Abstract: Publication date: January 2018 Source:Finite Fields and Their Applications, Volume 49 Author(s): Leyla Işık, Arne Winterhof Carlitz rank and index are two important measures for the complexity of a permutation polynomial f ( x ) over the finite field F q . In particular, for cryptographic applications we need both, a high Carlitz rank and a high index. In this article we study the relationship between Carlitz rank C r k ( f ) and index I n d ( f ) . More precisely, if the permutation polynomial is neither close to a polynomial of the form ax nor a rational function of the form a x − 1 , then we show that C r k ( f ) > q − max { 3 I n d ( f ) , ( 3 q ) 1 / 2 } . Moreover we show that the permutation polynomial which represents the discrete logarithm guarantees both a large index and a large Carlitz rank.

Abstract: Publication date: January 2018 Source:Finite Fields and Their Applications, Volume 49 Author(s): Daniele Bartoli, Giovanni Zini We determine all permutation trinomials of type x 2 p s + r + x p s + r + λ x r over the finite field F p t when ( 2 p s + r ) 4 < p t . This partially extends a previous result by Bhattacharya and Sarkar in the case p = 2 , r = 1 .

Abstract: Publication date: January 2018 Source:Finite Fields and Their Applications, Volume 49 Author(s): Nurdagül Anbar, Almasa Odz̆ak, Vandita Patel, Luciane Quoos, Anna Somoza, Alev Topuzoğlu The well-known Chowla and Zassenhaus conjecture, proved by Cohen in 1990, states that if p > ( d 2 − 3 d + 4 ) 2 , then there is no complete mapping polynomial f in F p [ x ] of degree d ≥ 2 . For arbitrary finite fields F q , a similar non-existence result was obtained recently by Işık, Topuzoğlu and Winterhof in terms of the Carlitz rank of f. Cohen, Mullen and Shiue generalized the Chowla–Zassenhaus–Cohen Theorem significantly in 1995, by considering differences of permutation polynomials. More precisely, they showed that if f and f + g are both permutation polynomials of degree d ≥ 2 over F p , with p > ( d 2 − 3 d + 4 ) 2 , then the degree k of g satisfies k ≥ 3 d / 5 , unless g is constant. In this article, assuming f and f + g are permutation polynomials in F q [ x ] , we give lower bounds for the Carlitz rank of f in terms of q and k. Our results generalize the above mentioned result of Işık et al. We also show for a special class of permutation polynomials f of Carlitz rank n ≥ 1 that if f + x k is a permutation over F q , with gcd ( k + 1 , q − 1 ) = 1 , then k ≥ ( q − n ) / ( n + 3 ) .

Abstract: Publication date: January 2018 Source:Finite Fields and Their Applications, Volume 49 Author(s): Ron Evans, Mark Van Veen Let F q be a field of q elements, where q is a power of an odd prime p. The polynomial f ( y ) ∈ F q [ y ] defined by f ( y ) : = ( 1 + y ) ( q + 1 ) / 2 + ( 1 − y ) ( q + 1 ) / 2 has the property that f ( 1 − y ) = ρ ( 2 ) f ( y ) , where ρ is the quadratic character on F q . This univariate identity was applied to prove a recent theorem of N. Katz. We formulate and prove a bivariate extension, and give an application to quadratic residuacity.

Abstract: Publication date: January 2018 Source:Finite Fields and Their Applications, Volume 49 Author(s): Hiroaki Taniguchi We construct a bilinear dual hyperoval S c ( S 1 , S 2 , S 3 ) from binary commutative presemifields S 1 = ( G F ( q ) , + , ∘ ) and S 2 = ( G F ( q ) , + , ⁎ ) , a binary presemifield S 3 = ( G F ( q ) , + , ⋆ ) which may not be commutative, and a non-zero element c ∈ G F ( q ) which satisfies some conditions. We also determine the isomorphism problems under the conditions that S 1 and S 2 are not isotopic, and c ≠ 1 . We also investigate farther on the isomorphism problem on the case that S 1 and S 2 are the Kantor commutative presemifields and S 3 is the Albert presemifield.

