In this paper, eight classes of complete permutation polynomials with Niho exponents are proposed. Based on certain polynomials over Fq2 whose (q−1)-th powers are 1 or monomials on the unit circle, three classes of complete permutation polynomials are obtained. In addition, we completely characterize the complete permutation properties of five classes of known permutation trinomials.

In this paper, we explicitly evaluate some special values of F2k−12k-hypergeometric series over finite fields. These values are based on expressions of certain character sums in terms of Gaussian hypergeometric series. We also find relation between the number of Fq-points on certain algebraic curves and F2k−12k-Gaussian hypergeometric series.

We define additive G-codes over finite fields. We prove that if C is an additive G-code over Fq with duality M then its dual with respect to this duality CM is an additive G-code. We prove that if M and M′ are two dualities, then CM and CM′ are equivalent codes. Finally, we study the existence of self-dual codes for a variety of dualities and relate them to formally self-dual and linear self-dual codes

In this paper, we propose a mechanism on how to construct long MDS self-dual codes from short ones. These codes are special types of generalized Reed-Solomon (GRS) codes or extended generalized Reed-Solomon codes. The main tool is utilizing additive structure or multiplicative structure on finite fields. By applying this method, more MDS self-dual codes can be constructed.

Wilson's Theorem states that the product of all nonzero elements of a finite field Fq is −1. In this article, we define some natural subsets S⊂Fq× (q odd) and find formulas for the product of the elements of S, denoted ∏S. These new formulas are appealing for the simple, natural description of the sets S, and for the simplicity of the product. An example is ∏{a∈Fq×:1−a and 3+a are nonsquares}=2 if

For any positive integers m and k, existing literature only determines the number of all Euclidean self-dual cyclic codes of length 2k over the Galois ring GR(4,m), such as in Kiah et al. (2012) [17]. Using properties for Kronecker products of matrices of a specific type and column vectors of these matrices, we give a simple and efficient method to construct all these self-dual cyclic codes precisely

We provide necessary conditions for polynomials FA,B,m,n(X)=Xn+m(1+AXm(q−1)+BXn(q−1))∈Fq[X], n,m odd, over Fq2, q even, to be permutations. The main tool is the study of algebraic curves associated with the polynomial FA,B,m,n(X) and in particular the investigation of their singularities.

A convolutional code C over Zpr[D] is a Zpr[D]-submodule of Zprn[D] where Zpr[D] stands for the ring of polynomials with coefficients in Zpr. In this paper, we study the list decoding problem of these codes when the transmission is performed over an erasure channel, that is, we study how much information one can recover from a codeword w∈C when some of its coefficients have been erased. We do that

We obtain new upper bounds on the number of distinct roots of lacunary polynomials over finite fields. Our focus will be on polynomials for which there is a large gap between consecutive exponents in the monomial expansion.

We prove that a non-trivial minimal blocking set with respect to hyperplanes in PG(r,2), r≥3, is a skeleton contained in some s-flat with odd s≥3. We also consider non-trivial minimal blocking sets with respect to lines and planes in PG(r,2), r≥3. Especially, we show that there are exactly two non-trivial minimal blocking sets with respect to lines and six non-trivial minimal blocking sets with respect

In [14], D. Skabelund constructed a maximal curve over Fq4 as a cyclic cover of the Suzuki curve. In this paper we explicitly determine the structure of the Weierstrass semigroup at any point P of the Skabelund curve. We show that its Weierstrass points are precisely the Fq4-rational points. Also we show that among the Weierstrass points, two types of Weierstrass semigroup occur: one for the Fq-rational

We give a classification of maximal elements of the set of finite groups that can be realized as the full automorphism groups of polarized abelian surfaces over finite fields.

In this note, we study a Li-type criterion for the zeta function associated to the function field K of an arbitrary genus over a finite field of constants. First, we define two types of generalized Li-type coefficients and relate them with the Generalized Riemann Hypothesis. Furthermore, we provide different analytic representations for these coefficients and derive some interesting consequences.

