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Over the past several years, there are numerous papers about permutation polynomials of the form x r h (x q − 1) over F q 2 . A bijection between the multiplicative subgroup μ q + 1 of F q 2 and the projective line PG (1 , q) = F q ∪ { ∞ } plays a very important role in the research. In this paper, we mainly construct permutation polynomials of the form x r h (x q − 1) over F q 3 from bijections of the projective plane PG (2 , q). A bijection from the multiplicative subgroup μ q 2 + q + 1 of F q 3 to PG (2 , q) is studied, which is a key theorem of this paper. On this basis, some explicit permutation polynomials of the form x r h (x q − 1) over F q 3 are constructed from the collineation of PG (2 , q) , d -homogeneous monomials, 2-homogeneous permutations. It is worth noting that although the bijections of PG (2 , q) are simple, the corresponding permutation polynomials over F q 3 are usually complex. [ABSTRACT FROM AUTHOR]

Locally repairable codes (LRCs) are a class of erasure codes that are widely used in distributed storage systems, which allow for efficient recovery of data in the case of node failures or data loss. In 2014, Tamo and Barg introduced Reed-Solomon-like (RS-like) Singleton-optimal (r , δ) -LRCs based on polynomial evaluation. These constructions rely on the existence of so-called good polynomial that is constant on each of some pairwise disjoint subsets of F q. In this paper, we extend the aforementioned constructions of RS-like LRCs and propose new constructions of (r , δ) -LRCs whose code length can be larger. These new (r , δ) -LRCs are all distance-optimal, namely, they attain an upper bound on the minimum distance that will be established in this paper. This bound is sharper than the Singleton-type bound in some cases owing to the extra conditions, it coincides with the Singleton-type bound for certain cases. Combining our constructions with known explicit good polynomials of special forms, we can get various explicit Singleton-optimal (r , δ) -LRCs with new parameters, whose code lengths are all larger than that constructed by the RS-like (r , δ) -LRCs introduced by Tamo and Barg. Note that the code length of classical RS codes and RS-like LRCs are both bounded by the field size. We explicitly construct the Singleton-optimal (r , δ) -LRCs with length n = q − 1 + δ for any positive integers r , δ ≥ 2 and (r + δ − 1) | (q − 1). We also show the existence of Singleton-optimal (r , δ) -LRCs with length q + δ over F q = F p a (a ≥ 3) provided p 2 | (r + δ − 1) , (r + δ − 1) | p a − 1 and p | δ. When δ is proportional to q , they are asymptotically longer than that constructed via elliptic curves whose length is at most q + 2 q. Besides, they allow more flexibility on the values of r and δ. [ABSTRACT FROM AUTHOR]

We prove two conjectures involving permutation polynomials in a paper of Dmytrenko, Lazebnik, Williford, in a low degree regime, using the theory of algebraic curves over finite fields. More precisely, we prove that Conjecture A holds whenever q ≥ max ⁡ { (2 k − 1) 2 + 1 , 1.823 (4 k 2 − 14 k + 12) } , whereas Conjecture B holds if q ≥ 2.233 (9 k 2 − 21 k + 12). Although one of these conjectures was already proved by Hou without any restriction on the degree of the polynomials, we consider the proof contained in this paper is more direct and less computational. [ABSTRACT FROM AUTHOR]

In recent years, several families of scattered polynomials have been investigated in the literature. However, most of them only exist in odd characteristic. In [9,24] , the authors proved that the trinomial f c (X) = X q + X q 3 + c X q 5 of F q 6 [ X ] is scattered under the assumptions that q is odd and c 2 + c = 1. They also explicitly observed that this is false when q is even. In this paper, we provide a different set of conditions on c for which this trinomial is scattered in the case of even q. Using tools of algebraic geometry in positive characteristic, we show that when q is even and sufficiently large, there are roughly q 3 elements c ∈ F q 6 such that f c (X) is scattered. Also, we prove that the corresponding MRD-codes and F q -linear sets of PG (1 , q 6) are not equivalent to the previously known ones. [ABSTRACT FROM AUTHOR]

In this paper, as an analogue of the integer case, we study detailedly the period and the rank of the generalized Fibonacci sequences of polynomials over a finite field modulo an arbitrary polynomial. We establish some formulas to compute them, and we also obtain some properties about their quotient. We find that the polynomial case is much more complicated than the integer case. [ABSTRACT FROM AUTHOR]

