464 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
Construction of isodual codes from polycirculant matrices
Double polycirculant codes are introduced here as a generalization of double
circulant codes. When the matrix of the polyshift is a companion matrix of a
trinomial, we show that such a code is isodual, hence formally self-dual.
Numerical examples show that the codes constructed have optimal or
quasi-optimal parameters amongst formally self-dual codes. Self-duality, the
trivial case of isoduality, can only occur over \F_2 in the double circulant
case. Building on an explicit infinite sequence of irreducible trinomials over
\F_2, we show that binary double polycirculant codes are asymptotically good
A Multi-level Blocking Distinct Degree Factorization Algorithm
We give a new algorithm for performing the distinct-degree factorization of a
polynomial P(x) over GF(2), using a multi-level blocking strategy. The coarsest
level of blocking replaces GCD computations by multiplications, as suggested by
Pollard (1975), von zur Gathen and Shoup (1992), and others. The novelty of our
approach is that a finer level of blocking replaces multiplications by
squarings, which speeds up the computation in GF(2)[x]/P(x) of certain interval
polynomials when P(x) is sparse. As an application we give a fast algorithm to
search for all irreducible trinomials x^r + x^s + 1 of degree r over GF(2),
while producing a certificate that can be checked in less time than the full
search. Naive algorithms cost O(r^2) per trinomial, thus O(r^3) to search over
all trinomials of given degree r. Under a plausible assumption about the
distribution of factors of trinomials, the new algorithm has complexity O(r^2
(log r)^{3/2}(log log r)^{1/2}) for the search over all trinomials of degree r.
Our implementation achieves a speedup of greater than a factor of 560 over the
naive algorithm in the case r = 24036583 (a Mersenne exponent). Using our
program, we have found two new primitive trinomials of degree 24036583 over
GF(2) (the previous record degree was 6972593)
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