Finite Fields and Their Applications最新文献

The a-number is an invariant of the isomorphism class of the p-torsion group scheme. We use the Cartier operator on $H^0(A_2,{\Omega }^1)$ to find a closed formula for the a-number of the form $A_2=v(Y^{\sqrt{q}}+Y-x^{\frac{\sqrt{q}+1}{2}})$ where $q=p^s$ over the finite field $F_{q^2}$. The application of the computed a-number in coding theory is illustrated by the relationship between the algebraic properties of the curve and the parameters of codes that are supported by it.

We consider information-theoretical private information retrieval (PIR) from a coded database with colluding servers. We target, for the first time, locally repairable storage codes (LRCs). We consider any number of local groups g, locality r, local distance δ and dimension k. Our main contribution is a PIR scheme for maximally recoverable (MR) LRCs based on linearized Reed–Solomon codes, which achieve the smallest field sizes among MR-LRCs for many parameter regimes. In our scheme, nodes are identified with codeword symbols and servers are identified with local groups of nodes. Only locally non-redundant information is downloaded from each server, that is, only r nodes (out of $r+\delta -1$) are downloaded per server. The PIR scheme achieves the (download) rate $R=(N-k-rt+1)/N$, where $N=gr$ is the length of the MDS code obtained after removing the local parities, and for any t colluding servers such that $k+rt\le N$. For an unbounded number of stored files, the obtained rate is strictly larger than those of known PIR schemes that work for any MDS code. Finally, the obtained PIR scheme can also be adapted when communication between the user and each server is performed via linear network coding, achieving the same rate as previous PIR schemes for this scenario but with polynomial finite field sizes, instead of exponential. Our rates are equal to those of PIR schemes for Reed–Solomon codes, but Reed–Solomon codes are incompatible with the MR-LRC property or linear network coding, thus our PIR scheme is less restrictive in its applications.

For a finite field $F_{q^r}$ with fixed q and r sufficiently large, we prove the existence of a primitive element outside of a set of r many affine hyperplanes for $q=4$ and $q=5$. This complements earlier results by Fernandes and Reis for $q\ge 7$. For $q=3$ the analogous result can be derived from a very recent bound on character sums of Iyer and Shparlinski. For $q=2$ the set consists only of a single element, and such a result is thus not possible.

Recently Zheng et al. [18] characterized the coefficients of $f\left(x\right)=x+a_1{x}^{s_1(2^m-1)+1}+a_2{x}^{s_2(2^m-1)+1}+a_3{x}^{s_3(2^m-1)+1}$ over $F_{2^{2m}}$ that lead $f\left(x\right)$ to be a permutation of $F_{2^{2m}}$ for $(s_1,s_2,s_3)=(\frac{1}{4},1,\frac{3}{4})$. They left open the question whether those conditions were also necessary. In this paper, we give a positive answer to that question, solving their conjecture.

In recent years, there have been a lot of research towards finding conditions under which the trinomial $x^r(x^{\alpha (q-1)}+x^{\beta (q-1)}+1)$ permutes $F_{2^{2m}}$ with $\alpha >\beta$ and r being positive integers. The authors of [6], [10], [24] have determined these conditions when $\alpha \le 5$ for certain values of β and r. In this paper, we work for $\alpha =6$ and determine four new classes of such permutation trinomials. Our contribution encompasses the investigation of these unexplored classes. Additionally, we analyze their quasi-multiplicative equivalence with already known permutation trinomials for $m\ge 1$. Through our research, we demonstrate that two of these determined classes are new, and for others, we explicitly compute the exponent for which they become equivalent.

We study the class of univariate polynomials ${\beta }_k\left(X\right)$, introduced by Carlitz, with coefficients in the algebraic function field $F_q\left(t\right)$ over the finite field $F_q$ with q elements. It is implicit in the work of Carlitz that these polynomials form an $F_q\left[t\right]$-module basis of the ring $\mathrm{Int}\left(F_q\right[t\left]\right)=\{f\in F_q(t\left)\right[X\left]\right|f\left(F_q\right[t\left]\right)\subseteq F_q\left[t\right]\}$ of integer-valued polynomials on the polynomial ring $F_q\left[t\right]$. This stands in close analogy to the famous fact that a $Z$-module basis of the ring $\mathrm{Int}\left(Z\right)$ is given by the binomial polynomials $\left(\begin{array}{c}X\\ k\end{array}\right)$.

