Solving equations over finite fields is an important problem from both theoretical and practice points of view. The problem of solving explicitly the equation Pa(X)=0 over the finite field FQ, where Pa(X):=Xq+1+X+a, Q=pn, q=pk, a∈FQ⁎ and p is a prime, arises in many different contexts including finite geometry, the inverse Galois problem [1], the construction of difference sets with Singer parameters

Our aim for this paper is to find the construction method for quasi-cyclic self-orthogonal codes over the finite field Fpm. We first explicitly determine the generators of α-constacyclic codes over the finite Frobenius non-chain ring Rp,m=Fpm[u,v]/〈u2=v2=0,uv=vu〉, where m is a positive integer, α=a+ub+vc+uvd is a unit of Rp,m, a,b,c,d∈Fpm, and a is nonzero. We then find a Gray map from Rp,m[x]/〈xn−α〉

Let p be a prime, k a positive integer and let Fq be the finite field of q=pk elements. Let f(x) be a polynomial over Fq and a∈Fq. We denote by Ns(f,a) the number of zeros of f(x1)+⋯+f(xs)=a. In this paper, we show that∑s=1∞Ns(f,0)xs=x1−qx−xMf′(x)qMf(x), where Mf′(x) stands for the derivative of Mf(x) andMf(x):=∏m∈Fq⁎Sf,m≠0(x−1Sf,m) with Sf,m:=∑x∈FqζpTr(mf(x)), ζp being the p-th primitive unit root

In this paper we give a modular interpretation of the k-th symmetric power L-function of the Kloosterman family of exponential sums in characteristics 2 and 3, and in the case of p=2 and k odd give the precise 2-adic Newton polygon. We also give a p-adic modular interpretation of Dwork's unit root L-function of the Kloosterman family, and give the precise 2-adic Newton polygon when k is odd. In a previous

In this paper we establish decomposition theorems for derivations of group rings. We provide a topological technique for studying derivations of a group ring A[G] in case G has finite conjugacy classes. As a result, we describe all derivations of algebra A[G] for the case when G is a finite group, or G is an FC-group. In addition, we describe an algorithm to explicitly calculate all derivations of

Let G be the additive group of a finite field. J. Li and D. Wan determined the exact number of solutions of the subset sum problem over G, by giving an explicit formula for the number of subsets of G of prescribed size whose elements sum up to a given element of G. They also determined a closed-form expression for the case where the subsets are required to contain only nonzero elements. In this paper

In this paper, two-to-one mappings and involutions without any fixed point on finite fields of even characteristic are investigated. First, we characterize a closed relationship between them by implicit functions and develop an AGW-like criterion for 2-to-1 mappings. Using this criterion, some new constructions of 2-to-1 mappings are proposed and eight classes of 2-to-1 mappings of the form (x2k+x+δ)s+cx

In this paper, we explicitly determine Hamming weight enumerators of several classes of multi-twisted codes over finite fields with at most two non-zero constituents, where each non-zero constituent has dimension 1. Among these classes of multi-twisted codes, we further identify two classes of optimal equidistant linear codes that have nice connections with the theory of combinatorial designs and several

A linearized polynomial f(x)∈Fqn[x] is called scattered if for any y,z∈Fqn, the condition zf(y)−yf(z)=0 implies that y and z are Fq-linearly dependent. In this paper two generalizations of the notion of a scattered linearized polynomial are provided and investigated. Let t be a nontrivial positive divisor of n. By weakening the property defining a scattered linearized polynomial, L-qt-partially scattered

The Critical Problem in matroid theory is the problem for finding the maximum dimension of a subspace that contains no element of a fixed subset S of Fqk. This problem was posed by H. Crapo and G.-C. Rota and has been one of the significant problems in matroid theory. It can be interpreted in terms of a linear code over a finite field as to find the critical exponent of a linear code. This paper introduces

Let n≥2 be an integer and let Fq be the finite field with q elements, where q is a prime power. Given Fq-affine hyperplanes A1,…,An of Fqn in general position, we study the existence and distribution of primitive elements of Fqn, avoiding each Ai. We obtain both asymptotic and concrete results, relating to past works on digits over finite fields.

