252 research outputs found
New results on permutation polynomials over finite fields
In this paper, we get several new results on permutation polynomials over
finite fields. First, by using the linear translator, we construct permutation
polynomials of the forms and
. These generalize the results obtained by
Kyureghyan in 2011. Consequently, we characterize permutation polynomials of
the form , which extends a theorem of Charpin and Kyureghyan obtained in
2009.Comment: 11 pages. To appear in International Journal of Number Theor
Upper bounds on cyclotomic numbers
In this article, we give upper bounds for cyclotomic numbers of order e over
a finite field with q elements, where e is a divisor of q-1. In particular, we
show that under certain assumptions, cyclotomic numbers are at most
, and the cyclotomic number (0,0) is at most
, where k=(q-1)/e. These results are obtained by
using a known formula for the determinant of a matrix whose entries are
binomial coefficients.Comment: 11 pages, minor revisio
Construction of Frequency Hopping Sequence Set Based upon Generalized Cyclotomy
Frequency hopping (FH) sequences play a key role in frequency hopping spread
spectrum communication systems. It is important to find FH sequences which have
simultaneously good Hamming correlation, large family size and large period. In
this paper, a new set of FH sequences with large period is proposed, and the
Hamming correlation distribution of the new set is investigated. The
construction of new FH sequences is based upon Whiteman's generalized
cyclotomy. It is shown that the proposed FH sequence set is optimal with
respect to the average Hamming correlation bound.Comment: 16 page
New families of optimal frequency hopping sequence sets
Frequency hopping sequences (FHSs) are employed to mitigate the interferences
caused by the hits of frequencies in frequency hopping spread spectrum systems.
In this paper, we present some new algebraic and combinatorial constructions
for FHS sets, including an algebraic construction via the linear mapping, two
direct constructions by using cyclotomic classes and recursive constructions
based on cyclic difference matrices. By these constructions, a number of series
of new FHS sets are then produced. These FHS sets are optimal with respect to
the Peng-Fan bounds.Comment: 10 page
Super-regular Steiner 2-designs
A design is additive under an abelian group (briefly, -additive) if,
up to isomorphism, its point set is contained in and the elements of each
block sum up to zero. The only known Steiner 2-designs that are -additive
for some have block size which is either a prime power or a prime power
plus one. Indeed they are the point-line designs of the affine spaces
, the point-line designs of the projective planes , and the
point-line designs of the projective spaces . In the attempt to find
new examples, possibly with a block size which is neither a prime power nor a
prime power plus one, we look for Steiner 2-designs which are strictly
-additive (the point set is exactly ) and -regular (any translate of
any block is a block as well) at the same time. These designs will be
called\break "-super-regular". Our main result is that there are infinitely
many values of for which there exists a super-regular, and therefore
additive, - design whenever is neither singly even nor of the
form . The case (mod 4) is a definite exception whereas
is at the moment a possible exception. We also find
super-regular - designs with and which are
not isomorphic to the point-line design of .Comment: 31 page
Super-regular Steiner 2-designs
A design is additive under an abelian group G (briefly, G-additive) if, up to isomorphism, its point set is contained in G and the elements of each block sum up to zero. The only known Steiner 2-designs that are G-additive for some G have block size which is either a prime power or a prime power plus one. Indeed they are the point-line designs of the affine spaces AG(n,q), the point-line designs of the projective planes PG(2,q), the point-line designs of the projective spaces PG(n,2) and a sporadic example of a 2-(8191,7,1) design. In the attempt to find new examples, possibly with a block size which is neither a prime power nor a prime power plus one, we look for Steiner 2-designs which are strictly G-additive (the point set is exactly G) and G-regular (any translate of any block is a block as well) at the same time. These designs will be called “G-super-regular”. Our main result is that there are infinitely many values of v for which there exists a super-regular, and therefore additive, 2-(v,k,1) design whenever k is neither singly even nor of the form 2n3≥12. The case k≡2 (mod 4) is a genuine exception whereas k=2n3≥12 is at the moment a possible exception. We also find super-regular 2-(pn,p,1) designs with p∈{5,7} and n≥3 which are not isomorphic to the point-line design of AG(n,p)
Dual Elliptic Primes and Applications to Cyclotomy Primality Proving
Two rational primes p, q are called dual elliptic if there is an elliptic
curve E mod p with q points. They were introduced as an interesting means for
combining the strengths of the elliptic curve and cyclotomy primality proving
algorithms. By extending to elliptic curves some notions of galois theory of
rings used in the cyclotomy primality tests, one obtains a new algorithm which
has heuristic cubic run time and generates certificates that can be verified in
quadratic time.
After the break through of Agrawal, Kayal and Saxena has settled the
complexity theoretical problem of primality testing, some interest remains for
the practical aspect of state of the art implementable proving algorithms
Partial Geometric Designs from Group Actions
In this paper, using group actions, we introduce a new method for
constructing partial geometric designs (sometimes referred to as
-designs). Using this new method, we construct several infinite
families of partial geometric designs by investigating the actions of various
linear groups of degree two on certain subsets of .
Moreover, by computing the stabilizers of such subsets in various linear groups
of degree two, we are also able to construct a new infinite family of balanced
incomplete block designs
All or Nothing at All
We continue a study of unconditionally secure all-or-nothing transforms
(AONT) begun in \cite{St}. An AONT is a bijective mapping that constructs s
outputs from s inputs. We consider the security of t inputs, when s-t outputs
are known. Previous work concerned the case t=1; here we consider the problem
for general t, focussing on the case t=2. We investigate constructions of
binary matrices for which the desired properties hold with the maximum
probability. Upper bounds on these probabilities are obtained via a quadratic
programming approach, while lower bounds can be obtained from combinatorial
constructions based on symmetric BIBDs and cyclotomy. We also report some
results on exhaustive searches and random constructions for small values of s.Comment: 23 page
Three new classes of optimal frequency-hopping sequence sets
The study of frequency-hopping sequences (FHSs) has been focused on the
establishment of theoretical bounds for the parameters of FHSs as well as on
the construction of optimal FHSs with respect to the bounds. Peng and Fan
(2004) derived two lower bounds on the maximum nontrivial Hamming correlation
of an FHS set, which is an important indicator in measuring the performance of
an FHS set employed in practice.
In this paper, we obtain two main results. We study the construction of new
optimal frequency-hopping sequence sets by using cyclic codes over finite
fields. Let be a cyclic code of length over a finite field
such that contains the one-dimensional subcode Two codewords of are said to be equivalent if
one can be obtained from the other through applying the cyclic shift a certain
number of times. We present a necessary and sufficient condition under which
the equivalence class of any codeword in
has size . This result addresses an open question raised by Ding {\it et
al.} in \cite{Ding09}. As a consequence, three new classes of optimal FHS sets
with respect to the Singleton bound are obtained, some of which are also
optimal with respect to the Peng-Fan bound at the same time. We also show that
the two Peng-Fan bounds are, in fact, identical.Comment: to appear in Designs, Codes and Cryptograph
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