We prove, for $k=q^s$, where s is a non-negative integer, that ${\beta }_k$ is irreducible in $\mathrm{Int}\left(F_q\right[t\left]\right)$ and that it is even absolutely irreducible, that is, all of its powers ${\beta }_k^m$ with $m>0$ factor uniquely as products of irreducible elements of this ring. As we show, this result is optimal in the sense that ${\beta }_k$ is not even irreducible if k is not a power of q.

Let G be a subgroup of the three dimensional projective group $\text{PGL}(3,q)$ defined over a finite field $F_q$ of order q, viewed as a subgroup of $\text{PGL}(3,K)$ where K is an algebraic closure of $F_q$. For $G\cong \text{PGL}(3,q)$ and for the seven nonsporadic, maximal subgroups G of $\text{PGL}(3,q)$, we investigate the (projective, irreducible) plane curves defined over K that are left invariant by G. For each, we compute the minimum degree $d\left(G\right)$ of G-invariant curves, provide a classification of all G-invariant curves of degree $d\left(G\right)$, and determine the first gap $\epsilon \left(G\right)$ in the spectrum of the degrees of all G-invariant curves. We show that the curves of degree $d\left(G\right)$ belong to a pencil depending on G, unless they are uniquely determined by G. For most examples of plane curves left invariant by a large subgroup of $\text{PGL}(3,q)$, the whole automorphism group of the curve is linear, i.e., a subgroup of $\text{PGL}(3,K)$. Although this appears to be a general behavior, we show that the opposite case can also occur for some irreducible plane curves, that is, the curve has a large group of linear automorphisms, but its full automorphism group is nonlinear.

We introduce a class of permutation polynomial over $F_{q^n}$ that can be written in the form $\frac{L\left(x\right)}{x^{q+1}}$ or $\frac{L\left(x^{q+1}\right)}{x}$ for some q-linear polynomial L over $F_{q^n}$. Specifically, we present those permutation polynomials explicitly as well as their inverses. In addition, more permutation polynomials can be derived in a more general form.

Matrix-product (MP) codes are a type of long codes formed by combining several commensurate constituent codes with a defining matrix. In this paper, we study the MP code when the defining matrix A satisfies the condition that $A{A}^{\top }$ is $(D,\tau )$-monomial. We give an explicit formula for calculating the dimension of the hull of a MP code. We present the necessary and sufficient conditions for a MP code to be dual-containing (DC), almost dual-containing (ADC), self-orthogonal (SO) and almost self-orthogonal (ASO), respectively. We theoretically determine the number of all possible ways involving the relationships among the constituent codes to yield a MP code that is DC, ADC, SO and ASO, respectively. We give alternative necessary and sufficient conditions for a MP code to be ADC and ASO, respectively, and show several cases where a MP code is not ADC or ASO. We give the construction methods of DC and ADC MP codes, including those with optimal minimum distance lower bounds. We introduce the notation of τ-optimal defining (τ-OD) matrices and provide the criteria for determining whether two types of $k\times k$ matrices are τ-OD matrices at $k=3$ and $k=4$, respectively. We give many examples of DC and ADC MP codes involving τ-OD matrices, some of which are optimal or almost optimal according to the Database [11]. By applying the generalized Steane's enlargement procedure to these DC MP codes, we obtain some good quantum codes that improve those available in the Database [7].

The classical way of extending an $[n,k,d]$ linear code $C$ is to add an overall parity-check coordinate to each codeword of the linear code $C$. This extended code, denoted by $\bar{C}(-1)$ and called the standardly extended code of $C$, is a linear code with parameters $[n+1,k,\overline{d}]$, where $\overline{d}=d$ or $\overline{d}=d+1$. This is one of the two extending techniques for linear codes in the literature. The standardly extended codes of some families of binary linear codes have been studied to some extent. However, not much is known about the standardly extended codes of nonbinary codes. For example, the minimum distances of the standardly extended codes of the nonbinary Hamming codes remain open for over 70 years. The first objective of this paper is to introduce the nonstandardly extended codes of a linear code and develop some general theory for this type of extended linear codes. The second objective is to study this type of extended codes of a number of families of linear codes, including cyclic codes and nonbinary Hamming codes. Four families of distance-optimal or dimension-optimal linear codes are obtained with this extending technique. The parameters of certain extended codes of many families of linear codes are settled in this paper.