Authors:Daniele Bartoli; Massimo Giulietti Pages: 1 - 16 Abstract: Publication date: May 2018 Source:Finite Fields and Their Applications, Volume 51 Author(s): Daniele Bartoli, Massimo Giulietti In this note we prove a conjecture by Li, Qu, Li, and Fu on permutation trinomials over F 3 2 k . In addition, new examples and generalizations of some families of permutation polynomials of F 3 k and F 5 k are given. We also study permutation quadrinomials of type A x q ( q − 1 ) + 1 + B x 2 ( q − 1 ) + 1 + C x q + x . Our method is based on the investigation of an algebraic curve associated with a fractional polynomial over a finite field.

Authors:Nikolay Yankov; Milena Ivanova; Moon Ho Lee Pages: 17 - 30 Abstract: Publication date: May 2018 Source:Finite Fields and Their Applications, Volume 51 Author(s): Nikolay Yankov, Milena Ivanova, Moon Ho Lee This paper studies and classifies optimal binary self-dual codes having an automorphism of order 7 with 9 cycles. This classification is done by applying a method for constructing binary self-dual codes with an automorphism of odd prime order p. There are exactly 69781 inequivalent binary self-dual [ 64 , 32 , 12 ] codes with an automorphism of type 7 − ( 9 , 1 ) . As for binary [ 66 , 33 , 12 ] self-dual codes with an automorphism of type 7 − ( 9 , 3 ) there are 1652432 such codes. We also construct more than 4 million new optimal codes of length 68 among which are the first known examples of the very elusive s-extremal self-dual codes. We prove the nonexistence of [ 70 , 35 , 14 ] codes with an automorphism of type 7 − ( 9 , 7 ) . Most of the constructed codes for all lengths have weight enumerators for which the existence was not known before.

Authors:Lisha Li; Shi Wang; Chaoyun Li; Xiangyong Zeng Pages: 31 - 61 Abstract: Publication date: May 2018 Source:Finite Fields and Their Applications, Volume 51 Author(s): Lisha Li, Shi Wang, Chaoyun Li, Xiangyong Zeng In this paper, we propose several classes of permutation polynomials with the form ( x p m − x + δ ) s 1 + ( x p m − x + δ ) s 2 + x over finite fields. The permutation behavior of the proposed polynomials is investigated by the AGW criterion and determination of the number of solutions to certain equations over finite fields.

Authors:Mostafa Alaoui Abdallaoui; Mohammed El Badry; Abdelfattah Haily Pages: 62 - 70 Abstract: Publication date: May 2018 Source:Finite Fields and Their Applications, Volume 51 Author(s): Mostafa Alaoui Abdallaoui, Mohammed El Badry, Abdelfattah Haily We consider group algebra of a metacyclic group of order p n q over a field of characteristic p, where p and q are distinct prime numbers. We determine the terms of the primary decomposition of this algebra. The central primitive idempotents are also calculated. In the end we give the characteristics (dimension and minimal weight) of codes generated by these idempotents.

Authors:Joe Gildea; Abidin Kaya; Rhian Taylor; Bahattin Yildiz Pages: 71 - 92 Abstract: Publication date: May 2018 Source:Finite Fields and Their Applications, Volume 51 Author(s): Joe Gildea, Abidin Kaya, Rhian Taylor, Bahattin Yildiz In this work, we establish a strong connection between group rings and self-dual codes. We prove that a group ring element corresponds to a self-dual code if and only if it is a unitary unit. We also show that the double-circulant and four-circulant constructions come from cyclic and dihedral groups, respectively. Using groups of order 8 and 16 we find many new construction methods, in addition to the well-known methods, for self-dual codes. We establish the relevance of these new constructions by finding many extremal binary self-dual codes using them, which we list in several tables. In particular, we construct 10 new extremal binary self-dual codes of length 68.

Authors:Cathy Swaenepoel Pages: 93 - 129 Abstract: Publication date: May 2018 Source:Finite Fields and Their Applications, Volume 51 Author(s): Cathy Swaenepoel Let p be a prime number and let q = p r . If C and D are large subsets of F q ⁎ we study the trace of products cd with c ∈ C and d ∈ D and show that it is well distributed in F p . We give an optimal condition (up to an absolute constant factor) on the size of the subsets C and D to ensure that the trace of products cd takes any given value in F p . We also give a condition (optimal up to an absolute constant factor in most cases) on the size of the subsets C and D to ensure that the trace of cd meets the set of k-th powers for k ≥ 1 , respectively the set of generators. Our method will enable us to take sets C and D whose size is substantially below q . Character sums and Gaussian sums over F p and F q will play an important role in the proofs. Some estimates lead to interesting combinatorial questions in finite fields.