Abstract: Publication date: January 2018 Source:Finite Fields and Their Applications, Volume 49 Author(s): Masaaki Homma, Seon Jeong Kim A basis of the ideal of the complement of a linear subspace in a projective space over a finite field is given. As an application, the second largest number of points of plane curves of degree d over the finite field of q elements is also given for d ≥ q + 1 .

Abstract: Publication date: January 2018 Source:Finite Fields and Their Applications, Volume 49 Author(s): E. Martínez-Moro, A. Piñera-Nicolás, I.F. Rúa We consider codes defined over an affine algebra A = R [ X 1 , … , X r ] / 〈 t 1 ( X 1 ) , … , t r ( X r ) 〉 , where t i ( X i ) is a monic univariate polynomial over a finite commutative chain ring R. Namely, we study the A − submodules of A l ( l ∈ N ). These codes generalize both the codes over finite quotients of polynomial rings and the multivariable codes over finite chain rings. Some codes over Frobenius local rings that are not chain rings are also of this type. A canonical generator matrix for these codes is introduced with the help of the Canonical Generating System. Duality of the codes is also considered.

Abstract: Publication date: January 2018 Source:Finite Fields and Their Applications, Volume 49 Author(s): Mireille Fouquet, Josep M. Miret, Javier Valera Volcanoes of ℓ-isogenies of elliptic curves are a special case of graphs with a cycle called crater. In this paper, given an elliptic curve E of a volcano of ℓ-isogenies, we present a condition over an endomorphism φ of E in order to determine which ℓ-isogenies of E are non-descending. The endomorphism φ is defined as the crater cycle of an m-volcano where E is located, with m ≠ ℓ . The condition is feasible when φ is a distortion map for a subgroup of order ℓ of E. We also provide some relationships among the crater sizes of volcanoes of m-isogenies whose elliptic curves belong to a volcano of ℓ-isogenies.

Abstract: Publication date: January 2018 Source:Finite Fields and Their Applications, Volume 49 Author(s): Sudhir R. Ghorpade, Prasant Singh We consider linear codes associated to Schubert varieties in Grassmannians. A formula for the minimum distance of these codes was conjectured in 2000 and after having been established in various special cases, it was proved in 2008 by Xiang. We give an alternative proof of this formula. Further, we propose a characterization of the minimum weight codewords of Schubert codes by introducing the notion of Schubert decomposable elements of certain exterior powers. It is shown that codewords corresponding to Schubert decomposable elements are of minimum weight and also that the converse is true in many cases. A lower bound, and in some cases, an exact formula, for the number of minimum weight codewords of Schubert codes is also given. From a geometric point of view, these results correspond to determining the maximum number of F q -rational points that can lie on a hyperplane section of a Schubert variety in a Grassmannian with its nondegenerate embedding in a projective subspace of the Plücker projective space, and also the number of hyperplanes for which the maximum is attained.

Abstract: Publication date: January 2018 Source:Finite Fields and Their Applications, Volume 49 Author(s): Hao Zhang We study the L-function of the Airy exponential sum associated to the polynomial f ( X ) = X d + λ X using Dwork's dual theory on cohomology level. We get explicit formulas for all the roots of the L-function in the ordinary case in terms of p-adic GKZ-hypergeometric functions. Our method can also be used to give explicit formulas for all the roots of the L-function of the exponential sum associated to f ( X ) = X d + λ d − 1 X d − 1 + ⋯ + λ 0 .

Abstract: Publication date: January 2018 Source:Finite Fields and Their Applications, Volume 49 Author(s): Yoshinori Hamahata We use periodic functions to introduce the generalized Dedekind sums in finite fields to describe the transformation formula for a family of certain functions.

