Distinct degree factorizations in additive arithmetical semigroups

The probability that an element of degreen has a given factorization pattern is computed within the context of a certain class of additive arithmetical semigroups. Concrete cases of these semigroups include the semigroup, of monic polynomials in one indeterminate over a finite fieldF _q , the multiplicative semigroups of ideals in principal orders within algebraic function fields overF _q and semigroups of integral divisors in algebraic function fields overF _q .     

Constructing quasi-cyclic codes from linear algebra theory

Let F _q be a finite field of cardinality q, l and m be positive integers and M _l (F _q ) the F _q -algebra of all l × l matrices over F _q . We investigate the relationship between monic factors of X ^m ? 1 in the polynomial ring M _l (F _q )[X] and quasi-cyclic (QC) codes of length lm and index l over F _q . Then we consider the idea of constructing QC codes from monic factors of X ^m ? 1 in polynomial rings over F _q -subalgebras of M _l (F _q ). This idea includes ideas of constructing QC codes of length lm and index l over F _q from cyclic codes of length m over a finite field F _q l, the finite chain ring F _q  + uF _q  + · · · + u ^{l ? 1} F _q (u ^l  = 0) and other type of finite chain rings.     

Quasirecognition by prime graph of the simple group <Superscript>2</Superscript><Emphasis Type="Italic">F</Emphasis><Subscript>4</Subscript>(<Emphasis Type="Italic">q</Emphasis>)

As the main result, we show that if G is a finite group such that Γ(G) = Γ(² F ₄(q)), where q = 2^2m+1 for some m ≧ 1, then G has a unique nonabelian composition factor isomorphic to ² F ₄(q). We also show that if G is a finite group satisfying |G| =|² F ₄(q)| and Γ(G) = Γ(² F ₄(q)), then G ≅ ² F ₄(q). As a consequence of our result we give a new proof for a conjecture of W. Shi and J. Bi for ² F ₄(q). The third author was supported in part by a grant from IPM (No. 87200022).     

Quasirecognition by prime graph of F4(q) where q?=?2n?>?2

The prime graph of a finite group G is denoted by Γ(G). In this paper as the main result, we show that if G is a finite group such that Γ(G) = Γ(F ₄(q)), where q = 2 ⁿ  > 2, then G has a unique nonabelian composition factor isomorphic to F ₄(q). We also show that if G is a finite group satisfying |G| = |F ₄(q)| and Γ(G) = Γ(F ₄(q)), where q = 2 ⁿ  > 2, then G @ F₄(q){G \cong F_4(q)}. As a consequence of our result we give a new proof for a conjecture of Shi and Bi for F ₄(q) where q = 2 ⁿ  > 2.     

M-P invertible matrices and unitary groups over Fq 2

The Moor-Penrose generalized inverses (M-P inverses for short) of matrices over a finite field F_q ² which is a generalization of the Moor-Penrose generalized inverses over the complex field, are studied in the present paper. Some necessary and sufficient conditions for anm xn matrixA over F_q ² having an M-P inverse are obtained, which make clear the set ofm xn matrices over F_q ² having M-P inverses and reduce the problem of constructing and enumerating the M-P invertible matrices to that of constructing and enumerating the non-isotropic subspaces with respect to the unitary group. Based on this reduction, both the construction problem and the enumeration problem are solved by borrowing the results in geometry of unitary groups over finite fields.     

Modular Counting of Rational Points over Finite Fields

Let F_q be the finite field of q elements, where q = p^h. Let f(x) be a polynomial over F_q in n variables with m nonzero terms. Let N(f) denote the number of solutions of f(x) = 0 with coordinates in F_q. In this paper we give a deterministic algorithm which computes the reduction of N(f) modulo p^b in O(n(8m)p^(h+b)p) bit operations. This is singly exponential in each of the parameters {h, b, p}, answering affirmatively an open problem proposed by Gopalan, Guruswami, and Lipton.     

M-P invertible matrices and unitary groups over Fq2

The Moor-Penrose generalized inverses (M-P inverses for short) of matrices over a finite field F_q ² which is a generalization of the Moor-Penrose generalized inverses over the complex field, are studied in the present paper. Some necessary and sufficient conditions for anm xn matrixA over F_q ² having an M-P inverse are obtained, which make clear the set ofm xn matrices over F_q ² having M-P inverses and reduce the problem of constructing and enumerating the M-P invertible matrices to that of constructing and enumerating the non-isotropic subspaces with respect to the unitary group. Based on this reduction, both the construction problem and the enumeration problem are solved by borrowing the results in geometry of unitary groups over finite fields.     

Zeta function of the projective curveaY 21 =bX 21 +cZ 21 over a class of finite fields, for odd primesl

Letp andl be rational primes such thatl is odd and the order ofp modulol is even. For such primesp andl, and fore = l, 2l, we consider the non-singular projective curvesaY ²¹ =bX ²¹ +cZ ²¹ defined over finite fields F_q such thatq = p ^α? l(mode).We see that the Fermat curves correspond precisely to those curves among each class (fore = l, 2l), that are maximal or minimal over F_q. We observe that each Fermat prime gives rise to explicit maximal and minimal curves over finite fields of characteristic 2. Fore = 2l, we explicitly determine the ζ -function(s) for this class of curves, over F_q, as rational functions in the variablet, for distinct cases ofa, b, andc, in F _q ^* . Theζ-function in each case is seen to satisfy the Weil conjectures (now theorems) for this concrete class of curves. Fore = l, 2l, we determine the class numbers for the function fields associated to each class of curves over F_q. As a consequence, when the field of definition of the curve(s) is fixed, this provides concrete information on the growth of class numbers for constant field extensions of the function field(s) of the curve(s).     

