Motivated by the classification of one-to-one rational functions of low degree in terms of their equivalence classes, we determine all many-to-one (including one-to-one) rational functions of degree two or three on the projective line explicitly in terms of their coefficients. Furthermore, we study the linear-fractional equivalence classes and the value sets of these rational functions. As an application, we characterize two classes of many-to-one quadranomials using their coefficients. These one-to-one quadranomials unify and generalize many results in the literature.
In this paper, we investigate the cardinality of the distance set Δ(A,B) for sets A and B⊂Fqd where A or B is contained in a k-dimensional affine subspace over a finite field. Assuming that B lies in a k-coordinate plane up to translations and rotations, we prove that if |A||B|>2qd, then |Δ(A,B)|>q2, where |Δ(A,B)| denotes the number of distinct distances between elements of A and B. In particular, we show that our result recovers the sharp (d+1)/2 threshold for the Erdős–Falconer distance problem in odd dimensions, where distances are determined by a single set. As a further application, we also obtain an improved result on the Box distance problem posed by Borges, Iosevich, and Ou, in the case where 2 is a square in Fq.
In this paper, based on the investigation of associated fractional polynomials over finite fields, we study the permutation property of trinomials and pentanomials with the form xrh(xpm−1) over Fp2m, where p∈{3,5}. More precisely, by choosing some irreducible trinomials and pentanomials over F3 and F5, three classes of permutation trinomials and six classes of permutation pentanomials are obtained by factorizing the corresponding plane algebraic curves and determining whether some low-degree equations have no solutions in the unit circle. In addition, three classes of permutation pentanomials are established by checking the fractional permutations via other arguments. Finally, we verify that the permutation trinomials and pentanomials proposed in this paper are quasi-multiplicative inequivalent to known ones.
BCH and LCD cyclic codes of length n=λ(qm+1) with λ|q−1 are studied. A complete characterization of q-cyclotomic cosets modulo n is established: Theorem 3.7 provides a necessary and sufficient condition for any 0≤γ<n to be a coset leader, and for odd m, the two largest coset leaders are explicitly determined (Theorem 3.10 and Theorem 3.11). Based on these results, the dimensions of several families of BCH codes are determined, and the lower bound on the minimum distance of C(q,n,2δ+1,n−δ+1) is raised to 2(δ+1) (Theorem 4.1, Theorem 4.6). Moreover, several of these codes are optimal. When m is odd, the necessary and sufficient condition for the BCH code C(q,n,δ,0) to be dually-BCH is proved (Theorem 4.9). Finally, an exact enumeration of all LCD cyclic codes of this length is derived (Theorem 5.8). All of the above results extend previous results restricted to λ=1.
Let Fq be the finite field of prime power order q and characteristic p. Let Q denote the quasi-Galois ring Fq[u]〈u2〉 with maximal ideal 〈u〉 of nilpotency index 2 and residue field Fq. Let R=Q×Fq be the mixed-alphabet ring. In this paper, we investigate the Galois hulls of the Gray images of three infinite families of linear codes over Q, constructed in Jose et al. [20]. We show that these codes are Galois self-orthogonal for every automorphism of Fq over Fp. We also obtain an infinite family of near-Griesmer, distance-optimal and Galois self-orthogonal codes over Fq. In certain special cases, codes in these families are double repetitions of the codes belonging to the well-known family of Solomon–Stiffler codes. Consequently, we investigate the Galois hulls of these special Solomon-Stiffler codes and show that these codes are Galois self-orthogonal for every automorphism of Fq over Fp. This also gives rise to three infinite families of Griesmer and Galois self-orthogonal codes over Fq. Leveraging this Galois self-orthogonality, we construct several new families of entanglement-assisted quantum error-correcting codes (EAQECCs). In addition, we identify three classes of EAQECCs with maximal entanglement that achieve the Griesmer-type bound on the lengths of EAQECCs constructed from linear codes over finite fields. Finally, we derive three infinite families of intersecting codes over Fq and explicitly determine their trellis complexities.
Most of the known (Almost) Perfect Nonlinear functions over an arbitrary finite field Fpn have the lowest possible algebraic degree (2) and can be represented by Dembowski-Ostrom (DO) polynomials. We define Generalized DO (GDO) polynomials as homogeneous polynomials of algebraic degree p over finite fields with characteristic p. We illustrate the similarities between DO and GDO polynomials when studied through generalized derivatives. We also discuss the differential uniformity and Walsh coefficients of GDO functions. Our main contribution is to use this framework in order to propose several new classes of Generalized Almost Perfect Nonlinear (GAPN) functions of algebraic degree p, the lowest possible over the finite field Fpn.
