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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 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]

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]

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]