1,541 research outputs found
On the Divisibility of Trinomials by Maximum Weight Polynomials over F2
Divisibility of trinomials by given polynomials over finite fields has been
studied and used to construct orthogonal arrays in recent literature. Dewar et
al.\ (Des.\ Codes Cryptogr.\ 45:1-17, 2007) studied the division of trinomials
by a given pentanomial over \F_2 to obtain the orthogonal arrays of strength
at least 3, and finalized their paper with some open questions. One of these
questions is concerned with generalizations to the polynomials with more than
five terms. In this paper, we consider the divisibility of trinomials by a
given maximum weight polynomial over \F_2 and apply the result to the
construction of the orthogonal arrays of strength at least 3.Comment: 10 pages, 1 figur
Characteristic and Ehrhart polynomials
Let A be a subspace arrangement and let chi(A,t) be the characteristic
polynomial of its intersection lattice L(A). We show that if the subspaces in A
are taken from L(B_n), where B_n is the type B Weyl arrangement, then chi(A,t)
counts a certain set of lattice points. One can use this result to study the
partial factorization of chi(A,t) over the integers and the coefficients of its
expansion in various bases for the polynomial ring R[t]. Next we prove that the
characteristic polynomial of any Weyl hyperplane arrangement can be expressed
in terms of an Ehrhart quasi-polynomial for its affine Weyl chamber. Note that
our first result deals with all subspace arrangements embedded in B_n while the
second deals with all finite Weyl groups but only their hyperplane
arrangements.Comment: 16 pages, 1 figure, Latex, to be published in J. Alg. Combin. see
related papers at http://www.math.msu.edu/~saga
Note on Ward-Horadam H(x) - binomials' recurrences and related interpretations, II
We deliver here second new recurrence formula,
were array is appointed by sequence of
functions which in predominantly considered cases where chosen to be
polynomials . Secondly, we supply a review of selected related combinatorial
interpretations of generalized binomial coefficients. We then propose also a
kind of transfer of interpretation of coefficients onto
coefficients interpretations thus bringing us back to
and Donald Ervin Knuth relevant investigation decades
ago.Comment: 57 pages, 8 figure
List-Decoding Gabidulin Codes via Interpolation and the Euclidean Algorithm
We show how Gabidulin codes can be list decoded by using a parametrization
approach. For this we consider a certain module in the ring of linearized
polynomials and find a minimal basis for this module using the Euclidean
algorithm with respect to composition of polynomials. For a given received
word, our decoding algorithm computes a list of all codewords that are closest
to the received word with respect to the rank metric.Comment: Submitted to ISITA 2014, IEICE copyright upon acceptanc
Finite Groebner bases in infinite dimensional polynomial rings and applications
We introduce the theory of monoidal Groebner bases, a concept which
generalizes the familiar notion in a polynomial ring and allows for a
description of Groebner bases of ideals that are stable under the action of a
monoid. The main motivation for developing this theory is to prove finiteness
theorems in commutative algebra and its applications. A major result of this
type is that ideals in infinitely many indeterminates stable under the action
of the symmetric group are finitely generated up to symmetry. We use this
machinery to give new proofs of some classical finiteness theorems in algebraic
statistics as well as a proof of the independent set conjecture of Hosten and
the second author.Comment: 24 pages, adds references to work of Cohen, adds more details in
Section
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