In this paper we develop further studies on nilpotent linearized polynomials (NLP's) over finite fields, a class of polynomials introduced by the second author. We characterize certain NLP's that are binomials and show that, in general, NLP's are also nilpotent over a particular tower of finite fields. We also develop results on the construction of permutation polynomials from NLP's, extending some past results. In particular, the latter yields polynomials that permutes certain infinite subfields of F ‾ q and have a very particular cycle structure. Finally, we provide a nice correspondence between certain NLP's and involutions in binary fields and, in particular, we discuss a general method to produce affine involutions over binary fields without fixed points. [ABSTRACT FROM AUTHOR]

In this paper we study the permutational property of polynomials of the form f (L (x)) + k (L (x)) ⋅ M (x) ∈ F q n [ x ] over the finite field F q n , where L , M ∈ F q [ x ] are q -linearized polynomials and k ∈ F q n [ x ] satisfies a generic condition. We specialize in the case where L (x) is the linearized q -associate of g t , a (x) = (x n − 1) / (x t − a) , t is a divisor of n and a ∈ F q satisfies a n / t = 1. This unifies many recent explicit constructions and provides new explicit constructions of permutation polynomials and their inverses. Moreover, we introduce a new algorithmic method to produce many permutation polynomials of F q n from permutations of F q t , by simply solving a system of independent equations of the form Tr q n / q t (δ i − 1 a i) = c i , where the a i 's are the coefficients of f. In fact, the same method can be applied to construct complete mappings of F q n from complete mappings of F q. [ABSTRACT FROM AUTHOR]

In recent years several papers have appeared that investigate the classical discrete logarithm problem for elliptic curves by means of the multivariate polynomial approach based on the celebrated summation polynomials, introduced by Semaev in 2004. With a notable exception by Petit et al. in 2016, all numerous papers on the subject have investigated only the composite-field case, leaving apart the laborious prime-field case. In this paper we propose a variation of Semaev's original approach that reduces to only one the relations to be found among points of the factor base, thus decreasing drastically the necessary Groebner basis computations. Our proposal holds for any finite field but it is particularly suitable for the prime-field case, where it outperforms both the original Semaev's method and the specialised algorithm by Petit et al.. [ABSTRACT FROM AUTHOR]

Planar functions, introduced by Dembowski and Ostrom, have attracted much attention in the last decade. As shown in this paper, we present a new class of planar functions of the form Tr (a x q + 1) + ℓ (x 2) on an extension of the finite field F q n / F q. Specifically, we investigate those functions on F q 2 / F q and construct several typical kinds of planar functions. We also completely characterize them on F q 3 / F q. When the degree of extension is higher, it will be proved that such planar functions do not exist given certain conditions. [ABSTRACT FROM AUTHOR]

Permutation rational functions over finite fields have attracted much attention in recent years. In this paper, we introduce a class of permutation rational functions over F q 2 , whose numerators are so-called q -quadratic polynomials. To this end, we will first determine the exact number of zeros of a special q -quadratic polynomial in F q 2 , by calculating character sums related to quadratic forms of F q 2 / F q. Then given some rational function, we can demonstrate whether it induces a permutation of F q 2 . [ABSTRACT FROM AUTHOR]

A linear code with complementary dual (or an LCD code) is defined to be a linear code which intersects its dual code trivially. Let I be an identity matrix and T be a Toeplitz matrix of the same order over a finite field. A Double Toeplitz code (or a DT code) is a linear code generated by a generator matrix of the form (I , T). In 2021, Shi et al. obtained necessary and sufficient conditions for a Double Toeplitz code to be LCD when T is symmetric and tridiagonal. In this paper, by using a result on factoring Dickson polynomials over finite fields, we determine when a Double Toeplitz code is LCD for T being a skew symmetric and tridiagonal matrix. In addition, using a concatenation, we construct LCD codes with arbitrary minimum distance from DT codes over extension fields, provided the length of which is increased if necessary. [ABSTRACT FROM AUTHOR]