Abstract: Publication date: November 2017 Source:Finite Fields and Their Applications, Volume 48 Author(s): Hidetaka Kitayama, Daisuke Shiomi In this paper, we give the necessary and sufficient condition that Fibonacci and Lucas polynomials are irreducible in F q [ T ] . As applications of our results, under the GRH, we prove that there are infinitely many irreducible Fibonacci and Lucas polynomials in F q [ T ] for some q.

Abstract: Publication date: November 2017 Source:Finite Fields and Their Applications, Volume 48 Author(s): Hyang-Sook Lee, Juhee Lee, Seongan Lim In the public key cryptography, we say that two public keys are duplicated if they share a private key in common. We point out that no duplicate public keys exist in the RSA public key scheme since there is a one-to-one correspondence between the set of problems and the set of solutions for integer factorization problem. Contrary to the integer factorization problem, there is no such one-to-one correspondence with Short Integer Solution (SIS)-type problems and this necessitates to study its effect on duplicate public keys of the schemes based on SIS. In this paper, we analyze the existence of duplicate public keys with four types of SIS problem: SIS, SIS with full rank solution set, basic Inhomogeneous SIS (ISIS), ISIS with the defining matrix A as a public parameter. As a result, we show that there is no provable way to exclude duplicate public keys of the schemes based on the basic SIS, basic ISIS, and SIS with a full rank solution set. However, we show that if A is given in the systematic form and the given set of solutions forms a matrix of rank ( m − n ) over Z q , then it guarantees duplication free public keys. We also prove that the schemes based on ISIS with the matrix A as a public parameter always guarantee duplication free public keys.

Abstract: Publication date: November 2017 Source:Finite Fields and Their Applications, Volume 48 Author(s): Hadi Kharaghani, Sara Sasani, Sho Suda For any positive integer m, the complete graph on 2 2 m ( 2 m + 2 ) vertices is decomposed into 2 m + 1 commuting strongly regular graphs, which give rise to a symmetric association scheme of class 2 m + 2 − 2 . Furthermore, the eigenmatrices of the symmetric association schemes are determined explicitly. As an application, the eigenmatrix of the commutative strongly regular decomposition obtained from the strongly regular graphs is derived.

Abstract: Publication date: November 2017 Source:Finite Fields and Their Applications, Volume 48 Author(s): Qiang Fu, Ruihu Li, Luobin Guo, Liangdong Lv The locality of locally repairable codes (LRCs) for a distributed storage system is the number of nodes that participate in the repair of failed nodes, which characterizes the repair cost. In this paper, we first determine the locality of MacDonald codes, then propose three constructions of LRCs with r = 1 , 2 and 3 . Based on these results, for 2 ≤ k ≤ 7 and n ≥ k + 2 , we give an optimal linear [ n , k , d ] code with small locality. The distance optimality of these linear codes can be judged by the codetable of M. Grassl for n < 2 ( 2 k − 1 ) and by the Griesmer bound for n ≥ 2 ( 2 k − 1 ) . Almost all the [ n , k , d ] codes ( 2 ≤ k ≤ 7 ) have locality r ≤ 3 except for the three codes, and most of the [ n , k , d ] code with n < 2 ( 2 k − 1 ) achieves the Cadambe–Mazumdar bound for LRCs.

Abstract: Publication date: November 2017 Source:Finite Fields and Their Applications, Volume 48 Author(s): Masaaki Homma, Seon Jeong Kim The numbers of F q -points of nonsingular hypersurfaces of a fixed degree in an odd-dimensional projective space are investigated, and an upper bound for them is given. Also we give the complete list of nonsingular hypersurfaces each of which realizes the upper bound. This is a natural generalization of our previous study of surfaces in projective 3-space.

Abstract: Publication date: Available online 11 September 2017 Source:Finite Fields and Their Applications Author(s): John B. Little This correction adds a necessary hypothesis in Proposition 2.4 of [1]. A counterexample to the previous version was pointed out to the author by Melda Görür and the author thanks her for bringing that to his attention. Theorem 3.2 and Corollary 3.3 are also reformulated so they do not refer to the new, more restricted Proposition 2.4.