We propose a unified construction for binary locally repairable codes (LRCs) with distance four. When there is a binary linear code [n,k,4] without all zero coordinates, we can construct a binary LRC [n,k,4] with very good locality r. Conditions for which these LRCs attain the Cadambe-Mazumdar bound are also presented. We prove that all our LRCs with locality r≤3 are optimal LRCs, and most of our LRCs

Constructions and equivalence of APN functions play a significant role in the research of cryptographic functions. On finite fields of characteristic 2, 6 families of power APN functions and 14 families of polynomial APN functions have been constructed in the literature. However, the study on the equivalence among the aforementioned APN functions is rather limited to the equivalence in the power APN

Every linear set in a projective space is the projection of a subgeometry, and most known characterizations of linear sets are given under this point of view. For instance, scattered linear sets of pseudoregulus type are obtained by considering a Desarguesian spread of a subgeometry and projecting from a vertex which is spanned by all but two director spaces. In this paper we introduce the concept

A function F:Fpn→Fpm, is a vectorial s-plateaued function if for each component function Fb(μ)=Trn(bF(x)),b∈Fpm⁎ and μ∈Fpn, the Walsh transform value |Fbˆ(μ)| is either 0 or pn+s2. In this paper, we explore the relation between (vectorial) s-plateaued functions and partial geometric difference sets. Moreover, we establish the link between three-valued cross-correlation of p-ary sequences and vectorial

Let p≠3 be a prime, s, m be positive integers, and λ be a nonzero element of the finite field Fpm. In [22] and [20], when the generator polynomials have one or two different irreducible factors, the Hamming distances of λ-constacyclic codes of length 3ps over Fpm have been considered. In this paper, we obtain that the Hamming distances of the repeated-root λ-constacyclic codes of length lps can be

We show that the points of a global function field, whose classes are 2-divisible in the Picard group, form a connected regular infinite graph, with the incidence relation generalizing the well known quadratic reciprocity law. We prove that for every global function field the diameter of this graph is precisely 2. In addition we develop an analog of global square theorem that concerns points 2-divisible

In this paper, we consider two classes of permutation trinomials with Niho-type exponents over the finite field F22m, where m is a positive integer. We transform the problem into investigating on some quartic equations (2k-th degree equations) over the subfield F2m in the first class (second class, respectively). We show that these equations have no solutions in F2m. Some sufficient conditions are

Let p≡1(mod4) be a prime. In this paper, with the help of Jacobsthal sums over finite fields, we study some permutation problems involving biquadratic residues modulo p.

Let V be a finite-dimensional vector space over a finite field and let f be a trilinear alternating form over V. For such forms, we introduce two new invariants. Together with a generalized radical polynomial used for classification of forms in dimension 8 over GF(2), they are sufficient to distinguish between all trilinear alternating forms in dimension 9 over GF(2). To prove the completeness of the

We introduce a new concept “geometric extending” for linear codes over finite fields and consider the extendability of divisible codes. As an application, we construct new Griesmer [n,5,d]q codes for 3q4−5q3+1≤d≤3q4−5q3+q2 with q≥3, combining the known geometric methods such as projective dual, geometric extending and geometric puncturing.

Let p>3 and consider a prime power q=ph. We completely characterize permutation polynomials of Fq2 of the type fa,b(X)=X(1+aXq(q−1)+bX2(q−1))∈Fq2[X]. In particular, using connections with algebraic curves over finite fields, we show that the already known sufficient conditions are also necessary.