Over any quadratic finite field we construct function fields of large genus that have simultaneously many rational places, small p-rank, and many automorphisms.

We compute the parameters of the linear codes that are associated with all projective embeddings of Grassmann varieties.

Let ψ be a character of Zp of order pm, and f(x)=xd+λxe be a binomial of degree d with (d,e)=1. The determination of the Newton slopes of the L-functions Lf,ψ(s) is interesting and still open for general d,e that coprime. If p≡e(modd) is large enough, an arithmetic polygon Pe,d is defined and shown to be the lower bound for the classical (ψ(1)−1)a(p−1)-adic Newton polygon of Lf,ψ(s). In addition, we

The kth Dickson polynomial of the first kind, Dk(x)∈Z[x], is determined by the formula: Dk(u+1/u)=uk+1/uk, where k≥0 and u is an indeterminate. These polynomials are closely related to Chebyshev polynomials and have been widely studied. Leonard Eugene Dickson proved in 1896 that Dk(x) is a permutation polynomial on Fpn, p prime, if and only if GCD(k,p2n−1)=1, and his result easily carries over to Chebyshev

The construction of quantum maximum distance separable (abbreviated to MDS) error-correcting codes has become one of the major concerns in quantum coding theory. In this paper, we further generalize the approach developed in the previous paper, and construct several new classes of Hermitian self-orthogonal generalized Reed-Solomon (GRS) codes. By employing these classical MDS codes, we obtain several

The differential-linear connectivity table (DLCT) of a vectorial Boolean function was recently introduced by Bar-On et al. at EUROCRYPT'19, whose value at a point is related to the autocorrelation value of its component functions. Further, in INDOCRYPT'19, we proposed a new construction method for vectorial Boolean functions with very low differential-linear uniformity using Maiorana–McFarland bent

Let p be a prime and n be a positive integer, and consider fb(X)=X+(Xp−X+b)−1∈Fpn(X), where b∈Fpn is such that Trpn/p(b)≠0. It is known that (i) fb permutes Fpn for p=2,3 and all n≥1; (ii) for p>3 and n=2, fb permutes Fp2 if and only if Trp2/p(b)=±1; and (iii) for p>3 and n≥5, fb does not permute Fpn. It has been conjectured that for p>3 and n=3,4, fb does not permute Fpn. We prove this conjecture

In this paper, permutation polynomials of the form xd+L(xs) over finite fields Fq3 are investigated, where q=2m and m is a positive integer. By means of the iterative method, the multivariate method and the resultant elimination, six classes of permutation trinomials are obtained from certain integers d, s and linearized polynomials L(x). It is also showed that the permutations proposed in this paper

Let C(u,v) denote the ternary cyclic code with two nonzeros αu and αv, where α is a generator of F3m⁎ and 0≤u,v≤3m−2. In this paper, we present a sufficient condition such that C(u,v) is an optimal ternary cyclic code. Based on this condition, we get several classes of optimal ternary cyclic codes by choosing the proper u and v. Moreover, we show that C(3m+12,3m−12+v) and C(1,v) have the same optimality

In this paper, for an odd prime p, by extending Li et al.'s construction [17], several classes of two-weight and three-weight linear codes over the finite field Fp are constructed from a defining set, and then their complete weight enumerators are determined by using Weil sums. Furthermore, we show that some examples of these codes are optimal or almost optimal with respect to the Griesmer bound. Our

Let Γ(n,k) be the Grassmann graph formed by the k-dimensional subspaces of a vector space of dimension n over a field F and, for t∈N∖{0}, let Δt(n,k) be the subgraph of Γ(n,k) formed by the set of linear [n,k]-codes having minimum dual distance at least t+1. We show that if |F|≥(nt) then Δt(n,k) is connected and it is isometrically embedded in Γ(n,k).

