minimal polynomial of one multiplication matrix. Then, we address the general case and give corresponding complexity results.  ...  ,x n ] be a 0-dimensional ideal of degree D where K is a field.  ...  BMS The BMS algorithm is one that can be used to find the minimal set w.r.t. a term ordering < of a linearly recurring relation generated by a given multi-dimensional array [18, 19, 17] .  ...

### Generalized Berlekamp-Massey decoding of algebraic-geometric codes up to half the Feng-Rao bound

S. Sakata, H.E. Jensen, T. Hoholdt
1995 IEEE Transactions on Information Theory
The Sakata algorithm is a generalization to N dimensions of the classical Berlekamp-Massey algorithm.  ...  Abstiuct-We treat a general class of algebraic-geometric codes and show how to decode these up to half the Feng-Rao bound, using an extension and modification of the Sakata algorithm.  ...  The complexity of finding a reduced minimal polynomial set for the array of known syndromes is O(al(m -g + 1)2). 3) To calculate the candidate values for SZ where O(7) = m + 1 costs at most al (m -g) operations  ...

### Efficiency improvement in an nD systems approach to polynomial optimization

Ivo Bleylevens, Ralf Peeters, Bernard Hanzon
2007 Journal of symbolic computation
The problem of finding the global minimum of a so-called Minkowski-norm dominated polynomial can be approached by the matrix method of Stetter and Möller, which reformulates it as a large eigenvalue problem  ...  This paper focuses on improving the efficiency of computing the action of the matrix on a vector.  ...  In this paper we present a technique which uses nD systems for finding the global minimum of a special class of dominated polynomials.  ...

### A Polycyclic Quotient Algorithm

E.H. Lo
1998 Journal of symbolic computation
A practical algorithm is one which will work well on average, even though in the worst case, the run time may not be polynomial.  ...  The algorithm involves a generalization of the Gr obner basis method of commutative ring theory to the integral group ring of a polycyclic group.  ...  We will start by nding a set P 0 which we will prove is a minimal Gr obner basis. Let M P be the set of all leading monomial vectors of the polynomial arrays in P.  ...

### An approach to checking link conflicts in the mapping of uniform dependence algorithms into lower dimensional processor arrays

Jenn-Yang Ke, Jong-Chuang Tsay
1999 IEEE transactions on computers
arrays.  ...  In order to enumerate integer solutions efficiently, a representation of the integer solutions is devised so that the size of the space enumerated is yPx nÀk .  ...  ACKNOWLEDGMENTS This research was supported by the National Science Council of the Republic of China under contract NSC88-2213-E-036-003.  ...

### Protection of Information from Imitation on the Basis of Crypt-Code Structures [article]

Dmitry Samoylenko and Mikhail Eremeev and Oleg Finko and Sergey Dichenko
2018 arXiv   pre-print
A system is offered for imitation resistant transmitting of encrypted information in wireless communication networks on the basis of redundant residue polynomial codes.  ...  The use of authentication codes as a means of one of the levels to detect erroneous blocks in the ciphertext in combination with the redundant residue polynomial codes of deductions makes it possible to  ...  the vector of redundant encrypted symbols {ϑ k+1 (z), ϑ k+2 (z), . . . , ϑ n (z)} ∈ B, where A is the array of blocks of the ciphertext, B is a finite array.  ...

### A Class of Factorial Designs with Unequal Cell-Frequencies

Gideon Schwarz
1960 The Annals of Mathematical Statistics
The fact that permitted us to look for an inverse of a linear polynomial in CLASS OF FACTORIAL DESIGNS 755 Ay among the set of linear polynomials in A,, is the degree of minimal poly- nomial of A,, : it  ...  In general, the inverse of any regular matrix P that is a polynomial P(A), where A has a minimal polynomial of degree r, can be written as Q(A), Q being of degree r — 1 at most.  ...

### An Adaptive CM Array Preconditioner for Blind Multi-User Separation [article]

Stanislaw Gorlow and João Paulo C. L. da Costa and and Martin Haardt
2018 arXiv   pre-print
With rigorous theoretical analysis of the array response based on the discrete-space Fourier transform we elaborate a solution that solves the problem by finding the roots of a polynomial equation.  ...  Based on the observation that the differential filter weights resemble a superposition of the array steering vectors, we cast the original task to a direction-of-arrival estimation problem.  ...  ACKNOWLEDGMENT The authors would like to thank the Brazilian research and innovation agencies FAPDF (Fundação de Apoio à Pesquisa do Distrito Federal), CAPES (Coordenação de Aperfeiçoamento de Pessoal  ...