Minimal cyclic codes of length pq

Explicit expressions for all the 3n+2 primitive idempotents in the ring R_pⁿq=GF(ℓ)[x]/(x^pnq−1), where p,q,ℓ are distinct odd primes, ℓ is a primitive root modulo pⁿ and q both, , are obtained. The dimension, generating polynomials and the minimum distance of the minimal cyclic codes of length pⁿq over GF(ℓ) are also discussed.     

Function Spaces with Exponential Weights I

In this paper we define weighted function spaces of type B^s_pq(u) and F^s_pq(u) on the Euclidean space IRⁿ, where u is a weight function of at most exponential growth. In particular, u(x) = exp(±|χ|) is an admissible weight. We prove some basic properties of these spaces, such as completeness and density of the smooth functions.     

The combinatorial construction for a class of optimal optical orthogonal codes

Optical orthogonal code (OOC) has good correlation properties. It has many important applications in a fiber-optic code-division multiple access channel. In this paper, a combinatorial construction for optimal(15p, 5,1) optical orthogonal codes withp congruent to 1 modulo 4 and greater than 5 is given by applying Weil's Theorem. From this, whenv is a product of primes congruent to 1 modulo 4 and greater than 5, an optimal (15v, 5, 1)-OOC can be obtained by applying a known recursive construction.     

Hasse invariants and anomalous primes for elliptic curves with complex multiplication

Let C be an elliptic curve defined over Q. Let p be a prime where C has good reduction. By definition, p is anomalous for C if the Hasse invariant at p is congruent to 1 modulo p. The phenomenon of anomalous primes has been shown by Mazur to be of great interest in the study of rational points in towers of number fields. This paper is devoted to discussing the Hasse invariants and the anomalous primes of elliptic curves admitting complex multiplication. The two special cases Y² = X³ + a₄X and Y² = X³ + a₆ are studied at considerable length. As corollaries, some results in elementary number theory concerning the residue classes of the binomial coefficients (_n²ⁿ) (Resp. (_n³ⁿ)) modulo a prime p = 4n + 1 (resp. p = 6n + 1) are obtained. It is shown that certain classes of elliptic curves admitting complex multiplication do not have any anomalous primes and that others admit only very few anomalous primes.     

A New Characterization of Semi-bent and Bent Functions on Finite Fields*

We present a new characterization of semi-bent and bent quadratic functions on finite fields. First, we determine when a GF(2)-linear combination of Gold functions Tr(x²ⁱ+1) is semi-bent over GF(2ⁿ), n odd, by a polynomial GCD computation. By analyzing this GCD condition, we provide simpler characterizations of semi-bent functions. For example, we deduce that all linear combinations of Gold functions give rise to semi-bent functions over GF(2^p) when p belongs to a certain class of primes. Second, we generalize our results to fields GF(pⁿ) where p is an odd prime and n is odd. In that case, we can determine whether a GF(p)-linear combination of Gold functions Tr(x^pi+1) is (generalized) semi-bent or bent by a polynomial GCD computation. Similar to the binary case, simple characterizations of these p-ary semi-bent and bent functions are provided. Parts of this paper were presented at the 2002 IEEE International Symposium on Information Theory [10]     

Generalizations of Arnold’s version of Euler’s theorem for matrices

A recent result, conjectured by Arnold and proved by Zarelua, states that for a prime number p, a positive integer k, and a square matrix A with integral entries one has ${\textrm tr}(A^{p^k}) \equiv {\textrm tr}(A^{p^{k-1}}) ({\textrm mod}{p^k})${\textrm tr}(A^{p^k}) \equiv {\textrm tr}(A^{p^{k-1}}) ({\textrm mod}{p^k}). We give a short proof of a more general result, which states that if the characteristic polynomials of two integral matrices A,  B are congruent modulo p ^k then the characteristic polynomials of A ^p and B ^p are congruent modulo p ^k+1, and then we show that Arnold’s conjecture follows from it easily. Using this result, we prove the following generalization of Euler’s theorem for any 2 × 2 integral matrix A: the characteristic polynomials of A ^Φ(n) and A ^Φ(n)-ϕ(n) are congruent modulo n. Here ϕ is the Euler function, ?_i=1^l p_i^a_i\prod_{i=1}^{l} p_i^{\alpha_i} is a prime factorization of n and $\Phi(n)=(\phi(n)+\prod_{i=1}^{l} p_i^{\alpha_i-1}(p_i+1))/2$\Phi(n)=(\phi(n)+\prod_{i=1}^{l} p_i^{\alpha_i-1}(p_i+1))/2.     