Let F q denote the finite field with q elements. Permutation polynomials and complete permutation polynomials over finite fields have been widely investigated in recent years due to their applications in cryptography, coding theory and combinatorial design. In this paper, several classes of (complete) permutation polynomials with the form (Tr m n (x) k + δ) s + L (x) and (Tr m n (x) k 1 + δ 1) s 1 + (Tr m n (x) k 2 + δ 2) s 2 + L (x) are proposed based on the AGW criterion and some techniques in solving equations over the finite field F 2 n , where L (x) = a Tr m n (x) + b x , a ∈ F 2 n and b ∈ F 2 m ⁎. We also determine the compositional inverse of these polynomials in some special cases. Besides, Mesnager (2014) [13] proposed a construction of bent functions by finding some triples of permutation polynomials satisfying a particular property named (A n). With the help of this approach, several classes of bent functions are presented. [ABSTRACT FROM AUTHOR]

Let q be a power of a prime p and F q be the finite field of order q. Let ϕ (x) be any polynomial in F q [ x ] and Φ (x) : = 1 ϕ (x). For any positive integer n , denote Φ (n) to be the n -th iterate of Φ and d n , ϕ to be the denominator of Φ (n). We call ϕ (x) ∈ F q [ x ] inversely stable over F q if d n , ϕ are distinct and irreducible over F q for all n. In this paper, we aim to find a class of inversely stable polynomials over F q. Actually, let ϕ (x) : = x p + a x + b ∈ F p [ x ] , it is proved that ϕ (x) is inversely stable over F p if and only if a = − 1 and b ≠ 0 ; moreover, if ϕ (x) is inversely stable over F p , then d n , ϕ is of degree p n for any positive integer n. Consequently, an infinite family of irreducible polynomials over F p is obtained. [ABSTRACT FROM AUTHOR]

By a result of Latimer and MacDuffee, there are a finite number of equivalence classes of n × n matrices over F q [ T ] with minimum polynomial p (X) , where p is an n th degree polynomial, irreducible over F q [ T ]. In this paper, we develop an algorithm for finding a canonical representative of each matrix class, for p (X) = X 2 − Γ X − Δ ∈ F q [ T ] [ X ]. [ABSTRACT FROM AUTHOR]

As a generalization of cyclic codes, constacyclic codes is an important and interesting class of codes due to their nice algebraic structures and various applications in engineering. This paper is devoted to the study of the q -polynomial approach to constacyclic codes. Fundamental theory of this approach will be developed, and will be employed to construct some families of optimal and almost optimal codes in this paper. [ABSTRACT FROM AUTHOR]

Permutation polynomials are an interesting subject of mathematics and have applications in other areas of mathematics and engineering. In this paper, we determine all permutation trinomials over F 2 m in Zieve's paper [30] . We prove a conjecture proposed by Gupta and Sharma in [8] and obtain some new permutation trinomials over F 2 m . Finally, we show that some classes of permutation trinomials with parameters are QM equivalent to some known permutation trinomials. [ABSTRACT FROM AUTHOR]

Let q = 2 m and F q 3 be the finite field with q 3 elements. In this paper, based on the multivariate method, resultant elimination, and transforming into dealing with some equations over finite fields, we propose two classes of permutation trinomials of F q 3 . We illustrate that these two classes of permutation trinomials are QM-inequivalent to all known permutation polynomials over F q 3 . Some well-known results can be covered by our theorems. [ABSTRACT FROM AUTHOR]

Let F q be a finite field with q elements, f ∈ F q [ x 1 , ... , x n ] a polynomial in n variables and let us denote by N (f) the number of roots of f in F q n. In this paper we consider the family of fully triangular polynomials, i.e., polynomials of the form f (x 1 , ... , x n) = a 1 x 1 d 1 , 1 + a 2 x 1 d 1 , 2 x 2 d 2 , 2 + ... + a n x 1 d 1 , n ⋯ x n d n , n − b , where d i , j > 0 for all 1 ≤ i ≤ j ≤ n. For these polynomials, we obtain explicit formulas for N (f) when the augmented degree matrix of f is row-equivalent to the augmented degree matrix of a linear polynomial or a quadratic diagonal polynomial. [ABSTRACT FROM AUTHOR]

