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

We give the correct statement of main Theorem 2.1.

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.

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.

Very recently, Tu et al. presented a sufficient condition on (a1,a2,a3), see Theorem 1.1, such that f(x)=x3⋅2m+a1x2m+1+1+a2x2m+2+a3x3 is a class of permutation polynomials over F2n with n=2m and m odd. In this present paper, we prove that the sufficient condition is also necessary.

Let R be a finite commutative ring and n a positive integer. In this paper, we study the unitary Cayley graph CMn(R) of the matrix ring over R. If F is a field, we use the additive characters of Mn(F) to determine three eigenvalues of CMn(F) and use them to analyze strong regularity and hyperenergetic graphs. We find conditions on R and n such that CMn(R) is strongly regular. Without explicitly having

Minimal linear codes have important applications in secret sharing schemes and secure multi-party computation, etc. In this paper, we study the minimality of a kind of linear codes over GF(p) from Maiorana-McFarland functions. We first obtain a new sufficient condition for this kind of linear codes to be minimal without analyzing the weights of its codewords, which is a generalization of some works

In this work, free multivariate skew polynomial rings are considered, together with their quotients over ideals of skew polynomials that vanish at every point (which includes minimal multivariate skew polynomial rings). We provide a full classification of such multivariate skew polynomial rings (free or not) over finite fields. To that end, we first show that all ring morphisms from the field to the

In this paper, we are interested in finding an algebraic structure of conjucyclic codes of length n over the finite field F4. We show that conjucyclic codes of length n over F4 are related to binary cyclic codes of length 2n and show that there is a canonical bijective correspondence between the two sets. We illustrate how the factorization of the polynomial x2n+1 plays a critical role in each setting

In the last decade there has been a great interest in extending results for codes equipped with the Hamming metric to analogous results for codes endowed with the rank metric. This work follows this thread of research and studies the characterization of systematic generator matrices (encoders) of codes with maximum rank distance. In the context of Hamming distance these codes are the so-called Maximum

Colored multiple zeta values are special values of multiple polylogarithms evaluated at Nth roots of unity. In this paper, we define both the symmetrized version of these values for all positive integers N and the finite version for N=3,4 and 6. We then show that both versions satisfy the double shuffle relations. Further, for N=3 and 4 we provide strong evidence for an isomorphism connecting the two

Recently, Tang et al. [12] (resp. Wu et al. [15]) obtained a necessary and sufficient condition for a finite commutative group ring FG to be a ⁎-clean ring under the classical involution (resp. the conjugate involution), where F denotes a finite field and G denotes a finite abelian group. It was shown in [12] (resp. [15]) that FG is ⁎-clean under the classical involution (resp. the conjugate involution)

In this paper, we give the dimension and the minimum distance of two subclasses of narrow-sense primitive BCH codes over Fq with designed distance δ=aqm−1−1(resp. δ=aqm−1q−1) for all 1≤a≤q−1, where q is a prime power and m>1 is a positive integer. As a consequence, we obtain an affirmative answer to two conjectures proposed by C. Ding in 2015. Furthermore, using the previous part, we extend some results

Huffman (2013) [12] studied Fq-linear codes over Fqm and he proved the MacWilliams identity for these codes with respect to ordinary and Hermitian trace inner products. Let S be a finite commutative Fq-algebra. An Fq-linear code over S of length n is an Fq-submodule of Sn. In this paper, we study Fq-linear codes over S. We obtain some bounds on minimum distance of these codes, and some large classes

Cyclic code is an interesting topic in coding theory and communication systems. In this paper, we investigate the ternary cyclic codes with parameters [3m−1,3m−1−2m,4] based on some results proposed by Ding and Helleseth in 2013. Six new classes of optimal ternary cyclic codes are presented by determining the solutions of certain equations over F3m.

The second generalized GK function fields Kn are a recently found family of maximal function fields over the finite field with q2n elements, where q is a prime power and n≥1 an odd integer. In this paper we construct many new maximal function fields by determining various Galois subfields of Kn. In case gcd⁡(q+1,n)=1 and either q is even or q≡1(mod4), we find a complete list of Galois subfields of

In this paper we study some conjectures on determinants with Jacobi symbol entries posed by Z.-W. Sun. For any positive integer n≡3(mod4), we show that(6,1)n=[6,1]n=(3,2)n=[3,2]n=0 and(4,2)n=(8,8)n=(3,3)n=(21,112)n=0 as conjectured by Sun, where(c,d)n=|(i2+cij+dj2n)|1≤i,j≤n−1 and[c,d]n=|(i2+cij+dj2n)|0≤i,j≤n−1 with (⋅n) the Jacobi symbol. We also prove that (10,9)p=0 for any prime p≡5(mod12), and [5

It is known that the dual of a weakly regular bent function is again weakly regular. On the other hand, the dual of a non-weakly regular bent function may not even be a bent function. In 2013, Çesmelioğlu, Meidl and Pott pointed out that the existence of a non-weakly regular bent function having weakly regular bent dual is an open problem. In this paper, we prove that for an odd prime p and n∈Z+, if

A set is primitive if no element of the set divides another. We consider primitive sets of monic polynomials over a finite field and find natural generalizations of many of the results known for primitive sets of integers. In particular, we show that primitive sets in the function field have lower density zero by showing that the sum ∑a∈A1qdeg⁡adeg⁡a, an analogue of a sum considered by Erdős, is uniformly

A locally recoverable (LRC) code is a code over a finite field Fq such that any erased coordinate of a codeword can be recovered from a small number of other coordinates in that codeword. We construct LRC codes correcting more than one erasure, which are subfield-subcodes of some J-affine variety codes. For these LRC codes, we compute localities (r,δ) that determine the minimum size of a set R‾ of

In this paper, a necessary and sufficient condition for the homogeneous distance on an arbitrary finite commutative principal ideal ring to be a metric is obtained. We completely characterize the lower bound of homogeneous distances of matrix product codes over any finite principal ideal ring where the homogeneous distance is a metric. Furthermore, the minimum homogeneous distances of the duals of

Up to linear transformations, we obtain a classification of permutation polynomials (PPs) of degree 8 over F2r with r>3. By Bartoli et al. (2017) [1], a polynomial f of degree 8 over F2r is exceptional if and only if f−f(0) is a linearized PP, which has already been classified. So it suffices to search for non-exceptional PPs of degree 8 over F2r, which exist only when r⩽9 by a previous result. This