Abstract: Publication date: November 2017 Source:Finite Fields and Their Applications, Volume 48 Author(s): Sandro Mattarei, Marco Pizzato A formula for the number of monic irreducible self-reciprocal polynomials, of a given degree over a finite field, was given by Carlitz in 1967. In 2011 Ahmadi showed that Carlitz's formula extends, essentially without change, to a count of irreducible polynomials arising through an arbitrary quadratic transformation. In the present paper we provide an explanation for this extension, and a simpler proof of Ahmadi's result, by a reduction to the known special case of self-reciprocal polynomials and a minor variation. We also prove further results on polynomials arising through a quadratic transformation, and through some special transformations of higher degree.

Abstract: Publication date: November 2017 Source:Finite Fields and Their Applications, Volume 48 Author(s): Bing He, Long Li, Ruiming Zhang In this paper we present a finite field analogue for one of the Appell series. We shall derive its transformations, reduction formulas as well as generating functions.

Abstract: Publication date: November 2017 Source:Finite Fields and Their Applications, Volume 48 Author(s): Masaaki Harada, Akihiro Munemasa A relationship between s-extremal singly even self-dual [ 24 k + 8 , 12 k + 4 , 4 k + 2 ] codes and extremal doubly even self-dual [ 24 k + 8 , 12 k + 4 , 4 k + 4 ] codes with covering radius meeting the Delsarte bound, is established. As an example of the relationship, s-extremal singly even self-dual [ 56 , 28 , 10 ] codes are constructed for the first time. In addition, we show that there is no extremal doubly even self-dual code of length 24 k + 8 with covering radius meeting the Delsarte bound for k ≥ 137 . Similarly, we show that there is no extremal doubly even self-dual code of length 24 k + 16 with covering radius meeting the Delsarte bound for k ≥ 148 . We also determine the covering radii of some extremal doubly even self-dual codes of length 80.

Abstract: Publication date: November 2017 Source:Finite Fields and Their Applications, Volume 48 Author(s): Nurdagül Anbar, Alp Bassa, Peter Beelen We give a complete characterization of all Galois subfields of the generalized Giulietti–Korchmáros function fields C n / F q 2 n for n ≥ 5 . Calculating the genera of the corresponding fixed fields, we find new additions to the list of known genera of maximal function fields.

Abstract: Publication date: November 2017 Source:Finite Fields and Their Applications, Volume 48 Author(s): Ravi Donepudi, Junxian Li, Alexandru Zaharescu Let F q be the finite field with q elements. Given an N-tuple Q ∈ F q N , we associate with it an affine plane curve C Q over F q . We consider the distribution of the quantity q − # C q , Q where # C q , Q denotes the number of F q -points of the affine curve C Q , for families of curves parameterized by Q. Exact formulae for first and second moments are obtained in several cases when Q varies over a subset of F q N . Families of Fermat type curves, Hasse–Davenport curves and Artin–Schreier curves are also considered and results are obtained when Q varies along a straight line.

Abstract: Publication date: November 2017 Source:Finite Fields and Their Applications, Volume 48 Author(s): Ziling Heng, Qin Yue Linear codes with a few weights are very important in coding theory and have attracted a lot of attention. In this paper, we present a construction of q-ary linear codes from trace and norm functions over finite fields. The weight distributions of the linear codes are determined in some cases based on Gauss sums. It is interesting that our construction can produce optimal or almost optimal codes. Furthermore, we show that our codes can be used to construct secret sharing schemes with interesting access structures and strongly regular graphs with new parameters.

Abstract: Publication date: November 2017 Source:Finite Fields and Their Applications, Volume 48 Author(s): Adel Alahmadi, S.P. Glasby, Cheryl E. Praeger Given a field F, a scalar λ ∈ F and a matrix A ∈ F n × n , the twisted centralizer code C F ( A , λ ) : = { B ∈ F n × n A B − λ B A = 0 } is a linear code of length n 2 over F. When A is cyclic and λ ≠ 0 we prove that dim C F ( A , λ ) = deg ( gcd ( c A ( t ) , λ n c A ( λ − 1 t ) ) ) where c A ( t ) denotes the characteristic polynomial of A. We also show how C F ( A , λ ) decomposes, and we estimate the probability that C F ( A , λ ) is nonzero when F is finite. Finally, we prove dim C F ( A , λ ) ⩽ n 2 / 2 for λ ∉ { 0 , 1 } and ‘almost all’ n × n matrices A over F.