Let F be a field of characteristic different from 2. Small-dimensional Jordan algebras over F have been extensively studied and classified. In the present paper we show that any two-dimensional Jordan algebras over a finite field are isomorphic if and only if they satisfy the same polynomial identities (the opposite happens in the case F is infinite, even if algebraically closed). We determine a finite generating set for the T-ideal of the polynomial identities of every two-dimensional Jordan algebra when F is finite, and linear bases for the corresponding relatively free algebras are also determined. [ABSTRACT FROM AUTHOR]

This paper investigates the permutation behavior of polynomials in the form b (x p m + a x + δ) s − a x over F p 2 m by employing the Akbary–Ghioca–Wang criterion, where b ∈ F p m ⁎ , a , δ ∈ F p 2 m with a q + 1 = 1. As a result, we present several classes of permutation polynomials of the form b (x 2 m + a x + δ) s + a x over F 2 2 m and three classes of permutation polynomials of the form b (x p m − x + δ) s + x over F p 2 m with infinitely many odd primes p. [ABSTRACT FROM AUTHOR]

In this paper, we investigate some classes of complete permutation polynomials (CPPs) with the form (L 1 (x)) t + L 2 (x) for some specific linearized polynomials L 1 (x) and L 2 (x) over finite fields. Some constructions of PPs and CPPs over finite fields using the AGW criterion are also proposed. We also obtain some constructions of sets of Mutually orthogonal Latin squares (MOLS) using permutation polynomials over finite fields. [ABSTRACT FROM AUTHOR]

In this paper, we propose several classes of permutation polynomials based on trace functions over finite fields of characteristic 2. The main result of this paper is obtained by determining the number of solutions of certain equations over finite fields. [ABSTRACT FROM AUTHOR]

In this paper, we propose several classes of permutation polynomials with the form ( x p m − x + δ ) s + L ( x ) over the finite field F p 2 m , where p is an odd prime, and L ( x ) is a linearized polynomial with coefficients in F p . The main method used in this paper is to determine the number of solutions of some equations over finite fields of odd characteristic. [ABSTRACT FROM AUTHOR]

Generalized quasi-cyclic (GQC) codes have been investigated as well as quasi-cyclic (QC) codes, e.g., on the construction of efficient low-density parity-check codes. While QC codes have the same length of cyclic intervals, GQC codes have different lengths of cyclic intervals. Similarly to QC codes, each GQC code can be described by an upper triangular generator polynomial matrix, from which the systematic encoder is constructed. In this paper, a complete theory of generator polynomial matrices of GQC codes, including a relation formula between generator polynomial matrices and parity-check polynomial matrices through their equations, is provided. This relation generalizes those of cyclic codes and QC codes. While the previous researches on GQC codes are mainly concerned with 1-generator case or linear algebraic approach, our argument covers the general case and shows the complete analogy of QC case. We do not use Gröbner basis theory explicitly in order that all arguments of this paper are self-contained. Numerical examples are attached to the dual procedure that extracts one from each other. Finally, we provide an efficient algorithm which calculates all generator polynomial matrices with given cyclic intervals. [ABSTRACT FROM AUTHOR]

Let F q be a finite field of order q. In this paper, for q even, we give a simple way to construct small degree polynomials over F q with no roots in the set μ q + 1 = { a ∈ F q 2 | a q + 1 = 1 } of the (q + 1) -th roots of unity. Further, we use these small degree polynomials over F q to construct and classify certain new families of permutation quadrinomials over F q 2 . [ABSTRACT FROM AUTHOR]

In this paper, we study some arithmetic properties of certain Toeplitz-type matrices over finite fields. For example, suppose { α 1 , ... , α n } are all the nonzero k -th powers in a finite field and for 1 ⩽ i , j ⩽ n , we set x i j = { 1 α i − α j if i ≠ j , 0 if i = j. We give an explicit formula for the characteristic polynomial of [ x i j ] 1 ⩽ i , j ⩽ n and confirm several conjectures posed by Zhi-Wei Sun. [ABSTRACT FROM AUTHOR]

Permutation polynomials of finite fields have many applications in Coding Theory, Cryptography and Combinatorics. In the first part of this paper we present a new family of local permutation polynomials based on a class of symmetric subgroups without fixed points, the so called e-Klenian groups. In the second part we use the fact that bivariate local permutation polynomials define Latin Squares, to discuss several constructions of Mutually Orthogonal Latin Squares (MOLS) and, in particular, we provide a new construction of MOLS on size a prime power. [ABSTRACT FROM AUTHOR]

