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1.
Let k 1 and
be a system of rational functions forming a strongly linearly independent set over a finite field
. Let
be arbitrarily prescribed elements. We prove that for all sufficiently large extensions
, there is an element
of prescribed order such that
is the relative trace map from
onto
We give some applications to BCH codes, finite field arithmetic and ordered orthogonal arrays. We also solve a question of Helleseth et~al. (Hypercubic 4 and 5-designs from Double-Error-Correcting codes, Des. Codes. Cryptgr. 28(2003). pp. 265–282) completely.classification 11T30, 11G20, 05B15 相似文献
2.
Let
be a 2-(v,k,1) design, and let G be a group of automorphisms of
. We show that if G is block primitive, then G does not admit a Ree group
as its socle. 相似文献
3.
Ákos Seress 《Designs, Codes and Cryptography》2005,34(2-3):265-281
A complete classification is given of finite primitive permutation groups which contain a regular subgroup of square-free order. Then a collection
of square-free numbers n is obtained such that there exists a vertex-primitive non-Cayley graph on n
vertices if and only if n is a member of
. 相似文献
4.
In this paper we classify point sets of minimum size of two types (1) point sets meeting all secants to an irreducible conic
of the desarguesian projective plane PG(2,q), q odd; (2) point sets meeting all external lines and tangents to a given irreducible conic
of the desarguesian projective plane PG(2,q), q even. 相似文献
5.
A set of linear maps
, V a finite vector space over a field K, is regular if to each
there corresponds a unique element
such that R(x)=y. In this context, Schur’s lemma implies that
is a field if (and only if) it consists of pairwise commuting elements. We consider when
is locally commutative: at some μ ∈V*, AB(μ)=BA(μ) for all
, and
has been normalized to contain the identity. We show that such locally commutative
are equivalent to commutative semifields, generalizing a result of Ganley, and hence characterizing commutative semifield spreads within the class of translation planes. This enables the determination of the orders |V| for which all locally commutative
on V are (globally) commutative. Similarly, we determine a sharp upperbound for the maximum size of the Schur kernel associated with strictly locally commutative
. We apply our main result to demonstrate the existence of a partial spread of degree 5, with nominated shears axis, that cannot be extend to a commutative semifield spread. Finally, we note that although local commutativity for a regular linear set
implies that the set of Lie products
consists entirely of singular maps, the converse is false. 相似文献
6.
We introduce [k,d]-sparse geometries of cardinality n, which are natural generalizations of partial Steiner systems PS(t,k;n), with d=2(k−t+1). We will verify whether Steiner systems are characterised in the following way. (*) Let
be a [k,2(k−t+1)]-sparse geometry of cardinality n, with
k \> t \> 1$$" align="middle" border="0">
. If
, then Γ is a S(t,k;n). If (*) holds for fixed parameters t, k and n, then we say S(t,k;n) satisfies, or has, characterisation (*). We could not prove (*) in general, but we prove the Theorems 1, 2, 3 and 4, which state conditions under which (*) is satisfied. Moreover, we verify characterisation (*) for every Steiner system appearing in list of the sporadic Steiner systems of small cardinality, and the list of infinite series of Steiner systems, both mentioned in the latest edition of the book ‘Design Theory’ by T. Beth, D. Jungnickel and H. Lenz. As an interesting application, one can use these results to build (almost) maximal binary codes in the following way. Every [k,d]-sparse geometry is associated with a [k,d]-sparse binary code of the same size (let
and link every block
with the code word
where ci=1 if and only if the point pi is a member of B), so one can construct maximal [k,d]-sparse binary codes using (partial) Steiner systems. These [k,d]-sparse codes can then be used as building bricks for binary codes having a bigger variety of weights (the weight of a code word is the sum of its entries). 相似文献
7.
For pairing based cryptography we need elliptic curves defined over finite fields
whose group order is divisible by some prime
with
where k is relatively small. In Barreto et al. and Dupont et al. [Proceedings of the Third Workshop on Security in Communication Networks (SCN 2002), LNCS, 2576, 2003; Building curves with arbitrary small Mov degree over finite fields, Preprint, 2002], algorithms for the construction of ordinary elliptic curves over prime fields
with arbitrary embedding degree k are given. Unfortunately, p is of size
.We give a method to generate ordinary elliptic curves over prime fields with p significantly less than
which also works for arbitrary k. For a fixed embedding degree k, the new algorithm yields curves with
where
or
depending on k. For special values of k even better results are obtained.We present several examples. In particular, we found some curves where
is a prime of small Hamming weight resp. with a small addition chain.AMS classification: 14H52, 14G50 相似文献
8.