On the matricial version of Fermat–Euler congruences

The congruences modulo the primary numbers n=p ^a are studied for the traces of the matrices A ⁿ and A ^n-φ(n), where A is an integer matrix and φ(n) is the number of residues modulo n, relatively prime to n. We present an algorithm to decide whether these congruences hold for all the integer matrices A, when the prime number p is fixed. The algorithm is explicitly applied for many values of p, and the congruences are thus proved, for instance, for all the primes p ≤ 7 (being untrue for the non-primary modulus n=6). We prove many auxiliary congruences and formulate many conjectures and problems, which can be used independently. Partially supported by RFBR, grant 05-01-00104. An erratum to this article is available at .     

Some cyclic codes of length 2<Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Explicit expressions for 4n + 2 primitive idempotents in the semi-simple group ring $R_{2p^{n}}\equiv \frac{GF(q)[x]}{p and q are distinct odd primes; n ≥ 1 is an integer and q has order \fracf(2pⁿ)2{\frac{\phi(2p^{n})}{2}} modulo 2p ⁿ . The generator polynomials, the dimension, the minimum distance of the minimal cyclic codes of length 2p ⁿ generated by these 4n + 2 primitive idempotents are discussed. For n = 1, the properties of some (2p, p) cyclic codes, containing the above minimal cyclic codes are analyzed in particular. The minimum weight of some subset of each of these (2p, p) codes are observed to satisfy a square root bound.     

Some infinite families of large sets of Kirkman triple systems

A large set of Kirkman triple systems of order v, denoted by LKTS(v), is a collection {(X, B_i) : 1 ≤ i ≤ v ? 2}, where every (X,B_i) is a KTS(v) and all B_i form a partition of all triples on X. Many researchers have studied the existence of LKTS(v) for a long time. In [13], the author introduced a concept—large set of generalized Kirkman systems (LGKS), which plays an important role in the discussion of LKTS. In this article, we give a new construction for LGKS and obtain some new results of LKTS, that is, there exists an LKTS(6u + 3) for u = qⁿ, where n ≥ 1, q ≡ 7 (mod 12) and q is a prime power. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 202–212, 2008     

On the value set of the Ramanujan function

We give lower bounds on the number of distinct values of the Ramanujan function τ(n), n ≦ x, and on the number of distinct residues of τ(n), n ≦ x, modulo a prime ℓ. We also show that for any prime ℓ the values τ(n), n ≦ ℓ⁴, form a finite additive basis modulo ℓ. Received: 6 October 2004     

On the k-error linear complexity of sequences with period 2pn over GF(q)

In this paper, we first optimize the structure of the Wei–Xiao–Chen algorithm for the linear complexity of sequences over GF(q) with period N =  2p ⁿ , where p and q are odd primes, and q is a primitive root modulo p ². The second, an union cost is proposed, so that an efficient algorithm for computing the k-error linear complexity of a sequence with period 2p ⁿ over GF(q) is derived, where p and q are odd primes, and q is a primitive root modulo p ². The third, we give a validity of the proposed algorithm, and also prove that there exists an error sequence e ^N , where the Hamming weight of e ^N is not greater than k, such that the linear complexity of (s + e) ^N reaches the k-error linear complexity c. We also present a numerical example to illustrate the algorithm. Finally, we present the minimum value k for which the k-error linear complexity is strictly less than the linear complexity.     

On the number of y-smooth natural numbers ≤x representable as a sum of two integer squares

LetB(x,y) be the sum taken over alln, 1≤n≤x, such that n can be represented as a sum of two squares of integers andn has no prime factors exceedingy. It is shown foru smaller than about .5log logx/log log logx thatB(x,x ^1/u)≈cxlog^-1/2 xσ(u), where σ(u satisfies a delay differential equation similar to the one satisfied by the Dickman function andc is a positive constant.     

On Tame Kernels and Ideal Class Groups

It is well known that there is a close connection between tame kernels and ideal class groups of number fields. However, the latter is a very difficult subject in number theory. In this paper, we prove some results connecting the p^n-rank of the tame kernel of a cyclic cubic field F with the p^n-rank of the coinvariants of μp^n×CI（δE,T） under the action of the Galois group, where E = F（ζp^n ） and T is the finite set of primes of E consisting of the infinite primes and the finite primes dividing p. In particular, if F is a cyclic cubic field with only one ramified prime and p = 3, n = 2, we apply the results of the tame kernels to prove some results of the ideal class groups of E, the maximal real subfield of E and F（ζ3）.     