Let F q denote the finite field of q elements with characteristic p. Let Z q denote the unramified extension of the p -adic integers Z p with residue field F q. In this paper, we investigate the q -divisibility for the number of solutions of a polynomial system in n variables over the finite Witt ring Z q / p m Z q , where the n variables of the polynomials are restricted to run through a box lifting F q n. It turns out that in general the answers do depend upon the box chosen. Based on the addition operation of Witt vectors, we prove a q -divisibility theorem for any box of low algebraic complexity, including the simplest Teichmüller box. This extends the classical Ax-Katz theorem over finite field F q (the case m = 1). Taking q = p to be a prime, our result extends and improves a recent theorem of Grynkiewicz for the unweighted case. [ABSTRACT FROM AUTHOR]

In this paper, two types of permutation polynomials, (x 2 m + x + δ) s + c x and x r (x q − 1 + a) , are studied. To formulate the necessary and sufficient conditions for the polynomials to be permutation polynomials over finite fields, the structures and properties of the field elements are analyzed. Meanwhile, the number of solutions to some equations over finite fields is investigated. For quadratic equations, the necessary and sufficient conditions are utilized thoroughly to study the permutation polynomials. But for cubic equations, only one direction is investigated by the existing literatures. This paper employs the properties of these equations in full strength to study (x 2 m + x + δ) s + c x over F 2 3 m ⁎. For the binomials of the form x r (x q − 1 + a) , two different methods are exploited to formulate the necessary and sufficient conditions over F q 3 , and some partial results are obtained for F q e . [ABSTRACT FROM AUTHOR]

In this paper, permutation polynomials of the form x d + L (x s) over finite fields F q 3 are investigated, where q = 2 m and m is a positive integer. By means of the iterative method, the multivariate method and the resultant elimination, six classes of permutation trinomials are obtained from certain integers d , s and linearized polynomials L (x). It is also showed that the permutations proposed in this paper are quasi-multiplicative inequivalent to all known permutation trinomials over F q 3 . [ABSTRACT FROM AUTHOR]

In this paper we give a modular interpretation of the k -th symmetric power L -function of the Kloosterman family of exponential sums in characteristics 2 and 3, and in the case of p = 2 and k odd give the precise 2-adic Newton polygon. We also give a p -adic modular interpretation of Dwork's unit root L -function of the Kloosterman family, and give the precise 2-adic Newton polygon when k is odd. In a previous paper, we gave an estimate for the q -adic Newton polygon of the symmetric power L -function of the Kloosterman family when p ≥ 5. We discuss how this restriction on primes was not needed, and also provide an improved estimate. [ABSTRACT FROM AUTHOR]

Abstract: Let p be a prime and q a power of p. For , let be the polynomial defined by the functional equation . When is a permutation polynomial (PP) of ? This turns out to be a challenging question with remarkable breath and depth, as shown in the predecessor of the present paper. We call a triple of positive integers desirable if is a PP of . In the present paper, we find many new classes of desirable triples whose corresponding PPs were previously unknown. Several new techniques are introduced for proving a given polynomial is a PP. [Copyright &y& Elsevier]

Abstract: Cyclic codes have been an interesting topic of both mathematics and engineering for decades. They are prominently used in consumer electronics, data transmission technologies, broadcast systems, and computer applications. Three classical approaches to the study and construction of cyclic codes are those based on the generator matrix, the generator polynomial and the idempotent. The objective of this paper is to develop another approach – the q-polynomial approach. Fundamental theory of this approach will be developed, and will be employed to construct a new family of cyclic codes in this paper. [Copyright &y& Elsevier]

In this paper we provide an algorithm to classify groups of points on abelian threefolds over finite fields. The classification is given in terms of the Weil polynomial of abelian varieties in a given F q -isogeny class. This work completes partial classification given in [9]. [ABSTRACT FROM AUTHOR]

