The study of n-dimensional inverted Kloosterman sums was suggested by Katz (1995) [7] who handled the case when from complex point of view. For general , the n-dimensional inverted Kloosterman sums were studied from both complex and p-adic point of view in our previous paper (2024) [10]. In this note, we study the algebraic degree of the inverted n-dimensional Kloosterman sum as an algebraic integer.
BCH codes form a special subclass of cyclic codes and have been extensively studied in the past decades. Determining the parameters of BCH codes, however, has been an important but difficult problem. Recently, in order to further investigate the dual codes of BCH codes, the concept of dually-BCH codes was proposed. In this paper, we study BCH codes of lengths and over the finite field , both of which are LCD codes. The dimensions of narrow-sense BCH codes of length with designed distance are determined, where and . Lower bounds on the minimum distances of the dual codes of narrow-sense BCH codes of length are developed for odd q, which are good in some cases. Moreover, sufficient and necessary conditions for the even-like subcodes of narrow-sense BCH codes of length being dually-BCH codes are presented, where q is odd and .
Permutation polynomials with low c-differential uniformity have important applications in cryptography and combinatorial design. In this paper, we investigate perfect c-nonlinear (PcN) and almost perfect c-nonlinear (APcN) polynomials over finite fields. Based on some known permutation polynomials, we present several classes of PcN or APcN polynomials by using the Akbary-Ghioca-Wang criterion.
n-to-1 mappings have many applications in cryptography, finite geometry, coding theory and combinatorial design. In this paper, we first use cyclotomic cosets to construct several kinds of n-to-1 mappings over . Then we characterize a new form of AGW-like criterion, and use it to present many classes of n-to-1 polynomials with the form over . Finally, by using monomials on the cosets of a subgroup of and another form of AGW-like criterion, we show some n-to-1 trinomials over .
An explicit formula for the number of solutions of the equation in the title is given when a certain condition, depending only on the exponent and the characteristic of the field, holds. This formula improves the one given by the authors in an earlier paper.
We study the irreducibility and Galois group of random polynomials over function fields. We prove that a random polynomial with i.i.d. coefficients taking values in the set with uniform probability, is irreducible with probability tending to as , where d and q are fixed. We also prove that with the same probability, the Galois group of this random polynomial contains the alternating group . Moreover, we prove that if we assume a version of the polynomial Chowla conjecture over , then the Galois group of this polynomial is actually equal to the symmetric group with probability tending to . We also study the other possible Galois groups occurring with positive limit probability. Finally, we study the same problems with n fixed and .
In this paper we find the largest automorphism group of a smooth cubic surface over any finite field of characteristic 2. We prove that if the order of the field is a power of 4, then the automorphism group of maximal order of a smooth cubic surface over this field is . If the order of the field of characteristic 2 is not a power of 4, then we prove that the automorphism group of maximal order of a smooth cubic surface over this field is the symmetric group of degree 6. Moreover, we prove that smooth cubic surfaces with such properties are unique up to isomorphism.
The notion of a Heffter array, which received much attention in the last decade, is equivalent to a pair of orthogonal Heffter systems. In this paper we study the existence problem of a set of r mutually orthogonal Heffter systems for any r. Such a set is equivalent to a resolvable partial linear space of degree r whose parallel classes are Heffter systems: this is a new combinatorial design that we call a Heffter space. We present a series of direct constructions of Heffter spaces with odd block size and arbitrarily large degree r obtained with the crucial use of finite fields. Among the applications we establish, in particular, that if is a prime power with kw odd and , then there are at least mutually orthogonal k-cycle systems of order q.
In this paper, we consider the moments of the trace of Frobenius of elliptic curves if the trace is restricted to a fixed arithmetic progression. We determine the asymptotic behavior for the ratio of the -th moment to the zeroeth moment as the size of the finite field goes to infinity. These results follow from similar asymptotic formulas relating sums and moments of Hurwitz class numbers where the sums are restricted to certain arithmetic progressions. As an application, we prove that the distribution of the trace of Frobenius in arithmetic progressions is equidistributed with respect to the Sato–Tate measure.
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