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

Boolean functions, and bent functions in particular, are considered up to so-called EA-equivalence, which is the most general known equivalence relation preserving bentness of functions. However, for a special type of bent functions, so-called Niho bent functions there is a more general equivalence relation called o-equivalence which is induced from the equivalence of o-polynomials. In the present

Functions with low differential uniformity have relevant applications in cryptography. Recently, functions with low c-differential uniformity attracted lots of attention. In particular, so-called APcN and PcN functions (generalization of APN and PN functions) have been investigated. Here, we provide a characterization of such functions via quadratic polynomials as well as non-existence results.

We study the asymptotic proportion of smooth plane curves over a finite field Fq which are tangent to every line defined over Fq. This partially answers a question raised by Charles Favre. Our techniques include applications of Poonen's Bertini theorem and Schrijver's theorem on perfect matchings in regular bipartite graphs. Our main theorem implies that a random smooth plane curve over Fq admits a

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