特征2的有限域中一些特殊元素的存在性(II)

设q=2s.s,n为正整数,Fqn为qn元素的有限域.在本文中,我们考虑Fqn中一些特殊元素的存在性.主要结果是:当下面的条件之一成立时,在Fqn中存在ξ使得ξ和ξ+ξ-1都是本原元并且ξ+ξ-1还是一个正规元:1.当n|(q-1)时,n37,s>6,或者2.当n|■(q-1)时,n≥34,s>6.进一步,如果n是奇数,则当下列条件之一成立时,存在ξ∈Fqn使得ξ和ξ+ξ-1都是Fqn的本原正规元:1.当n|(q-1)时,n≥257,s>9,或者2.当n■(q-1)时,n≥43,s≥9.     

The Maximum or Minimum Number of Rational Points on Genus Three Curves over Finite Fields

We show that for all finite fields F_q, there exists a curve C over F_q of genus 3 such that the number of rational points on C is within 3 of the Serre–Weil upper or lower bound. For some q, we also obtain improvements on the upper bound for the number of rational points on a genus 3 curve over F_q.with an Appendix by Jean-Pierre Serre     

An algorithm to compute the number of points on elliptic curves ofj-invariant 0 or 1728 over a finite field

We present an algorithm to compute the number ofF _q -rational points on elliptic curves defined over a finite fieldF _q , withj-invariant 0 or 1728. This algorithm takesO(log³ p) bit operations, werep is the characteristic ofF _q .     

On the group extension of the transformation associated to non-archimedean continued fractions

Summary Let F_q be a finite field with q elements. We consider formal Laurent series of F_q -coefficients with their continued fraction expansions by F_q -polynomials. We prove some arithmetic properties for almost every formal Laurent series with respect to the Haar measure. We construct a group extension of the non-archimedean continued fraction transformation and show its ergodicity. Then we get some results as an application of the individual ergodic theorem. We also discuss the convergence rate for limit behaviors.     

A conjecture of Abe-Iiyori and the Ree groups 2 F 4(q), II

In this paper the author has solved a problem of Abe and liyori for the finite simple groups ² F ₄(q) and ² F ₄(2)′.     

A conjecture of Abe-Iiyori and the Ree groups 2F4(q), II

In this paper the author has solved a problem of Abe and liyori for the finite simple groups ² F ₄(q) and ² F ₄(2)′.     

Equalities on the number of conjugacy classes of the Sylow p-subgroups of GL(n,q)

We denote by Gn the group of the upper unitriangular matrices over F_q, the finite field with q = p^t elements, and r(G_n) the number of conjugacy classes of G_n. In this paper, we obtain the value of r(G_n) modulo (q² -1)(q -1). We prove the following equalities     

The Cardinality of Sets of k -Independent Vectors over Finite Fields

A set of vectors is k-independent if all its subsets with no more than k elements are linearly independent. We obtain a result concerning the maximal possible cardinality Ind _q (n, k) of a k-independent set of vectors in the n-dimensional vector space F _q ⁿ over the finite field F _q of order q. Namely, we give a necessary and sufficient condition for Ind _q (n, k) = n + 1. We conclude with some pertinent remarks re applications of our results to codes, graphs and hypercubes.     

Newtonian and Schinzel sequences in a domain

A Schinzel or F sequence in a domain is such that, for every ideal I with norm q, its first q terms form a system of representatives modulo I, and a Newton or N sequence such that the first q terms serve as a test set for integer-valued polynomials of degree less than q. Strong F and strong N sequences are such that one can use any set of q consecutive terms, not only the first ones, finally a very well F ordered sequence, for short, a V.W.F sequence, is such that, for each ideal I with norm q, and each integer s,{u_sq,…,u_(s+1)q−1} is a complete set of representatives modulo I. In a quasilocal domain, V.W.F sequences and N sequences are the same, so are strong F and strong N sequences. Our main result is that a strong N sequence is a sequence which is locally a strong F sequence, and an N sequence a sequence which is locally a V.W.F. sequence. We show that, for F sequences there is a bound on the number of ideals of a given norm. In particular, a sequence is a strong F sequence if and only if it is a strong N sequence and for each prime p, there is at most one prime ideal with finite residue field of characteristic p. All results are refined to sequences of finite length.     

On Recognizability of the Group E8(q) by the Set of Orders of Elements

We prove that if a finite group G has the same set of orders of elements as the group E ₈(q), then O ³(G/F(G)) is isomorphic to E ₈(q).     

Primitive Elements with Zero Traces

Let F_q denote the finite field of order q, a power of a prime p, and n be a positive integer. We resolve completely the question of whether there exists a primitive element of F_qⁿ which is such that it and its reciprocal both have zero trace over F_q. Trivially, there is no such element when n<5: we establish existence for all pairs (q, n) (n≥5) except (4, 5), (2, 6), and (3, 6). Equivalently, with the same exceptions, there is always a primitive polynomial P(x) of degree n over F_q whose coefficients of x and of x^n-1 are both zero. The method employs Kloosterman sums and a sieving technique.     

Some constructions of linearly optimal group codes

We continue here the research on (quasi)group codes over (quasi)group rings. We give some constructions of [n,n-3,3]_q-codes over F_q for n=2q and n=3q. These codes are linearly optimal, i.e. have maximal dimension among linear codes having a given length and distance. Although codes with such parameters are known, our main results state that we can construct such codes as (left) group codes. In the paper we use a construction of Reed-Solomon codes as ideals of the group ring F_qG where G is an elementary abelian group of order q.