Double Toeplitz (DT) codes are codes with a generator matrix of the form (I,T) with T a Toeplitz matrix, that is to say constant on the diagonals parallel to the main. When T is tridiagonal and symmetric we determine its spectrum explicitly by using Dickson polynomials, and deduce from there conditions for the code to be LCD. Using a special concatenation process, we construct optimal or quasi-optimal

In this paper we consider estimating the number of solutions to multiplicative equations in finite fields when the variables run through certain sets with high additive structure. In particular, we consider estimating the multiplicative energy of generalized arithmetic progressions in prime fields and of boxes in arbitrary finite fields. We obtain sharp bounds in more general scenarios than previously

An orthomorphism over a finite field Fq is a permutation θ:Fq→Fq such that the map x↦θ(x)−x is also a permutation of Fq. The degree of an orthomorphism of Fq, that is, the degree of the associated reduced permutation polynomial, is known to be at most q−3. We show that this upper bound is achieved for all prime powers q∉{2,3,5,8}. We do this by finding two orthomorphisms in each field that differ on

The arrangement of all Galois lines for the Artin–Schreier–Mumford curve in the projective 3-space is described. Surprisingly, there exist infinitely many Galois lines intersecting this curve.

Let q=2n, and let E/Fqℓ be a generalized Galbraith–Lin–Scott (GLS) binary curve, with ℓ≥2 and (ℓ,n)=1. We show that the GLS endomorphism on E/Fqℓ induces an efficient endomorphism on the Jacobian JacH(Fq) of the genus-g hyperelliptic curve H corresponding to the image of the GHS Weil-descent attack applied to E/Fqℓ, and that this endomorphism yields a factor-n speedup when using standard index-calculus

In this work, we introduce a natural notion concerning finite vector spaces. A family of k-dimensional subspaces of Fqn, which forms a partial spread, is called almost affinely disjoint if any (k+1)-dimensional subspace containing a subspace from the family non-trivially intersects with only a few subspaces from the family. The central question discussed in the paper is the polynomial growth (in q)

We obtain an explicit combinatorial formula for the number of solutions (x1,…,xr)∈(Fpab)r to the diagonal equation x1k+⋯+xrk=α over the finite field Fpab, with k=pab−1b(pa−1) and b>1, by using the number of r-walks in NEPS of complete graphs.

Quasi-subfield polynomials were introduced by Huang et al. together with a new algorithm to solve the Elliptic Curve Discrete Logarithm Problem (ECDLP) over finite fields of small characteristic. In this paper we provide both new quasi-subfield polynomial families and a new theorem limiting their existence. Our results do not allow to derive any speedup for the new ECDLP algorithm compared to previous

A conjecture of Le says that the Deligne polytope Δd is generically ordinary if p≡1(modD(Δd)), where D(Δd) is a combinatorial constant determined by Δd. In this paper a counterexample is given to show that the conjecture is not true in general.

In this paper, we study the conditions for a convolutional code to be MDP in terms of the size of the base field Fq as well as the openness of the MDP property in a given family of convolutional codes. Given (n,k,δ), our main result is an explicit bound depending on (n,k,δ) such that if q is greater than this bound, there exists a (n,k,δ) MDP convolutional code. A similar result is also offered for

In this paper, we resolve a conjecture of Green and Liebeck (2019) [3] on codes in PGL(2,q). To be specific, we show that: if D is a dihedral subgroup of order 2(q+1) in G=PGL(2,q), and A={g∈G:gq+1=1,g2≠1}, then λG=A⋅D, where λ=q or q−1 according as q is even or odd.

In this paper we introduce the notion of λ-constacyclic codes over finite rings R for arbitrary element λ of R. We study the non-invertible-element constacyclic codes (NIE-constacyclic codes) over finite principal ideal rings (PIRs). We determine the algebraic structures of all NIE-constacyclic codes over finite chain rings, give the unique form of the sets of the defining polynomials and obtain their

In this work, we give a new technique for constructing self-dual codes over commutative Frobenius rings using λ-circulant matrices. The new construction was derived as a modification of the well-known four circulant construction of self-dual codes. Applying this technique together with the building-up construction, we construct singly-even binary self-dual codes of lengths 56, 58, 64, 80 and 92 that

In this paper, we show an analogue of Kural, McDonald and Sah's result on Alladi's formula for global function fields. Explicitly, we show that for a global function field K, if a set S of prime divisors has a natural density δ(S) within prime divisors, then−limn→∞⁡∑1≤deg⁡D≤nD∈D(K,S)μ(D)|D|=δ(S), where μ(D) is the Möbius function on divisors and D(K,S) is the set of all effective distinguishable divisors

We show that the Ree unital R(q) has an embedding in a projective plane over a field F if and only if q=3 and F8 is a subfield of F. In this case, the embedding is unique up to projective linear transformations. Besides elementary calculations, our proof uses the classification of the maximal subgroups of the simple Ree groups.

