Finite Fields and Their Applications最新文献

Let $F_q$ denote the finite field with q elements. Permutation polynomials and complete permutation polynomials over finite fields have been widely investigated in recent years due to their applications in cryptography, coding theory and combinatorial design. In this paper, several classes of (complete) permutation polynomials with the form ${({\mathrm{Tr}}_m^n{\left(x\right)}^k+\delta )}^s+L\left(x\right)$ and ${({\mathrm{Tr}}_m^n{\left(x\right)}^{k_1}+{\delta }_1)}^{s_1}+{({\mathrm{Tr}}_m^n{\left(x\right)}^{k_2}+{\delta }_2)}^{s_2}+L\left(x\right)$ are proposed based on the AGW criterion and some techniques in solving equations over the finite field $F_{2^n}$, where $L\left(x\right)=a{\mathrm{Tr}}_m^n\left(x\right)+bx$, $a\in F_{2^n}$ and $b\in F_{2^m}^{⁎}$. We also determine the compositional inverse of these polynomials in some special cases. Besides, Mesnager (2014) [13] proposed a construction of bent functions by finding some triples of permutation polynomials satisfying a particular property named $\left(A_n\right)$. With the help of this approach, several classes of bent functions are presented.

We study the superzeta functions on function fields as constructed by Voros (see [11, Chapter 10, p.91]) in the case of the classical Riemann zeta function. Furthermore, we study special values of those functions, relate them to the Li coefficients, deduce some interesting summation formulas, and prove some results about the regularized product of the zeros of zeta functions on function fields.

A linear code with complementary dual (or an LCD code) is defined to be a linear code which intersects its dual code trivially. Let I be an identity matrix and T be a Toeplitz matrix of the same order over a finite field. A Double Toeplitz code (or a DT code) is a linear code generated by a generator matrix of the form $(I,T)$. In 2021, Shi et al. obtained necessary and sufficient conditions for a Double Toeplitz code to be LCD when T is symmetric and tridiagonal. In this paper, by using a result on factoring Dickson polynomials over finite fields, we determine when a Double Toeplitz code is LCD for T being a skew symmetric and tridiagonal matrix. In addition, using a concatenation, we construct LCD codes with arbitrary minimum distance from DT codes over extension fields, provided the length of which is increased if necessary.

We prove the Hausdorff dimension of various limsup sets over the field of formal power series. Typically, the upper bound is easier to establish by considering the natural covering of the underlying set. To establish the lower bound, we identify a suitable set that serves as a subset of several limsup sets by selecting appropriate values for the involved parameters. To be precise, given a fixed integer m and for all integers $0\le i\le m-1$, let ${\alpha }_i>0$ be a real number. Define the set$F_m({\alpha }_0,\dots ,{\alpha }_{m-1})\stackrel{\text{def}}{=}\{x\in I:\mathrm{deg}{A}_{n+i}=\lfloor n{\alpha }_i\rfloor +c_i,0\le i\le m-1\phantom{\stackrel{\text{def}}{=}\{x\in I}\text{for infinitely many}n\in N\},$ where $c_i\in N$ are fixed, and the partial quotients $A_i\left(x\right)$ are polynomials of strictly positive degree. We then determine the Hausdorff dimension of this set which establishes an optimal lower bound for various sets of interest, including the key results in [6], [8], [9]. Some new applications of our theorem include the lower bound of the Hausdorff dimension of the formal power series analogues of the sets considered in [2], [20]. We also prove their upper bounds to provide the comprehensive Hausdorff dimension analysis of these sets.

The main ingredient of the proof lies in the introduction of m probability measures consistently distributed over the Cantor-type subset of