F_q[x]上的素数定理和Dirichlet定理及最小范数不可约多项式

讨论了F_q[x]上的zeta函数和L函数的解析性质,并在不假定黎曼猜想的情况下,导出了F_q[x]上的多项式环及其算术级数中不可约多项式的分布.然后,通过一系列的技术性处理,给出了算术级数中不可约多项式的最小范数的估计.成功地把素数定理及Dirichlet定理推广到了F_q[x]中,最重要的是,对应于最小素数问题,得到的最小范数的估计值本质上要比有理整数环上假定黎曼猜想情况下所推得的结果还好.     

A note on the intersection of two distinct constacyclic codes

This note gives a counterexample of a fact applied to the proof of two results in the papers of Dinh and Liu et al. Under suitable conditions, we give a sufficient condition for this fact to be correct. By means of the sufficient condition, we give the correct proof of the two results.     

On the number of Diophantine m-tuples in finite fields

We use a new argument to improve the error term in the asymptotic formula for the number of Diophantine m-tuples in finite fields, which is due to A. Dujella and M. Kazalicki (2021) and N. Mani and S. Rubinstein-Salzedo (2021).     

The black-box Niederreiter algorithm and its implementation over the binary field

The most time-consuming part of the Niederreiter algorithm for factoring univariate polynomials over finite fields is the computation of elements of the nullspace of a certain matrix. This paper describes the so-called ``black-box' Niederreiter algorithm, in which these elements are found by using a method developed by Wiedemann. The main advantages over an approach based on Gaussian elimination are that the matrix does not have to be stored in memory and that the computational complexity of this approach is lower. The black-box Niederreiter algorithm for factoring polynomials over the binary field was implemented in the C programming language, and benchmarks for factoring high-degree polynomials over this field are presented. These benchmarks include timings for both a sequential implementation and a parallel implementation running on a small cluster of workstations. In addition, the Wan algorithm, which was recently introduced, is described, and connections between (implementation aspects of) Wan's and Niederreiter's algorithm are given.

     

A Bertini type theorem for pencils over finite fields

We study the question of finding smooth hyperplane sections to a pencil of hypersurfaces over finite fields.