Codes over
that are closed under addition, and multiplication with elements from Fq are called Fq-linear codes over
. For m 1, this class of codes is a subclass of nonlinear codes. Among Fq-linear codes, we consider only cyclic codes and call them Fq-linear cyclic codes (Fq LC codes) over
The class of Fq LC codes includes as special cases (i) group cyclic codes over elementary abelian groups (q=p, a prime), (ii) subspace subcodes of Reed–Solomon codes (n=qm–1) studied by Hattori, McEliece and Solomon, (iii) linear cyclic codes over Fq (m=1) and (iv) twisted BCH codes. Moreover, with respect to any particular Fq-basis of
, any FqLC code over
can be viewed as an m-quasi-cyclic code of length mn over Fq. In this correspondence, we obtain transform domain characterization of Fq LC codes, using Discrete Fourier Transform (DFT) over an extension field of
The characterization is in terms of any decomposition of the code into certain subcodes and linearized polynomials over
. We show how one can use this transform domain characterization to obtain a minimum distance bound for the corresponding quasi-cyclic code. We also prove nonexistence of self dual Fq LC codes and self dual quasi-cyclic codes of certain parameters using the transform domain characterization.AMS classification 94B05 相似文献
9.
The number of Fq
-rational points of a plane non-singular algebraic curve
defined over a finite field Fq
is computed, provided that the generic point of
is not an inflexion and that
is Frobenius non-classical with respect to conics.
Received: 18 March 2003 相似文献
10.
A.YA. Dorofeev L.S. Kazarin V.M. Sidelnikov M.E. Tuzhilin 《Designs, Codes and Cryptography》2005,37(3):391-404
We consider a finite matrix group
with 34· 216 elements, which is a subgroup of the infinite group
, where
is the regular representation of the quaternion group and C is a matrix that transforms the regular representation Q to its cellwise-diagonal form. There is a number of ways to define the matrix C. Our aim is to make the group
similar in a certain sense to a finite group. The eventual choice of an appropriate matrix C done heuristically.
We study the structure of the group
and use this group to construct spherical orbit codes on the unit Euclidean sphere in R8. These codes have code distance less than 1. One of them has 32· 28 = 2304 elements and its squared Euclidean code distance is 0.293.
Communicated by: V. A. Zinoviev 相似文献
11.
Structure of Degenerate Block Algebras 总被引:13,自引:0,他引:13
Given a non-trivial torsion-free abelian group (A,+,Q), a field F of characteristic 0, and a non-degenerate bi-additive skew-symmetric map
: A
A
F, we define a Lie algebra
=
(A,
) over F with basis {ex | x
A/{0}} and Lie product [ex,ey] =
(x,y)ex+y. We show that
is endowed uniquely with a non-degenerate symmetric invariant bilinear form and the derivation algebra Der
of
is a complete Lie algebra. We describe the double extension D(
, T) of
by T, where T is spanned by the locally finite derivations of
, and determine the second cohomology group H2(D(
, T),F) using anti-derivations related to the form on D(
, T). Finally, we compute the second Leibniz cohomology groups HL2(
, F) and HL2(D(
, T), F).2000 Mathematics Subject Classification: 17B05, 17B30This work was supported by the NNSF of China (19971044), the Doctoral Programme Foundation of Institution of Higher Education (97005511), and the Foundation of Jiangsu Educational Committee. 相似文献
12.
The main result in Cossidente and Siciliano (J. Number Theory, Vol. 99 (2003) pp. 373–382) states that if a Singer subgroup of PGL(3,q) is an automorphism group of a projective, geometric
irreducible, non-singular plane algebraic curve
then either
or
. In the former case
is projectively equivalent to the curve
with equation Xq+1Y+Yq+1+X=0 studied by Pellikaan. Furthermore, the curve
has a very nice property from Finite Geometry point of view: apart from the three distinguished points fixed by the Singer
subgroup, the set of its
-rational points can be partitioned into finite projective planes
. In this paper, the full automorphism group of such curves is determined. It turns out that
is the normalizer of a Singer group in
. 相似文献
13.
F. Ben SaÏd 《The Ramanujan Journal》2005,9(1-2):63-75
Let IN be the set of positive integers,
= {b1 < ⋅s < bh}⊂IN, N∊IN and N≥ bh.