It is known that N(n), the maximum number of mutually orthogonal latin squares of order n, satisfies the lower bound N(n)≥n1/14.8 for large n. For h≥2, relatively little is known about the quantity N(hn), which denotes the maximum number of ‘HMOLS’ or mutually orthogonal latin squares having a common equipartition into n holes of a fixed size h. We generalize a difference matrix method that had been

An element α∈Fqn is normal over Fq if B={α,αq,αq2,⋯,αqn−1} forms a basis of Fqn as a vector space over Fq. It is well known that α∈Fqn is normal over Fq if and only if gα(x)=αxn−1+αqxn−2+⋯+αqn−2x+αqn−1 and xn−1 are relatively prime over Fqn, that is, the degree of their greatest common divisor in Fqn[x] is 0. Using this equivalence, the notion of k-normal elements was introduced in Huczynska et al

Linear codes with a few weights can be applied to communication, consumer electronics and data storage system. In addition, the weight hierarchy of a linear code has many applications such as on the type II wire-tap channel, dealing with t-resilient functions and trellis or branch complexity of linear codes and so on. In this paper, we present a formula for computing the weight hierarchies of linear

A ring R is called clean if every element of R is the sum of a unit and an idempotent. Motivated by a question proposed by Lam on the cleanness of von Neumann Algebras, Vaš introduced a more natural concept of cleanness for ⁎-rings, called the ⁎-cleanness. More precisely, a ⁎-ring R is called a ⁎-clean ring if every element of R is the sum of a unit and a projection (⁎-invariant idempotent). Let F

We study and compare natural generalizations of Euclid's algorithm for polynomials with coefficients in a finite field. This leads to gcd algorithms together with their associated continued fraction maps. The gcd algorithms act on triples of polynomials and rely on two-dimensional versions of the Brun, Jacobi–Perron and fully subtractive continued fraction maps, respectively. We first provide a unified

Alahmadi et al. (2017) [2] introduced the notion of twisted centralizer codes, CFq(A,γ), defined asCFq(A,γ)={X∈Fqn×n:AX=γXA}, for A∈Fqn×n, and γ∈Fq. Moreover, Alahmadi et al. (2017) [3] also investigated the dimension of such codes and obtained upper and lower bounds for the dimension, and the exact value of the dimension only for cyclic or diagonalizable matrices A. Generalizing and sharpening Alahmadi

Prompted by a question of Jim Propp, this paper examines the cyclic sieving phenomenon (CSP) in certain cyclic codes. For example, it is shown that, among dual Hamming codes over Fq, the generating function for codedwords according to the major index statistic (resp. the inversion statistic) gives rise to a CSP when q=2 or q=3 (resp. when q=2). A byproduct is a curious characterization of the irreducible

Motivated by recent developments in the metrical theory of continued fractions for real numbers concerning the growth of consecutive partial quotients, we consider its analogue over the field of formal Laurent series. Let An(x) be the nth partial quotient of the continued fraction expansion of x in the field of formal Laurent series. We consider the sets of x such that deg⁡An+1(x)+⋯+deg⁡An+k(x)≥Φ(n)

We introduce and investigate some properties of lifted polynomials over finite fields. When f(t) is a lifted polynomial of degree 2 we give an exact count of the number of f-sequences that represent an f-subgroup, improving a previous result.

Flag codes are multishot network codes consisting of sequences of nested subspaces (flags) of a vector space Fqn, where q is a prime power and Fq, the finite field of size q. In this paper we study the construction on Fq2k of full flag codes having maximum distance (optimum distance full flag codes) that can be endowed with an orbital structure provided by the action of a subgroup of the general linear

Nonlinear feedback shift registers (NFSRs) are widely used in stream cipher design as building blocks. The cascade connection of NFSRs, known as an important architecture, has been adopted in Grain family of stream ciphers. In this paper, a new sufficient condition under which an NFSR cannot be decomposed into the cascade connection of two smaller NFSRs is presented, which is easy to be verified from

We obtain divisibility conditions on the multiplicative orders of elements of the form ζ+ζ−1 in a finite field by exploiting a link to the arithmetic of real quadratic fields.