Abstract: Publication date: November 2017 Source:Finite Fields and Their Applications, Volume 48 Author(s): Kaloyan Slavov Let X be an absolutely irreducible hypersurface of degree d in A n , defined over a finite field F q . The Lang–Weil bound gives an interval that contains # X ( F q ) . We exhibit an explicit interval, which does not contain # X ( F q ) , and which overlaps with the Lang–Weil interval. In particular, we sharpen the best known nontrivial lower bound for # X ( F q ) . The proof uses a combinatorial probabilistic technique.

Abstract: Publication date: November 2017 Source:Finite Fields and Their Applications, Volume 48 Author(s): David Covert, Doowon Koh, Youngjin Pi We study the k-resultant modulus set problem in the d-dimensional vector space F q d over the finite field F q with q elements. Given E ⊂ F q d and an integer k ≥ 2 , the k-resultant modulus set, denoted by Δ k ( E ) , is defined as Δ k ( E ) = { ‖ x 1 ± x 2 ± ⋯ ± x k ‖ ∈ F q : x j ∈ E , j = 1 , 2 , … , k } , where ‖ α ‖ = α 1 2 + ⋯ + α d 2 for α = ( α 1 , … , α d ) ∈ F q d . In this setting, the k-resultant modulus set problem is to determine the minimal cardinality of E ⊂ F q d such that Δ k ( E ) = F q or F q ⁎ . This problem is an extension of the Erdős–Falconer distance problem. In particular, we investigate the k-resultant modulus set problem with the restriction that the set E ⊂ F q d is contained in a specific algebraic variety. Energy estimates play a crucial role in our proof.

Abstract: Publication date: November 2017 Source:Finite Fields and Their Applications, Volume 48 Author(s): Nazar Arakelian Let F be a plane singular curve defined over a finite field F q . Via results of [11] and [1], the linear system of plane curves of a given degree passing through the singularities of F provides potentially good bounds for the number of points on a non-singular model of F . In this note, the case of a curve with two singularities such that the sum of their multiplicities is precisely the degree of the curve is investigated in more depth. In particular, such plane models are completely characterized, and for p > 3 , a curve of this type attaining one of the obtained bounds is presented.

Abstract: Publication date: November 2017 Source:Finite Fields and Their Applications, Volume 48 Author(s): Andrea Luigi Tironi Let X n be a hypersurface in P n + 1 with n ≥ 2 defined over a finite field. The main result of this note is the classification, up to projective equivalence, of hypersurfaces X n without a linear component when the number of their rational points achieves the Homma–Kim bound.

Abstract: Publication date: November 2017 Source:Finite Fields and Their Applications, Volume 48 Author(s): Maosheng Xiong, Haode Yan Let F ( x ) = x 2 2 k + 2 k + 1 be a power function over the finite field GF ( 2 n ) , where k is a positive integer and n = 4 k . It is known by a recent result of Bracken and Leander that F ( x ) is a highly nonlinear differentially 4-uniform function. Building upon this work, in this paper we determine the differential spectrum of F ( x ) .

Abstract: Publication date: November 2017 Source:Finite Fields and Their Applications, Volume 48 Author(s): Parinyawat Choosuwan, Somphong Jitman, Patanee Udomkavanich In this paper, the determinants of n × n matrices over commutative finite chain rings and over commutative finite principal ideal rings are studied. The number of n × n matrices over a commutative finite chain ring R of a fixed determinant a is determined for all a ∈ R and positive integers n. Using the fact that every commutative finite principal ideal ring is a product of commutative finite chain rings, the number of n × n matrices of a fixed determinant over a commutative finite principal ideal ring is shown to be multiplicative, and hence, it can be determined. These results generalize the case of matrices over the ring of integers modulo m.

