Finite Fields and Their Applications最新文献

In this work, we propose two criteria for linear codes obtained from the Plotkin sum construction being symplectic self-orthogonal (SO) and linear complementary dual (LCD). As specific constructions, several classes of symplectic SO codes with good parameters including symplectic maximum distance separable codes are derived via ℓ-intersection pairs of linear codes and generalized Reed-Muller codes. Also symplectic LCD codes are constructed from general linear codes. Furthermore, we obtain some binary symplectic LCD codes, which are equivalent to quaternary trace Hermitian additive complementary dual codes that outperform the best-known quaternary Hermitian LCD codes reported in the literature. In addition, we prove that symplectic SO and LCD codes obtained in these ways are asymptotically good.

This article explores additive codes with one-rank hull, offering key insights and constructions. The article introduces a novel approach to finding one-rank hull codes over finite fields by establishing a connection between self-orthogonal elements and solutions of quadratic forms. It also provides a precise count of self-orthogonal elements for any duality over the finite field $F_q$, particularly odd primes. Additionally, construction methods for small rank hull codes are introduced. The highest possible minimum distance among additive one-rank hull codes is denoted by $d_1{[n,k]}_{p^e,M}$. The value of $d_1{[n,k]}_{p^e,M}$ for $k=1,2$ and $n\ge 2$ with respect to any duality M over any finite field $F_{p^e}$ is determined. Furthermore, the new quaternary one-rank hull codes are identified over non-symmetric dualities with better parameters than symmetric ones.

Suppose that F is a finite field and $R=M_n\left(F\right)$ is the ring of n-square matrices over F. Here we characterize when the Cayley graph of the additive group of R with respect to the set of invertible elements of R, called the unitary Cayley graph of R, is well-covered. Then we apply this to characterize all finite rings with identity whose unitary Cayley graph is well-covered or Cohen-Macaulay.

We give a new proof of a result of DiPippo and Wan for counting points of bounded height on projective spaces over global function fields. The new proof adapts the geometry of numbers arguments used by Schanuel in the number field case.

This paper corrects an error in the proof of Theorem 1.4 (3) of our earlier paper, Further Improvements to the Chevalley-Warning Theorems. The error originally appeared in Heath-Brown's paper, On Chevalley-Warning Theorems, which invalidates the proof of Theorem 2 (iii) in that paper. In this paper, we use a new method to give a correct proof of Theorem 1.4 (3). The correction in this paper also fixes the proof of Theorem 2 (iii) in Heath-Brown's paper. The proof in this paper provides slightly stronger estimates for some of the inequalities that were used in Further Improvements to the Chevalley-Warning Theorems.

The arithmetic crosscorrelation of binary m-sequences with coprime periods $2^{n_1}-1$ and $2^{n_2}-1$ ($\gcd (n_1,n_2)=1$) is determined. The result shows that the absolute value of arithmetic crosscorrelation of such binary m-sequences is not greater than $2^{\min (n_1,n_2)}-1$.

Starting with the multiplication of elements in $F_q^2$ which is consistent with that over $F_{q^2}$, where q is a prime power, via some identification of the two environments, we investigate the c-differential uniformity for bivariate functions $F(x,y)=\left(G\right(x,y),H(x,y\left)\right)$. By carefully choosing the functions $G(x,y)$ and $H(x,y)$, we present several constructions of bivariate functions with low c-differential uniformity, in particular, many PcN and APcN functions can be produced from our constructions.

In this paper, we study constacyclic codes of length $n=7p^s$ over a finite field of characteristics p, where $p\ne 7$ is an odd prime number and s a positive integer. The previous methods in the literature that were used to compute the Hamming distances of repeated-root constacyclic codes of lengths $n{p}^s$ with $1\le n\le 6$ cannot be applied to completely determine the Hamming distances of those with $n=7$. This is due to the high computational complexity involved and the large number of unexpected intermediate results that arise during the computation. To overcome this challenge, we propose a computer-assisted method for determining the Hamming distances of simple-root constacyclic codes of length 7, and then utilize it to derive the Hamming distances of the repeated-root constacyclic codes of length $7p^s$. Our method is not only straightforward to implement but also efficient, making it applicable to these codes with larger values of n as well. In addition, all self-orthogonal, dual-containing, self-dual, MDS and AMDS codes among them will also be characterized.

A subspace of $F_q^n$ is called a cyclically covering subspace if for every vector of $F_q^n$, operating a certain number of cyclic shifts on it, the resulting vector lies in the subspace. In this paper, we study the problem of under what conditions $F_q^n$ is itself the only covering subspace of $F_q^n$, symbolically, $h_q\left(n\right)=0$, which is an open problem posed in Cameron et al. (2019) [3] and Aaronson et al. (2021) [1]. We apply the primitive idempotents of the cyclic group algebra to attack this problem; when q is relatively prime to n, we obtain a necessary and sufficient condition under which $h_q\left(n\right)=0$, which completely answers the problem in this case. Our main result reveals that the problem can be fully reduced to that of determining the values of the trace function over finite fields. As consequences, we explicitly determine several infinitely families of $F_q^n$ which satisfy $h_q\left(n\right)=0$.

In this paper we study the permutational property of polynomials of the form $f\left(L\right(x\left)\right)+k\left(L\right(x\left)\right)\cdot M\left(x\right)\in F_{q^n}\left[x\right]$ over the finite field $F_{q^n}$, where $L,M\in F_q\left[x\right]$ are q-linearized polynomials and $k\in F_{q^n}\left[x\right]$ satisfies a generic condition. We specialize in the case where $L\left(x\right)$ is the linearized q-associate of $g_{t,a}\left(x\right)=(x^n-1)/(x^t-a)$, t is a divisor of n and $a\in F_q$ satisfies $a^{n/t}=1$. This unifies many recent explicit constructions and provides new explicit constructions of permutation polynomials and their inverses. Moreover, we introduce a new algorithmic method to produce many permutation polynomials of $F_{q^n}$ from permutations of $F_{q^t}$, by simply solving a system of independent equations of the form ${\mathrm{Tr}}_{q^n/q^t}\left({\delta }^{i-1}{a}_i\right)=c_i$, where the $a_i$'s are the coefficients of f. In fact, the same method can be