2-Designs overGF(q)

We give a construction of a series of 2-(n, 3,q ²+q+1;q) designs of vector spaces over a finite fieldGF(q) of odd characteristic. These designs correspond to those constructed by Thomas and the author for even characteristic. As a natural generalization we give a collection ofm-dimensional subspaces which possibly become a 2-(n, m, λ; q) design.     

2-designs overGF(2 m )

At – (n, k, ; q) design is a collection ofk-dimensional subspaces of ann-dimensional vector space overGF(q) with the property that anyt-dimensional subspace is contained in exactly members of . It is also called a design over a finite field or aq-analoguet-design. The first nontrivial example fort 2 was given by S. Thomas. Namely, he constructed a series of 2 – (n, 3, 7; 2) design for alln 7 satisfying (n, 6) = 1. Under the same restriction onn, we show that the base field of Thomas' design is extensible toGF(2 ^m ), i.e., we construct a 2 – (n, 3, 2^2m + 2 ^m + 1; 2 ^m ) design for allm 1.Dedicated to Professor Tuyosi Oyama on his 60th Birthday     

On even (unitary) perfect polynomials over F2

We give all even perfect (resp. unitary perfect) polynomials over the prime field ${\mathbb{F}}_2$ of the form $x^a{(x+1)}^b{M}_1^{h_1}?M_r^{h_r}$, where each $M_i$ is a Mersenne irreducible polynomial, $h_i=2^{n_i}?1$ (resp. $h_i=2^{n_i}$) and $a,b,r,n_i\in \mathbb{N}$. In particular, we characterize nine of the eleven known “sporadic” even perfect polynomials over ${\mathbb{F}}_2$ that have the above form.     

On computing logarithms overGF(2 p )

In this paper we present a new, heuristic method for computing logarithms overGF(2 ^p ). When 2 ^p ?1 is a Mersenne prime ≦2³¹?1 it works in very short running times on a general purpose computer. It is based on the interdependent relations $$f_{rs} (t) = t^{ - 2^r } f(t)^{2^s } $$ and $$\log f_{rs} (t) = - 2^r + 2^s \log f(t)$$ , wheref andf _rs are polynomials overGF(2). Its cryptographic significance is discussed and it can be considered as an attempt to swindle MITRE Corporation which reportedly is using a public key distribution system, based on the presumed difficulty in computing logarithms overGF(2¹²⁷).     

TheGF(p)-dimension of the codes generated by the classical point-line geometries overGF(p)

In this note we develop a combinatorial-algebro-geometric approach to give a new self-contained analysis of the structure and dimension of theGF(p)-codes generated by the point-line geometriesAG ₁ (n, p) andPG ₁ (n, p) of dimensionn overGF(p) as coordinate field. The novelty of this paper is a beautiful representation of these codes as nil-ideals of the modular group algebra , which leads to an easy derivation of the dimension formula.Dedicated to Professor Konrad Jacobs, Erlangen, on the occasion of this 65th birthday.     

On the number of irreducible factors of degreek dividing a given polynomial overGF(q)

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Sur l'équationf(x)f(y)f(−x−y)=g(x)g(y)g(−x−y)

Sans résumé

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Action of U_q(g) on Its Positive Part U_q~+(g)

In this paper,two kinds of skew derivations of a type of Nichols algebras are intro- duced,and then the relationship between them is investigated.In particular they satisfy the quantum Serre relations.Therefore,the algebra generated by these derivations and correspond- ing automorphisms is a homomorphic image of the Drinfeld-Jimbo quantum enveloping algebra U q (g),which proves the Nichols algebra becomes a U q (g)-module algebra.But the Nichols alge- bra considered here is exactly U + q (g),namely,the posi...     

On class numbers of algebraic function fields defined byy 2=x 5 +ax overGF(p)

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