Abstract: Publication date: November 2017 Source:Finite Fields and Their Applications, Volume 48 Author(s): Adam Tyler Felix, Pär Kurlberg We study number theoretic properties of the map x ↦ x x ( mod p ) , where x ∈ { 1 , 2 , … , p − 1 } , and improve on some recent upper bounds, due to Kurlberg, Luca, and Shparlinski, on the number of primes p < N for which the map only has the trivial fixed point x = 1 . A key technical result, possibly of independent interest, is the existence of subsets N q ⊂ { 2 , 3 , … , q − 1 } such that almost all k-tuples of distinct integers n 1 , n 2 , … , n k ∈ N q are multiplicatively independent (if k is not too large), and N q = q ⋅ ( 1 + o ( 1 ) ) as q → ∞ . For q a large prime, this is used to show that the number of solutions to a certain large and sparse system of F q -linear forms { L n } n = 2 q − 1 “behaves randomly” in the sense that { v ∈ F q d : L n ( v ) = 1 , n = 2 , 3 , … , q − 1 } ∼ q d ( 1 − 1 / q ) q ∼ q d / e . (Here d = π ( q − 1 ) and the coefficients of L n are given by the exponents in the prime power factorisation of n.)

Abstract: Publication date: November 2017 Source:Finite Fields and Their Applications, Volume 48 Author(s): Gerardo Vega A characterization of a class of optimal three-weight cyclic codes of dimension 3 over any finite field was recently presented in [10]. Almost immediately thereafter, several classes of cyclic codes with either optimal three weights or a few weights, were given in [3], showing at the same time that one of these classes can be constructed as a generalization of the sufficient numerical conditions of the characterization given in [10]. The main purpose of this work is to show that these numerical conditions that were found in [3], are also necessary. That is, the main contribution in this work is to present an extended characterization of a class of optimal three-weight cyclic codes, in terms of a weight distribution table. Now, note that the kind of characterizations that are given in terms of a weight distribution table are, in general, very difficult to establish. On the other hand, an interesting feature of the present work, in clear contrast with these two preceding works, is that we use some new and non-conventional methods in order to achieve our goals. In fact, through these non-conventional methods, we not only were able to extend the characterization in [10], but also present a less complex proof of such extended characterization, which avoids the use of some of the sophisticated – but at the same time complex – theorems, that are the key arguments of the proofs given in [10] and [3]. Furthermore, we also find the parameters for the dual code of any cyclic code in our extended characterization class. In fact, after the analysis of some examples, it seems that such dual codes always have the same parameters as the best known linear codes.

Abstract: Publication date: November 2017 Source:Finite Fields and Their Applications, Volume 48 Author(s): Maria Montanucci, Giovanni Zini The Deligne–Lusztig curves associated to the algebraic groups of type A 2 2 , B 2 2 , and G 2 2 are classical examples of maximal curves over finite fields. The Hermitian curve H q is maximal over F q 2 , for any prime power q, the Suzuki curve S q is maximal over F q 4 , for q = 2 2 h + 1 , h ≥ 1 , and the Ree curve R q is maximal over F q 6 , for q = 3 2 h + 1 , h ≥ 0 . In this paper we show that S 8 is not Galois covered by H 64 . We also prove an unpublished result due to Rains and Zieve stating that R 3 is not Galois covered by H 27 . Furthermore, we determine the spectrum of genera of Galois subcovers of H 27 , and we point out that some Galois subcovers of R 3 are not Galois subcovers of H 27 .