Determining the deep holes of a linear code is essential for its decoding. The deep hole problem of RS, PRS, and TRS codes has attracted significant attention from researchers. Reed-Solomon codes RSk(D) with an evaluation set D subset of F-q are based on polynomials of degree at most k - 1. In this paper, we focus on a code C(D, k) of length n and dimension k over F-q defined by evaluation polynomials from the subspace span(Fq) {1, x, ... , x(k-2), x(k)}. We investigate the deep hole problem of the code C(D, k) of non-Reed-Solomon type. For a general evaluation set D, we determine the covering radius of the code C(D, k) and establish fundamental results on its deep holes. For D = F-q with even q >= 8, we completely classify its deep holes except for the case k = q-4. For D = F-q with odd q, we obtain a complete classification for q+1/2 <= k <= q - 1. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
In this paper, we investigate a generalized Kloosterman sum of the form K( psi, chi;a,b)= sum x in F d <^> * psi(x) * chi(ax + b * x <^> - 1) , where q is a prime character of F,, and is a multiplicative character of Fx. While existing work on generalized Kloosterman sums has mainly focused on bounding their magnitudes, we determine their exact value distributions in two specific cases: (i) when q = 2 <^> n with n even and of order 3; (ii) when qp where p is an odd prime and has order 2. Additionally, based on the proposed distribution, we present a novel construction of a polyphase sequence set of length q1 and size q & sup2;-1, with a maximum correlation magnitude is upper bounded by 2sqrt(q) Notably, when p = 2 (with n even) or when p is an odd prime, the maximum correlation magnitude of the proposed sequence set is exactly 2sqrt(q) (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, Al training, and similar technologies.
Let q be a prime power and m > 1 be any integer. Let F-q(m) be the finite field of order q(m) and theta is an element of F-q(m) be such that F-q(m) = F-q(theta). We obtain a nontrivial bound for the mixed character sum Sigma(x is an element of Fq)chi(theta + x) psi(x), where chi and psi are multiplicative and additive characters of F-q(m) and F-q, respectively, using function field methods. As an application of our main result, we prove that for fixed m and sufficiently large prime powers q, that satisfy certain conditions, F-q(m) /F-q possesses the weak line property for primitive normal elements. In particular, our result is a strengthening of existing results. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Bugeaud, Mignotte, and Siksek proved that the only perfect powers in the Fibonacci sequence are 0, 1, 8, and 144. In this paper, we study the polynomial analogue of this problem. In particular, we give a complete characterization of the Fibonacci polynomials that are perfect powers or powerful over finite fields, where infinitely many such examples occur. We also give similar characterizations for some of Horadam's generalized Lucas polynomial sequences, which include Fibonacci, Lucas, Chebyshev, and Jacobsthal polynomials.
A subspace U of Fqn is called cyclically covering if the whole space Fqn is the union of the cyclic shifts of U. The case Fqn itself is the only covering subspace, is of particular interest. Recently, Huang solved this problem completely under the condition gcd(n,q)=1 using primitive idempotents and trace functions, and explicitly posed the non-coprime case as an open question. This paper provides a complete answer to Huang's question. We prove that if n=pkm where p=char(Fq) and gcd(m,p)=1, then hq(pkm)=0 if and only if hq(m)=0. This result fully reduces the non-coprime case to the coprime case settled by Huang. Our proof employs the structure theory of cyclic group algebras in modular characteristic.
In this paper, we characterize all minimal value set binomials over F-q, that is, binomials whose size of the set of images is the smallest possible. With this information, we also classify all quadrinomial curves with separated variables that are F-q-Frobenius nonclassical for the morphism of lines. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Butson Hadamard matrices are complex Hadamard matrices with entries in the complex roots of unity of given order. There is an interesting code in phase space related to these matrices called Butson Hadamard codes in (Armario et al. 2023). We study the covering radius of these codes for the homogeneous weight, a weight of fundamental importance in codes over rings. It is defined uniquely, up to scaling, for a commutative ring alphabet that is Quasi Frobenius. Two upper bounds on the covering radius are derived by an orthogonal array argument. A lower bound relies on the existence of bent sequences in the sense of (Shi et al. 2022). This latter bound generalizes a bound of (Armario et al. 2025) for the Hamming weight.