Abstract In this paper, several classes of complete permutation polynomials with the form (x p m − x + δ) s + a x p m + b x over F p n are proposed by the AGW criterion and determining the number of solutions of some equations. Our results also enrich constructions of known permutation polynomials. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we construct several new classes of complete permutation monomials a − 1 x d over a finite field F q n with exponents d = q n − 1 q − 1 + 1 , q p − 1 − 1 q − 1 + 1 , and q q − 1 − 1 q − 1 + 1 , respectively, where q = p k is a power of a prime number p. Our approach uses the AGW criterion (the multiplicative case) together with Dickson permutation polynomials and a class of exceptional polynomials respectively. One of our results confirms Conjecture 4.18 by G. Wu, N. Li, T. Helleseth, Y. Zhang in [42] under the assumption that the characteristic p is primitive modulo a prime number n + 1. Moreover, we show that Conjecture 4.18 is false in general using our approach and a counterexample is provided. We also re-confirm Conjecture 4.20 in [42] that was proved recently in [24] , and extend some of these recent results to more general n 's and more general a 's. [ABSTRACT FROM AUTHOR]

Abstract Subspace codes and particularly cyclic subspace codes have attracted a lot of research due to their applications in error correction for random network coding. In this paper, we explore the parameters of a class of the cyclic subspace codes whose subspace polynomials are more generalized. Therefore, some corollaries that extend the previously known results are obtained. Finally, a construction of k -dimensional cyclic subspace codes of size q N − 1 q − 1 and minimal distance 2 k − 2 is provided. [ABSTRACT FROM AUTHOR]

Abstract In this paper some new links between the nonlinearity and differential uniformity of some large classes of functions are established. Differentially two-valued functions and quadratic functions are mainly treated. A lower bound for the nonlinearity of monomial δ -uniform permutations is obtained, for any δ , as well as an upper bound for differentially two-valued functions. Concerning quadratic functions, significant relations between nonlinearity and differential uniformity are exhibited. In particular, we show that the quadratic differentially 4-uniform permutations should be differentially two-valued and possess the best known nonlinearity. [ABSTRACT FROM AUTHOR]

Abstract Let F q be the finite field with q elements and let p = char F q. It was conjectured that for integers e ≥ 2 and 1 ≤ a ≤ p e − 2 , the polynomial X q − 2 + X q 2 − 2 + ⋯ + X q a − 2 is a permutation polynomial of F q e if and only if (i) a = 2 and q = 2 , or (ii) a = 1 and gcd (q − 2 , q e − 1) = 1. In the present paper we confirm this conjecture. [ABSTRACT FROM AUTHOR]

Abstract Two vectorial Boolean functions are "CCZ-equivalent" if there exists an affine permutation mapping the graph of one to the other. It preserves many of the cryptographic properties of a function such as its differential and Walsh spectra, which is why it could be used by Dillon et al. to find the first APN permutation on an even number of variables. However, the meaning of this form of equivalence remains unclear. In fact, to the best of our knowledge, it is not known how to partition a CCZ-equivalence class into its Extended-Affine (EA) equivalence classes; EA-equivalence being a simple particular case of CCZ-equivalence. In this paper, we characterize CCZ-equivalence as a property of the zeroes in the Walsh spectrum of a function F : F 2 n → F 2 m or, equivalently, of the zeroes in its Difference Distribution Table. We use this framework to show how to efficiently upper bound the number of distinct EA-equivalence classes in a given CCZ-equivalence class. More importantly, we prove that it is possible to go from a specific member of any EA-equivalence class to a specific member of another EA-equivalence class in the same CCZ-equivalence class using an operation called twisting ; so that CCZ-equivalence can be reduced to the association of EA-equivalence and twisting. Twisting a function is a simple process and its possibility is equivalent to the existence of a particular decomposition of the function considered. Using this knowledge, we revisit several results from the literature on CCZ-equivalence and show how they can be interpreted in light of our new framework. Our results rely on a new concept, the "thickness" of a space (or linear permutation), which can be of independent interest. [ABSTRACT FROM AUTHOR]

Abstract Let q be a power of a prime and F q be a finite field with q elements. In this paper, we propose four families of infinite classes of permutation trinomials having the form c x − x s + x q s , and investigate the relationship between two types of widely discussed permutation polynomials. Based on this relation, many classes of permutation polynomials having the form (x q k − x + δ) s + c x without restriction on δ are derived. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we propose several classes of complete permutation polynomials over a finite field based on certain polynomials over its subfields or subsets. In addition, a class of complete permutation trinomials with Niho exponents is studied, and the number of these complete permutation trinomials is also determined. [ABSTRACT FROM AUTHOR]