Abstract: Publication date: November 2017 Source:Finite Fields and Their Applications, Volume 48 Author(s): Shudi Yang, Xiangli Kong, Chunming Tang Recently, linear codes constructed from defining sets have been studied extensively. They may have excellent parameters if the defining set is chosen properly. Let m > 2 be a positive integer. For an odd prime p, let r = p m and Tr be the absolute trace function from F r onto F p . In this paper, we give a construction of linear codes by defining the code C D = { ( Tr ( a x ) ) x ∈ D : a ∈ F r } , where D = { x ∈ F r : Tr ( x ) = 1 , Tr ( x 2 ) = 0 } . Its complete weight enumerator and weight enumerator are determined explicitly by employing cyclotomic numbers and Gauss sums. However, we find that the code is optimal with respect to the Griesmer bound provided that m = 3 . In fact, it is MDS when m = 3 . Moreover, the codes presented have higher rate compared with other codes, which enables them to have essential applications in areas such as association schemes and secret sharing schemes.

Abstract: Publication date: November 2017 Source:Finite Fields and Their Applications, Volume 48 Author(s): Arnisa Rasri, Yotsanan Meemark In this work, we define and study the algebraic Cayley directed graph over a finite local ring. Its vertex set is the unit group of a finite extension of a finite local ring R and its adjacency condition is that the quotient is a monic primary polynomial. We investigate its connectedness and diameter bound, and we also show that our graph is an expander graph. In addition, if a local ring has nilpotency two, then we obtain a better view of our graph from the lifting of the graph over its residue field.

Abstract: Publication date: November 2017 Source:Finite Fields and Their Applications, Volume 48 Author(s): Ismail Aydogdu, Irfan Siap, Roger Ten-Valls Recently some special type of mixed alphabet codes that generalize the standard codes has attracted much attention. Besides Z 2 Z 4 -additive codes, Z 2 Z 2 [ u ] -linear codes are introduced as a new member of such families. In this paper, we are interested in a new family of such mixed alphabet codes, i.e., codes over Z 2 Z 2 [ u 3 ] where Z 2 [ u 3 ] = { 0 , 1 , u , 1 + u , u 2 , 1 + u 2 , u + u 2 , 1 + u + u 2 } is an 8-element ring with u 3 = 0 . We study and determine the algebraic structures of linear and cyclic codes defined over this family. First, we introduce Z 2 Z 2 [ u 3 ] -linear codes and give standard forms of generator and parity-check matrices and later we present generators of both cyclic codes and their duals over Z 2 Z 2 [ u 3 ] . Further, we present some examples of optimal binary codes which are obtained through Gray images of Z 2 Z 2 [ u 3 ] -cyclic codes.

Abstract: Publication date: November 2017 Source:Finite Fields and Their Applications, Volume 48 Author(s): Jingxue Ma, Gennian Ge Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, two conjectures on permutation polynomials proposed recently by Wu and Li [19] are settled. Moreover, a new class of permutation trinomials of the form x + γ Tr q n / q ( x k ) is also presented, which generalizes two examples of [10].

Abstract: Publication date: November 2017 Source:Finite Fields and Their Applications, Volume 48 Author(s): Robert W. Fitzgerald, Yasanthi Kottegoda We complete A. Klapper's work on the invariant of a one-term trace form over a finite field of odd characteristic. We apply this to computing the probability of a successful impersonation attack on an authentication code proposed by C. Ding et al. (2005).

Abstract: Publication date: November 2017 Source:Finite Fields and Their Applications, Volume 48 Author(s): José María Grau, Antonio M. Oller-Marcén In this paper we compute the sum of the k-th powers of all the elements of a finite commutative unital ring, thus generalizing known results for finite fields, the rings of integers modulo n or the ring of Gaussian integers modulo n. As an application, we focus on quotient rings of the form ( Z / n Z ) [ x ] / ( f ( x ) ) for a polynomial f ∈ Z [ x ] .

Abstract: Publication date: September 2017 Source:Finite Fields and Their Applications, Volume 47 Author(s): Everett W. Howe The defect of a curve over a finite field is the difference between the number of rational points on the curve and the Weil–Serre upper bound for the number of points on the curve. We present algorithms for constructing curves of genus 5, 6, and 7 with small defect. Our aim is to be able to produce, in a reasonable amount of time, curves that can be used to populate the online table of curves with many points found at manypoints.org .