We define a graph structure associated in a natural way to finite fields that nevertheless distinguishes between different models of isomorphic fields. Certain basic notions in finite field theory have interpretations in terms of standard graph properties. We show that the graphs are connected and provide an estimate of their diameter. An accidental graph isomorphism is uncovered and proved. The smallest

We introduce a new multivariate encryption scheme inspired by random linear codes. The construction is similar to that of UOV, one of the oldest and most trusted multivariate signature schemes, but with a parameterization nothing like that of UOV. The structure of the scheme admits many generic modifications providing an array of security and performance properties. The scheme also supports an embedding

We revisit theoretical background on OSIDH (Oriented Supersingular Isogeny Diffie-Hellman protocol), which is an isogeny-based key-exchange protocol proposed by Colò and Kohel at NutMiC 2019. We give a proof of a fundamental theorem for OSIDH. The theorem was stated by Colò and Kohel without proof. Furthermore, we consider parameters of OSIDH, give a sufficient condition on the parameters for the protocol

In a previous paper (see [3]) we determined the values of the next-to-minimal Hamming weights of affine cartesian codes of order d, for almost all values of d. In the present paper we determine the next-to-minimal weights for the missing values of d, assuming that the sets in the cartesian product which appears in the definition of the code are subfields of Fq.

Divisible design digraphs are constructed from skew balanced generalized weighing matrices and generalized Hadamard matrices. Commutative and non-commutative association schemes are shown to be attached to the constructed divisible design digraphs.

In this paper, X is an algebraic curve of genus g≥2 defined over an algebraically closed field K of positive characteristic p, G is an automorphism group of X which fixes K element-wise, and, for a point P∈X, GP is the subgroup of G which fixes P. The question “how large GP can be compared to g” has been the subject of several papers. We are concerned with the case where the second ramification group

In this paper, a Roos like bound on the minimum distance for skew cyclic codes over a general field is provided. The result holds in the Hamming metric and in the rank metric. The proofs involve arithmetic properties of skew polynomials and an analysis of the rank of parity-check matrices. For the rank metric case, a way to arithmetically construct codes with a prescribed minimum rank distance, using

In the article we propose a new compression method (to 2⌈log2⁡(q)⌉+3 bits) for the Fq2-points of an elliptic curve Eb:y2=x3+b (for b∈Fq2⁎) of j-invariant 0. It is based on Fq-rationality of some generalized Kummer surface GKb. This is the geometric quotient of the Weil restriction Rb:=RFq2/Fq(Eb) under the order 3 automorphism restricted from Eb. More precisely, we apply the theory of conic bundles

In this paper we enumerate irreducible polynomials of degree m and related objects over a finite field Fq with respect to their trace and norm. Our approach makes use of both generating functions and exponential sums, and we improve all previously known results in the literature when m>q.

In [12], A. Pasini and S. Yoshiara studied the distance regular graphs constructed from the Yoshiara dual hyperovals. In this note, we prove that the incidence graphs of the semibiplanes constructed from dimensional dual hyperovals are distance regular graphs if the dual hyperovals are doubly dual hyperovals (DDHOs). This generalizes the result in [12].

Sequences with perfect linear complexity profile were defined more than thirty years ago in the study of measures of randomness for binary sequences. More recently apwenian sequences, first with values ±1, then with values in {0,1}, were introduced in the study of Hankel determinants of automatic sequences. We explain that these two families of sequences are the same up to indexing, and give consequences

Consider a set of n points on a plane. A line containing exactly 3 out of the n points is called a 3-rich line. The classical orchard problem asks for a configuration of the n points on the plane that maximizes the number of 3-rich lines. In this note, using the group law in elliptic curves over finite fields, we exhibit several (infinitely many) group models for orchards wherein the number of 3-rich

In this paper we study a class of multishot network codes given by families of nested subspaces (flags) of a vector space Fqn, being q a prime power and Fq the finite field of q elements. In particular, we focus on flag codes having maximum distance (optimum distance flag codes). We explore the existence of these codes from spreads, based on the good properties of the latter ones. For n=2k, we show

Polycyclic codes are a powerful generalization of cyclic and constacyclic codes. Their algebraic structure is studied here by the theory of invariant subspaces from linear algebra. As an application, a bound on the minimum distance of these codes is derived which outperforms, in some cases, the natural analogue of the BCH bound.