Abstract Self-reciprocal and self-conjugate-reciprocal polynomials over finite fields have been of interest due to their rich algebraic structures and wide applications. Self-reciprocal irreducible monic factors of x n − 1 over finite fields and their applications have been quite well studied. In this paper, self-conjugate-reciprocal irreducible monic (SCRIM) factors of x n − 1 over finite fields of square order are focused on. The characterization of such factors is given together with the enumeration formula. In many cases, recursive formulas for the number of SCRIM factors of x n − 1 are given as well. As applications, Hermitian complementary dual cyclic codes over finite fields and Hermitian self-dual cyclic codes over finite chain rings of prime characteristic are discussed. [ABSTRACT FROM AUTHOR]

Abstract A convolutional code C over Z p r ((D)) is a Z p r ((D)) -submodule of Z p r n ((D)) that admits a polynomial set of generators, where Z p r ((D)) stands for the ring of (semi-infinity) Laurent series. In this paper we study several structural properties of its dual C ⊥. We use these results to provide a constructive algorithm to build an explicit generator matrix of C ⊥. Moreover, we show that the transpose of such a matrix is a parity-check matrix (also called syndrome former) of C. [ABSTRACT FROM AUTHOR]

Abstract: The present paper deals with counting the number and orders of regular semisimple elements, together with the number of primary classes of . The approach used in this paper depends essentially on partitions of positive integers ⩽n. We give the numbers of regular semisimple elements and primary classes of for and see that the number of regular semisimple elements is an integral polynomial in q, while the number of primary classes is a rational polynomial in q. [Copyright &y& Elsevier]

Abstract Riemann's hypothesis on function fields over a finite field implies the Hasse–Weil bound for the number of zeros of an absolutely irreducible bi-variate polynomial over a finite field. The Hasse–Weil bound has extensive applications in the arithmetic of finite fields. In this paper, we use the Hasse–Weil bound to prove two results on permutation polynomials over F q where q is sufficiently large. To facilitate these applications, the absolute irreducibility of certain polynomials in F q [ X , Y ] is established. [ABSTRACT FROM AUTHOR]

Abstract Guruswami and Resch proved that a random F q -linear rank-metric code is list decodable with list decoding radius attaining the Gilbert–Varshamov bound [8]. Furthermore, in Hamming metric, random linear self-orthogonal codes can be list decoded up to the Gilbert–Varshamov bound with polynomial list size [11]. Motivated by these two results and the potential applications of self-orthogonal rank-metric codes in network coding and cryptography [20] , [18] and [5] , we focus on investigating their list decodability. In this paper, we prove that with high probability, a random F q -linear self-orthogonal rank-metric code over F q n × m can be list decoded up to the Gilbert–Varshamov bound with polynomial list size. In addition, we show that an F q m -linear self-orthogonal rank-metric code of rate up to the Gilbert–Varshamov bound with exponential list size. [ABSTRACT FROM AUTHOR]

Abstract The Carlitz–Wan conjecture, which is now a theorem, asserts that for any positive integer n , there is a constant C n such that if q is any prime power > C n with GCD (n , q − 1) > 1 , then there is no permutation polynomial of degree n over the finite field with q elements. From the work of von zur Gathen, it is known that one can take C n = n 4. On the other hand, a conjecture of Mullen, which asserts essentially that one can take C n = n (n − 2) has been shown to be false. In this paper, we use a precise version of Weil bound for the number of points of affine algebraic curves over finite fields to obtain a refinement of the result of von zur Gathen where n 4 is replaced by a sharper bound. As a corollary, we show that Mullen's conjecture holds in the affirmative if n (n − 2) is replaced by n 2 (n − 2) 2. [ABSTRACT FROM AUTHOR]

In this paper, we completely determine the isotopism classes of the Budaghyan–Helleseth commutative semifields constructed in Budaghyan and Helleseth (2011) [4] . [ABSTRACT FROM AUTHOR]

In this paper, two classes of permutation trinomials with Niho-type exponents are proposed. The coefficients in the first class are completely determined, and a sufficient condition is established to characterize the coefficients in the second class. [ABSTRACT FROM AUTHOR]