Abstract: Publication date: September 2017 Source:Finite Fields and Their Applications, Volume 47 Author(s): Theodoulos Garefalakis, Giorgos Kapetanakis We consider the problem of enumerating polynomials over F q , that have certain coefficients prescribed to given values and permute certain substructures of F q . In particular, we are interested in the group of N-th roots of unity and in the submodules of F q . We employ the techniques of Konyagin and Pappalardi to obtain results that are similar to their results in Konyagin and Pappalardi (2006) [8]. As a consequence, we prove conditions that ensure the existence of low-degree permutation polynomials of the mentioned substructures of F q .

Abstract: Publication date: September 2017 Source:Finite Fields and Their Applications, Volume 47 Author(s): Kanat Abdukhalikov In this paper we study those bent functions which are linear on elements of spreads, their connections with ovals and line ovals, and we give descriptions of their dual bent functions. In particular, we give a geometric characterization of Niho bent functions and of their duals, we give explicit formula for the dual bent function and present direct connections with ovals and line ovals. We also show that bent functions which are linear on elements of inequivalent spreads can be EA-equivalent.

Abstract: Publication date: September 2017 Source:Finite Fields and Their Applications, Volume 47 Author(s): Nicholas M. Katz We find some new one-parameter families of exponential sums in every odd characteristic whose geometric and arithmetic monodromy groups are G 2 .

Abstract: Publication date: September 2017 Source:Finite Fields and Their Applications, Volume 47 Author(s): Satoshi Yoshiara It is shown that the Kasami function defined on F 2 n with n even is plateaued. This generalizes a result [3, Theorem 11], where the restriction ( n , 3 ) = 1 is assumed. The result is used to establish the CCZ-inequivalence of the Kasami function defined on F 2 n with n even to the other known monomial APN functions [4].

Abstract: Publication date: September 2017 Source:Finite Fields and Their Applications, Volume 47 Author(s): F.E. Brochero Martínez, Theodoulos Garefalakis, Lucas Reis, Eleni Tzanaki In this paper, we find a lower bound for the order of the group 〈 θ + α 〉 ⊂ F ‾ q ⁎ , where α ∈ F q , θ is a generic root of the polynomial F A , r ( X ) = b X q r + 1 − a X q r + d X − c ∈ F q [ X ] and a d − b c ≠ 0 .

Abstract: Publication date: September 2017 Source:Finite Fields and Their Applications, Volume 47 Author(s): A. Tomasi, A. Meneghetti, M. Sala We quantify precisely the distribution of the output of a binary random number generator (RNG) after conditioning with a binary linear code generator matrix by showing the connection between the Walsh spectrum of the resulting random variable and the weight distribution of the code. Previously known bounds on the performance of linear binary codes as entropy extractors can be derived by considering generator matrices as a selector of a subset of that spectrum. We also extend this framework to the case of non-binary codes.

Abstract: Publication date: September 2017 Source:Finite Fields and Their Applications, Volume 47 Author(s): Masamichi Kuroda, Shuhei Tsujie Almost perfect nonlinear (APN) functions on finite fields of characteristic two have been studied by many researchers. Such functions have useful properties and applications in cryptography, finite geometries and so on. However, APN functions on finite fields of odd characteristic do not satisfy desired properties. In this paper, we modify the definition of APN function in the case of odd characteristic, and study its properties.

Abstract: Publication date: September 2017 Source:Finite Fields and Their Applications, Volume 47 Author(s): Nian Li Permutation polynomials with a few terms attract researchers' interest in recent years due to their simple algebraic form and some additional extraordinary properties. In this paper, by analyzing the quadratic factors of a fifth-degree polynomial and a seventh-degree polynomial over the finite field F 3 2 k , two conjectures on permutation trinomials over F 3 2 k proposed recently by Li, Qu, Li and Fu are settled, where k is a positive integer.