In this work, we investigate hyperelliptic curves of type C:y2=x2g+1+axg+1+bx over the finite field Fq,q=pn,p>2. For the case of g=3 we propose an algorithm to compute the number of points on the Jacobian of the curve with complexity O˜(log4⁡p) over Fp. In case of g=4 we present a point counting algorithm with complexity O˜(log8⁡q) over Fq. The Jacobian JC splits over an extension of the field Fq on

Let p be a prime and n be a positive integer. Let fb(X)=X+(Xp−X+b)−1, where b∈Fpn is such that Trpn/p(b)≠0. In 2008, Yuan et al. [12] showed that for p=2,3, fb permutes Fpn for all n≥1. Using the Hasse-Weil bound, we show that when p>3 and n≥5, fb does not permute Fpn. For p>3 and n=2, we prove that fb permutes Fp2 if and only if Trp2/p(b)=±1. We conjecture that for p>3 and n=3,4, fb does not permute

Let q be a prime power, let Fq be the finite field with q elements and let d1,…,dk be positive integers. In this note we explore the number of solutions (z1,…,zk)∈F‾qk of the equationL1(x1)+⋯+Lk(xk)=b, with the restrictions zi∈Fqdi, where each Li(x) is a non zero polynomial of the form ∑j=0miaijxqj∈Fq[x] and b∈F‾q. We characterize the elements b for which the equation above has a solution and, in affirmative

In this paper, we consider two classes of permutation trinomials with Niho-type exponents over the finite field F22m. Some sufficient conditions are obtained to characterize the coefficients of the permutation trinomials. Our numerical result suggests that those sufficient conditions for one class are also necessary.

The Rσ,t-transform introduced by Bassa and Menares can be used to construct families of irreducible polynomials in Fq[x]. This iterative construction is a generalization of Cohen's R-transform. For this transform, Chapman proved that under some conditions, the polynomials in the resulting family are completely normal. In this paper we establish conditions ensuring that the polynomials obtained by using

Given two subsets A,B of the d dimensional vector space over the finite field Fq of q elements, we show that the sumsetA+B={a+b|a∈A,b∈B} contains k distinct vectors of the form (v1λ1j,…,vdλdj), where j=0,…,k−1, with nonzero vectors (v1,…,vd),(λ1,…,λd)∈Fqd, whenever#A#B⩾cq2d(1−1/k), for some constant c>0 which depends only on k. This improves the previous result of O. Ahamadi and I.E. Shparlinski (2007)

Let X be a del Pezzo surface of degree 2 or greater over a finite field Fq. The image Γ of the Galois group Gal(F‾q/Fq) in the group Aut(Pic(X‾)) is a cyclic subgroup preserving the anticanonical class and the intersection form. The conjugacy class of Γ in the subgroup of Aut(Pic(X‾)) preserving the anticanonical class and the intersection form is a natural invariant of X. We say that the conjugacy

Let l and k be two integers such that l|k. Define Tlk(X):=X+Xpl+⋯+Xpk−2l+Xpk−l and Slk(X):=X−Xpl+⋯+(−1)(k/l−1)Xpk−l, where p is any prime. This paper gives explicit representations of all solutions in Fpn to the affine equations Tlk(X)=a and Slk(X)=a, a∈Fpn. The case p=2 was solved very recently in [10]. The results of this paper reveal another solution.

We are interested in extending normal bases of F2n/F2 to bases of F2nd/F2 which allow fast arithmetic in F2nd. This question has been studied by Thomson and Weir in 2018 in case d is equal to 2. We construct efficient extended bases in case d is equal to 3 and 4. We also give conditions under which Thomson-Weir construction can be combined with ours.