Inspired by the seminal work of André-Louis Cholesky – whose contributions remain crucial in broader sciences even after more than a century – Cooper, Hanna and Whitlatch (2024) developed a theory of positive matrices over finite fields, and Khare and Vishwakarma (2025) described a general Cholesky factorization for a dense sub-family of the cone of Hermitian matrices over real/complex fields, whose leading principal minors (LPM) are nonzero. Building on these works, we develop a parallel theory in the finite field setting. Specifically (i) we extend the general Cholesky factorization to the LPM cone over finite fields which has asymptotic density 1. We show that (ii) this factorization is compatible with the entrywise Frobenius map, recently studied in the context of positivity preservers by Guillot et al. (2025) [3]. We also (iii) leverage the Cholesky-structures to define meaningful group operations on the matrix cone, and as an application (iv) enumerate sub-cones of LPM matrices using our general Cholesky factorizations.
An essential problem in algebraic coding theory is to determine the structure and the Hamming distance of codes. For some natural number B <= 9, this problem has been solved for the repeated-root constacyclic codes of length Bps over the finite field Fpm of characteristic p. In this work, we study the structure and the Hamming distance of all repeated-root constacyclic codes of length 10ps over Fpm . As an application, we find all the optimal or nearly optimal MDS, almost-MDS, and near-MDS codes with respect to the Singleton bound among these codes. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
In coding theory, cyclic codes and MDS codes are important families that have been extensively studied. The objective of this paper is to study MDS cyclic codes, which represent the intersection of these two classes. We focus on cyclic codes with defining sets of the forms {b, b + 1, & centerdot; & centerdot; & centerdot;, b + delta - 2, s} or {s, b, b + 1, & centerdot; & centerdot; & centerdot;, b + delta - 2}. We establish necessary and sufficient conditions for these cyclic codes to be MDS. Our results extend the main theorems of [Li et al. IEEE-TIT, 2025] and strengthen several cases, yielding many new classes of non-GRS MDS cyclic codes. Our approach differs significantly from that of the aforementioned paper. More generally, we regard the targeted cyclic codes as subcodes of Reed-Solomon codes and utilize fundamental properties of polynomials to derive our conclusions. As illustrative examples, we provide several infinite families of non-GRS MDS cyclic codes of prime length over Fq and of length q + 1 over Fq2 . (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Let Fqn be the finite extension of the finite field Fq of degree n, where q is a prime power and n is a positive integer. Suppose E, C and Dare nonempty proper subsets of Fq & lowast;n such that for each y E E there is at least one (c, d) E C & times; D such that y = cd. We show that if the three subsets are large enough, an arbitrary element in Fq can be assigned as trace for some y E E. We extend the idea to k nonempty proper subsets C1, C2, ... , Ck of Fq & lowast;n, where at least one element (c1, c2, ... , ck) E C1 & times; C2 & times; & centerdot; & centerdot; & centerdot; & times; Ck exists for each y E E such that y = c1c2 & centerdot; & centerdot; & centerdot; ck. Conditions obtained are applied to obtain a new criterion for the existence of r-primitive elements in Fqn with arbitrary traces in Fq. This criterion significantly reduces the number and complexity of the calculations required to solve the case for any r such that ( q case of 3-primitive elements completely as an example and give another concise proof for the case of 2-primitive elements. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies. n-1 r ) =/ 1, 2. We solve the
In this paper, we study the asymptotic properties of exponential sums on smooth projective curves over finite fields. Specifically, as an analogue of the original Ihara constant, we introduce a version of the Ihara constant for exponential sums and also a certain relative version of the Ihara constant, and give the relationship between these three constants (here we employ Deligne's modification of the exponential sums in the definition). In particular, we prove that these constants satisfy the analogue of the lower bound of Bassa-Beelen-Garc & iacute;a-Stichtenoth obtained for the original Ihara constant, by using the tower they constructed, when the cardinality of the finite field is not a prime. Consequently, all these constants are equal and reach the optimal Drinfeld-Vladut bound when the cardinality of the finite field is a square. In addition, we use results on infinite class field towers to show that these constants are positive in general. Finally by using the analytic technique of explicit formula, we also obtain an analogue of Tsfasman's basic inequality for these (Deligne's modified) exponential sums. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