Existence and multiplicity of solutions for a nonlinear Neumann problem

We consider a Neumann problem of the type -εΔu+F ^′(u(x))=0 in an open bounded subset Ω of R ⁿ , where F is a real function which has exactly k maximum points. Using Morse theory we find that, for ε suitably small, there are at least 2k nontrivial solutions of the problem and we give some qualitative information about them. Received: October 30, 1999 Published online: December 19, 2001     

A highly accurate numerical solution of a biharmonic equation

The coefficients for a nine–point high–order accuracy discretization scheme for a biharmonic equation ∇ ⁴u = f(x, y) (∇² is the two–dimensional Laplacian operator) are derived. The biharmonic problem is defined on a rectangular domain with two types of boundary conditions: (1) u and ∂²u/∂n² or (2) u and ∂u/part;n (where ∂/part;n is the normal to the boundary derivative) are specified at the boundary. For both considered cases, the truncation error for the suggested scheme is of the sixth-order O(h⁶) on a square mesh (h_x = h_y = h) and of the fourth-order O(h⁴_xh²_xh²_y h⁴_y) on an unequally spaced mesh. The biharmonic equation describes the deflection of loaded plates. The advantage of the suggested scheme is demonstrated for solving problems of the deflection of rectangular plates for cases of different boundary conditions: (1) a simply supported plate and (2) a plate with built-in edges. In order to demonstrate the high–order accuracy of the method, the numerical results are compared with exact solutions. © John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 375–391, 1997     

Extensions and the Induced Characters of Cyclic p-Groups

For a prime p, we denote by B_n the cyclic group of order pⁿ. Let φ be a faithful irreducible character of B_n, where p is an odd prime. We study the p-group G containing B_n such that the induced character φ^G is also irreducible. The purpose of this article is to determine the subgroup N_G(N_G(B_n)) of G under the hypothesis [N_G(B_n):B_n]⁴ ≦ pⁿ.     

Pseudo-Paley graphs and skew Hadamard difference sets from presemifields

Let (K, + ,*) be an odd order presemifield with commutative multiplication. We show that the set of nonzero squares of (K, *) is a skew Hadamard difference set or a Paley type partial difference set in (K, +) according as q is congruent to 3 modulo 4 or q is congruent to 1 modulo 4. Applying this result to the Coulter–Matthews presemifield and the Ding–Yuan variation of it, we recover a recent construction of skew Hadamard difference sets by Ding and Yuan [7]. On the other hand, applying this result to the known presemifields with commutative multiplication and having order q congruent to 1 modulo 4, we construct several families of pseudo-Paley graphs. We compute the p-ranks of these pseudo-Paley graphs when q = 3⁴, 3⁶, 3⁸, 3¹⁰, 5⁴, and 7⁴. The p-rank results indicate that these graphs seem to be new. Along the way, we also disprove a conjecture of René Peeters [17, p. 47] which says that the Paley graphs of nonprime order are uniquely determined by their parameters and the minimality of their relevant p-ranks. Dedicated to Dan Hughes on the occasion of his 80th birthday.     

Everywhere H?lder continuity of the spatial gradient of weak solutions to nonlinear parabolic systems

We consider weak solutions to the parabolic system ∂u ⁱ∂t−D _α A _i ^α (∇u)=B _i(∇u) in (i=1,...,) (Q=Ω×(0,T), R ⁿ a domain), where the functionsB _i may have a quadratic growth. Under the assumptionsn≤2 and ∇u ɛL _loc ^4+δ (Q; R ^nN ) (δ>0) we prove that ∇u is locally H?lder continuous inQ.     

On the several differences between primes

Enumeration of the primes with difference 4 between consecutive primes, is counted up to 5×10¹⁰, yielding the counting function π_2,4(5 × 10¹⁰) = 118905303. The sum of reciprocals of primes with gap 4 between consecutive primes is computedB ₄(5×10¹⁰)=1.197054473029 andB ₄=1.197054±7×10^?6. And Enumeration of the primes with difference 6 between consecutive primes, is counted up to 5×10¹⁰, yielding the counting function π_2,6(5 × 10¹⁰) = 215868063. The sum of reciprocals of primes with gap 6 between consecutive primes is computedB ₆(5×10¹⁰)=0.93087506039231 andB ₆=1.135835±1.2×10^?6.     

Inner Functions and Cyclic Composition Operators on H(Bn)

The present paper shows that the algebra generated by {C|  Aut(B_n)} is cyclic on H²(B_n), and any nonconstant function f  H²(B_n) is a cyclic vector of . In addition, the hypercyclic and cyclic composition operators will be discussed.