Given a global function field K of characteristic p, for all effective divisors D in the divisor group of K we count the number of cyclic extensions F⊃K of degree p where the relative discriminant DiscK(F)=(p−1)D.

Linear codes over finite extension fields have widespread applications in theory and practice. In some scenarios, the decoder has a sequential access to the codeword symbols, giving rise to a hierarchical erasure structure. In this paper we develop a mathematical framework for hierarchical erasures over extension fields, provide several bounds and constructions, and discuss potential applications in

In this paper, explicit equations for algebraic curves with genus 4, 5, and 10 already studied in characteristic zero, are analyzed in positive characteristic p. We show that these curves have an interesting behaviour on the number of their rational places. Namely, they are either maximal or minimal over the finite field with p2 elements for infinitely many p's. The key tool is the investigation of

Nested code pairs play a crucial role in the construction of ramp secret sharing schemes [15] and in the CSS construction of quantum codes [14]. The important parameters are (1) the codimension, (2) the relative minimum distance of the codes, and (3) the relative minimum distance of the dual set of codes. Given values for two of them, one aims at finding a set of nested codes having parameters with

Group codes are right or left ideals in a group algebra of a finite group over a finite field. Following the ideas of a paper on binary group codes by Bazzi and Mitter in 2006, we prove that group codes over finite fields of any characteristic are asymptotically good.

In this paper we analyze the intersection between the norm-trace curve over Fq3 and the curves of the form y=ax3+bx2+cx+d, giving a complete characterization of the intersection between the curve and the parabolas (a=0), as well as sharp bounds for the other cases. This information is used for the determination of the weight distribution of some one-point AG codes arising from the curve.

In this paper, we find three classes of complete permutation polynomials over finite fields of even characteristic. The first class of quadrinomials is complete in the sense of addition. The second and third classes of binomials and trinomials are complete in multiplication. Moreover, a result related to the complete property in multiplication of a special class of polynomials is also given.

In this paper, we focus on computing the higher slope Hasse polynomials of L-functions of certain exponential sums associated to the following family of Laurent polynomials f(x1,…,xn+1)=∑i=1naixn+1(xi+1xi)+an+1xn+1+1xn+1, where ai∈Fq⁎, i=1,2,…,n+1. We find a simple formula for the Hasse polynomial of the slope one side and study the irreducibility of these Hasse polynomials. We will also provide a

Let q be a prime power and Fq the finite field with q elements. In this paper, it is shown that the minimal polynomial of a modified de Bruijn sequence of order n over Fq cannot be the product of an irreducible polynomial of degree n over Fq and any polynomial of degree k over Fq if n≥4k, which in fact improves several previous results ([5], [13]). Based on this result, a non-trivial lower bound 5n/4

Almost perfect nonlinear (APN) and almost bent (AB) functions are integral components of modern block ciphers and play a fundamental role in symmetric cryptography. In this paper, we describe a procedure for searching for quadratic APN functions with coefficients in F2 over the finite field F2n and apply this procedure to classify all such functions over F2n with n≤9. We discover two new APN functions

In this paper we study the set of -rational solutions of equations defined by polynomials evaluated in power-sum polynomials with coefficients in . This is done by means of applying a methodology which relies on the study of the geometry of the set of common zeros of symmetric polynomials over the algebraic closure of . We provide improved estimates and existence results of -rational solutions to the

We present a new structure theorem for finite fields of odd order that relates multiplicative and additive properties in an interesting way. This theorem has several applications, including an improved understanding of Dickson and Chebyshev polynomials and some formulas with a number-theoretic flavor. This paper is an abridged version of two articles by the author.

A recovering set for a coordinate position i in a code is a set Ri of other coordinate positions such that the value at the ith position can be recovered by accessing the values at coordinate positions in Ri. A locally recoverable (LRC) code with multiple recovering sets is a code in which for every coordinate position there are more than one recovering set. Such codes have been generally studied with