=
0(
,N) is the set (introduced by J.-L. Nicolas, I.Z. Ruzsa and A. Sárközy) such that
{1,..., N} =
and p(
,n)≡ 0 (mod 2) for n∊IN and n > N, where p(
,n) denotes the number of partitions of n with parts in
. Let us denote by σ (
,n) the sum of the divisors of n belonging to
. In a paper jointly written with J.-L. Nicolas, we have recently proved that, for all k≥ 0, the sequence (σ(
,2k n))n≥ 1mod 2k+1 is periodic with an odd period qk. In this paper, we will characterize for any fixed odd positive integer q, the sets
and the integers N such that q0 = q, and those for which qk = q for all k≥ 0. Moreover, a set
=
0(
,N) is constructed with the property that its period, i.e. the period of (σ(
,n))n≥ 1mod 2, is 217, and for which the counting function is asymptotically equal to that of
0({1,2,3,4,5},5) which is a set of period 31.Dedicated to Professor J.-L. Nicolas on the occasion of his 60th birthday2000 Mathematics Subject Classification: Primary—11P81, 11P83Research supported by MIRA 2002 program no 0203012701, Number Theory, Lyon-Monastir. 相似文献
14.
Let G: = G(1,n,q) denote the Grassmannian of lines in PG(n,q), embedded as a point-set in PG(N, q) with
For n = 2 or 3 the characteristic function
of the complement of G is contained in the linear code generated by characteristic functions of complements of n-flats in PG(N, q). In this paper we prove this to be true for all cases (n, q) with q = 2 and we conjecture this to be true for all remaining cases (n, q). We show that the exact polynomial degree of
is
for δ: = δ(n, q) = 0 or 1, and that the possibility δ = 1 is ruled out if the above conjecture is true. The result deg(
for the binary cases (n,2) can be used to construct quantum codes by intersecting G with subspaces of dimension at least
相似文献
15.
We characterize the finite Veronesean
of all Hermitian varieties of PG(n,q2) as the unique representation of PG(n,q2) in PG(d,q), d n(n+2), where points and lines of PG(n,q2) are represented by points and ovoids of solids, respectively, of PG(d,q), with the only condition that the point set of PG(d,q) corresponding to the point set of PG(n,q2) generates PG(d,q). Using this result for n=2, we show that
is characterized by the following properties: (1)
; (2) each hyperplane of PG(8,q) meets
in q2+1, q3+1 or q3+q2+1 points; (3) each solid of PG(8,q) having at least q+3 points in common with
shares exactly q2+1 points with it.51E24 相似文献
16.
We consider type II codes over finite rings
. It is well-known that their gth complete weight enumerator polynomials are invariant under the action of a certain finite subgroup of
, which we denote Hk,g. We show that the invariant ring with respect to Hk,g is generated by such polynomials. This is carried out by using some closely related results concerning theta series and Siegel modular forms with respect to
. 相似文献
17.
In this paper, we continue our investigation on “Extremal problems under dimension constraints” introduced [1]. The general problem we deal with in this paper can be formulated as follows. Let
be an affine plane of dimension k in
. Given
determine or estimate
.Here we consider and solve the problem in the special case where
is a hyperplane in
and the “forbidden set”
. The same problem is considered for the case, where
is a hyperplane passing through the origin, which surprisingly turns out to be more difficult. For this case we have only partial results.AMS Classification: 05C35, 05B30, 52C99 相似文献
18.
Gretchen L. Matthews 《Designs, Codes and Cryptography》2005,37(3):473-492
We consider the quotient of the Hermitian curve defined by the equation yq + y = xm over
where m > 2 is a divisor of q+1. For 2≤ r ≤ q+1, we determine the Weierstrass semigroup of any r-tuple of
-rational points
on this curve. Using these semigroups, we construct algebraic geometry codes with minimum distance exceeding the designed
distance. In addition, we prove that there are r-point codes, that is codes of the form
where r ≥ 2, with better parameters than any comparable one-point code on the same curve. Some of these codes have better parameters
than comparable one-point Hermitian codes over the same field. All of our results apply to the Hermitian curve itself which
is obtained by taking m=q +1 in the above equation
Communicated by: J.W.P. Hirschfeld 相似文献
19.
Let
be the Poisson point process with intensity 1 in Rd and let
be
. We obtain a strong invariance principle for the total length of the nearest-neighbor graph on
. 相似文献
20.
The article [6] contains the result that if a finite generalized quadrangle of order s has an ovoid
that is translation with respect to two opposite flags, but not with respect to any two non-opposite flags, then is self-polar and
is the set of absolute points of a polarity. In particular, if is the classical generalized quadrangle Q(4, q) then
is a Suzuki-Tits ovoid. In this article, we remove the need to assume that is Q(4, q) in order to conclude that
is a Suzuki-Tits ovoid by showing that the initial assumptions in fact imply that is Q(4, q). At the same time, we also relax the requirement that have order s.Received: 14 May 2004 相似文献