Any permutation polynomial is an n-cycle permutation. When n is a specific small positive integer, one can obtain efficient permutations, such as involutions, triple-cycle permutations and quadruple-cycle permutations. These permutations have important applications in cryptography and coding theory. Inspired by the AGW Criterion, we propose criteria for n-cycle permutations, which mainly are of the

In this article, we consider the decoding problem of Grassmann codes using majority logic. We show that for two points of the Grassmannian, there exists a canonical geodesic between these points once a complete flag is fixed. These geodesics are used to construct a large set of parity checks orthogonal on a coordinate of the code, resulting in a majority decoding algorithm.

We prove a Pólya-Vinogradov type variation of the Chebotarev density theorem for function fields over finite fields valid for “incomplete intervals” I⊂Fp, provided (p1/2log⁡p)/|I|=o(1). Applications include density results for irreducible trinomials in Fp[x], i.e. the number of irreducible polynomials in the set {f(x)=xd+a1x+a0∈Fp[x]}a0∈I0,a1∈I1 is ∼|I0|⋅|I1|/d provided |I0|>p1/2+ϵ, |I1|>pϵ, or |I1|>p1/2+ϵ

For any smooth Hurwitz curve Hn:XYn+YZn+XnZ=0 over the finite field Fp, an explicit description of its Weierstrass points for the morphism of lines is presented. As a consequence, bounds on the number of rational points of Hn are obtained via Stöhr-Voloch Theory. Further, the full automorphism group Aut(Hn), as well as the genera of all Galois subcovers of Hn, with n≠3,pr, are computed. Finally, a

An f-subgroup is a linear recurring sequence subgroup, a multiplicative subgroup of a field whose elements can be generated (without repetition) by a linear recurrence relation, where the relation has characteristic polynomial f. It is called non-standard if it can be generated in a non-cyclic way (that is, not in the order αi,αi+1,αi+2… for a zero α of f), and standard otherwise. We will show that

We investigate the moment and the distribution of L(1,χP), where χP varies over quadratic characters associated to irreducible polynomials P of degree 2g+1 over Fq[T] as g→∞. In the first part of the paper, we compute the integral moments of the class number hP associated to quadratic function fields with prime discriminants P, and this is done by adapting to the function field setting some of the

Reed-Solomon codes have gained a lot of interest due to its encoding simplicity, well structuredness and list-decoding capability [6] in the classical setting. This interest also translates to other metric setting, including the insertion and deletion (insdel for short) setting which is used to model synchronization errors caused by positional information loss in communication systems. Such interest

Let Fq be a finite field of q elements. We show that the normalized Jacobi sum q−(m−1)/2J(χ1,…,χm) (χ1⋯χm nontrivial) is asymptotically equidistributed on the unit circle, when χ1∈A1,…,χm∈Am run through arbitrary sets of nontrivial multiplicative characters of Fq×, if #A1≥q12+ϵ, #A2≥(log⁡q)1δ−1 for ϵ>δ>0 fixed and q→∞ or if #A1#A2/q→∞. This extends previous results of Xi, Z. Zheng, and the authors

Let Fq be a finite field with q elements, where q is a power of 2. In this paper we study the positive integers n for which the irreducible factors of the polynomial xn−1 over Fq are all binomials and trinomials. In particular, we completely describe these integers for q=2,4.

The present paper deals with the problem of finding elements α and β in a finite field Fq, such that both are primitive and β is a rational function of α. Recently Cohen, Sharma and Sharma found a sufficient condition for the existence of such elements. In the present paper we present other conditions that guarantee the existence of such pair of elements, or prove it doesn't exist. We also improve

Given a polynomial system F over a finite field k which is not necessarily of dimension zero, we consider the Weil descent F′ of F over a subfield k′. We prove a theorem which relates the last fall degrees of F1 and F1′, where the zero set of F1 corresponds bijectively to the set of k-rational points of F, and the zero set of F1′ is the set of k′-rational points of the Weil descent F′. As an application