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

We give an improvement on a character sum estimate in Fq[t] and answer a question of Shparlinski.

Self-dual codes, which are codes that are equal to their orthogonal, are a widely studied family of codes. Various techniques involving circulant matrices and matrices from group rings have been used to construct such codes. Moreover, families of rings have been used, together with a Gray map, to construct binary self-dual codes. In this paper, we introduce a new bordered construction over group rings

Let q be a prime power. For u=(u1,…,un),v=(v1,…,vn)∈Fq2n let 〈u,v〉:=∑i=1nuiqvi be the Hermitian form of Fq2n. Fix an n×n matrix M over Fq2. Set Num(M):={〈u,Mu〉|u∈Fq2n,〈u,u〉=1} (the numerical range of M introduced by Coons, Jenkins, Knowles, Luke and Rault (case q a prime q≡3(mod4)) and by the author (arbitrary q)). When n=2 we prove an upper bound for |Num(M)|. We describe Num(M) for several classes

Let p be an odd prime and consider the finite field Fp2. Given a linear code C⊂Fp2n, we use algebraic number theory to construct an associated lattice ΛC⊂OLn for L an algebraic number field and OL the ring of integers of L. We attach a theta series θΛC to the lattice ΛC and prove a relation between θΛC and the complete weight enumerator evaluated on weight one theta series.

Let q be a perfect power of a prime number p and E(Fq) be an elliptic curve over Fq given by the equation y2=x3+Ax+B. For a positive integer n we denote by #E(Fqn) the number of rational points on E (including infinity) over the extension Fqn. Under a mild technical condition, we show that the sequence {#E(Fqn)}n>0 contains at most 10200 perfect squares. If the mild condition is not satisfied, then

In this work, we use the concept of distance between self-dual codes, which generalizes the concept of a neighbor for self-dual codes. Using the k-neighbors, we are able to construct extremal binary self-dual codes of length 68 with new weight enumerators. We construct 143 extremal binary self-dual codes of length 68 with new weight enumerators including 42 codes with γ=8 in their W68,2 and 40 with

Lower and upper bounds on the size of resolving sets for the point-hyperplane incidence graph of the finite projective space PG(n,q) are presented.

In this paper, three classes of binary linear codes with few weights are proposed from vectorial Boolean power functions, and their weight distributions are completely determined by solving certain equations over finite fields. In particular, a class of simplex codes and a class of first-order Reed-Muller codes can be obtained from our construction by taking the identity map, whose dual codes are Hamming

This paper is a contribution to the study of 2-designs admitting a flag-transitive automorphism group. We prove that if D is a non-trivial non-symmetric 2-(v,k,λ) design with (r,λ)=1 and G=PSL(3,q) acts flag-transitively on D, then up to isomorphism D is a unique 2-(8,4,3) design, or a unique 2-(q2+q+1,q,q−1) design with q=2n.

We give the correct statement of main Theorem 2.1.

Artin's primitive root conjecture for function fields was proved by Bilharz in his thesis in 1937, conditionally on the proof of the Riemann hypothesis for function fields over finite fields, which was proved later by Weil in 1948. In this paper, we provide a simple proof of Artin's primitive root conjecture for function fields which does not use the Riemann hypothesis for function fields but rather

Let PG(3,q) be the projective space of dimension three over the finite field with q elements. Consider a twisted cubic in PG(3,q). The structure of the point-plane incidence matrix in PG(3,q) with respect to the orbits of points and planes under the action of the stabilizer group of the twisted cubic is described. This information is used to view generalized doubly-extended Reed-Solomon codes of codimension

We study a construction method of binary reversible self-dual codes in this paper. Reversible codes have good properties in applications, and it is interesting to note that a class of reversible codes is closely connected to BCH codes and LCD codes. We first characterize binary reversible self-dual codes. Using these characteristics of reversible self-dual codes, we find an explicit method for constructing

Generalized quasi-twisted (GQT) codes form a generalization of quasi-twisted (QT) codes and generalized quasi-cyclic (GQC) codes. By the Chinese remainder theorem, the GQT codes can be decomposed into a direct sum of some linear codes over Galois extension fields, which leads to the trace representation of the GQT codes. Using this trace representation, we first prove the minimum distance bound for

Let p be a prime such that p≡1(mod3). Let c be a cubic residue (modp) such that cp−13≡1(modp). In this paper, we present a refinement of Müller's algorithm for computing a cube root of c [11], which also improves Williams' [14], [15] Cipolla-Lehmer type algorithms. Under the assumption that a suitable irreducible polynomial of degree 3 is given, Müller gave a cube root algorithm which requires 8.5log⁡p

The study of Cameron-Liebler line classes in PG(3,q) arose from classifying specific collineation subgroups of PG(3,q). Recently, these line classes were considered in new settings. In this point of view, we will generalize the concept of Cameron-Liebler line classes to AG(3,q). In this article we define Cameron-Liebler line classes using the constant intersection property towards line spreads. The

We give a complete description of the distance relation on the graph of 4-ary simplex codes of dimension 2. This is a connected graph of diameter 3. For every vertex we determine the sets of all vertices at distance i∈{1,2,3} and describe their symmetries.

An important problem on almost perfect nonlinear (APN) functions is the existence of APN permutations on even-degree extensions of F2 larger than 6. Browning et al. (2010) gave the first known example of an APN permutation on the degree-6 extension of F2. The APN permutation is CCZ-equivalent to the previously known quadratic Kim κ-function (Browning et al. (2009)). Aside from the computer based CCZ-inequivalence

An [n,k,d]q code is a linear code of length n, dimension k and minimum weight d over the field of q elements. An [n,k,d]q code C is called extendable if C can be extended to an [n+1,k,d+1]q code. We give a new extension theorem for ternary linear codes. As an application, we prove the non-existence of [512,6,340]3 code, which is a new result.

Let R be a finite principal ideal ring and m,n,d positive integers. In this paper, we study the matrix graph over R which is the graph whose vertices are m×n matrices over R and two matrices A and B are adjacent if and only if 0

This paper is dedicated to a question whether the currently known families of quadratic APN polynomials are pairwise different up to CCZ-equivalence. We reduce the list of these families to those CCZ-inequivalent to each other. In particular, we prove that the families of APN trinomials (constructed by Budaghyan and Carlet in 2008) and multinomials (constructed by Bracken et al. 2008) are contained

Let Fq be the finite field with q elements, and T a positive integer. In this article, we find an asymptotic formula for the total number of monic irreducible binomials in Fq[x] of degree less or equal to T, when T is large enough. We also show explicit lower and upper bounds for the number of binomials in the case when T is small.

In this paper, we propose the concepts of intersection distribution and non-hitting index, which can be viewed from two related perspectives. The first one concerns a point set S of size q+1 in the classical projective plane PG(2,q), where the intersection distribution of S indicates the intersection pattern between S and the lines in PG(2,q). The second one relates to a polynomial f over a finite

By using the piecewise method, Lagrange interpolation formula and Lucas' theorem, we determine explicit expressions of the inverses of a class of reversed Dickson permutation polynomials and some classes of generalized cyclotomic mapping permutation polynomials over finite fields of characteristic three.