Authors:Peter Beelen; Mrinmoy Datta Pages: 130 - 145 Abstract: Publication date: May 2018 Source:Finite Fields and Their Applications, Volume 51 Author(s): Peter Beelen, Mrinmoy Datta Let F be any field and A 1 , … , A m be finite subsets of F . We determine the maximum number of common zeroes a linearly independent family of r polynomials of degree at most d of F [ x 1 , … , x m ] can have in A 1 × … × A m . In the case when F is a finite field, our results resolve the problem of determining the generalized Hamming weights of affine Cartesian codes. This is a generalization of the work of Heijnen and Pellikaan where these were determined for the generalized Reed–Muller codes. Finally, we determine the duals of affine Cartesian codes and compute their generalized Hamming weights as well.

Authors:Jingjie Lv; Jian Gao Pages: 146 - 167 Abstract: Publication date: May 2018 Source:Finite Fields and Their Applications, Volume 51 Author(s): Jingjie Lv, Jian Gao We provide a concatenated structure for 2-dimension λ-quasi-twisted (λ-Q2DT) codes over F q , which is a generalization of the concatenated structure for 2-dimension quasi-cyclic (Q2DC) codes. Further, by trace representations of λ-constacyclic codes, λ-quasi-twisted codes, 2-dimension λ-constacyclic codes and λ-Q2DT codes, we obtain a general lower bound of the minimum distance for λ-Q2DT codes of length mℓk, where m , ℓ and k are relatively prime to q.

Authors:José Manuel Rodríguez Caballero Pages: 183 - 190 Abstract: Publication date: May 2018 Source:Finite Fields and Their Applications, Volume 51 Author(s): José Manuel Rodríguez Caballero A q-analog P n ( q ) of the sum of divisors of n was introduced by C. Kassel and C. Reutenauer in a combinatorial setting and by T. Hausel, E. Letellier, F. Rodriguez-Villegas in a Hodge-theoretic setting. We study the reduction modulo 3 of the polynomial P n ( q ) with respect to the ideal ( q 2 + q + 1 ) F 3 [ q ] .

Authors:A.S. Sivatski Pages: 191 - 203 Abstract: Publication date: May 2018 Source:Finite Fields and Their Applications, Volume 51 Author(s): A.S. Sivatski Let F be a finite field of characteristic distinct from 2, f and g quadratic forms over F, dim f = dim g = n . A particular case of Chevalley's theorem claims that if n ≥ 5 , then f and g have a common zero. We give an algorithm, which establishes whether f and g have a common zero in the case n ≤ 4 . The most interesting case is n = 4 . In particular, we show that if n = 4 and det ( f + t g ) is a squarefree polynomial of degree different from 2, then f and g have a common zero. We investigate the orbits of pairs of 4-dimensional forms ( f , g ) under the action of the group GL 4 ( F ) , provided f and g do not have a common zero. In particular, it turns out that for any polynomial p ( t ) of degree at most 4 up to the above action there exist at most two pairs ( f , g ) such that det ( f + t g ) = p ( t ) , and the forms f, g do not have a common zero. The proofs heavily use Brumer's theorem and the Hasse–Minkowski theorem.

Authors:Weiqiong Wang; Jennifer Nguyen Pages: 204 - 217 Abstract: Publication date: May 2018 Source:Finite Fields and Their Applications, Volume 51 Author(s): Weiqiong Wang, Jennifer Nguyen The subset sum problem is an important theoretical problem with many applications, such as in coding theory, cryptography, graph theory and other fields. One of the many aspects of this problem is to answer the solvability of the k-subset sum problem. It has been proven to be NP-hard in general. However, if the evaluation set has some special algebraic structure, it is possible to obtain some good conclusions. Zhu, Wan and Keti proposed partial results of this problem over two special kinds of evaluation sets. We generalize their conclusions in this paper, and propose asymptotical results of the solvability of the k-subset sum problem by using estimates of additive character sums over the evaluation set, together with the Brun sieve and the new sieve proposed by Li and Wan. We also apply the former two examples as application of our results.

Authors:Lucas Reis Pages: 218 - 237 Abstract: Publication date: May 2018 Source:Finite Fields and Their Applications, Volume 51 Author(s): Lucas Reis Let F q be the finite field with q elements, where q is a power of a prime p. Recently, a particular action of the group GL 2 ( F q ) on irreducible polynomials in F q [ x ] has been introduced and many questions concerning the invariant polynomials have been discussed. In this paper, we give a natural extension of this action on the polynomial ring F q [ x 1 , … , x n ] and study the algebraic properties of the invariant elements.

Authors:Anuradha Sharma; Varsha Chauhan; Harshdeep Singh Pages: 270 - 297 Abstract: Publication date: May 2018 Source:Finite Fields and Their Applications, Volume 51 Author(s): Anuradha Sharma, Varsha Chauhan, Harshdeep Singh Let F q denote the finite field of order q, let m 1 , m 2 , ⋯ , m ℓ be positive integers satisfying gcd ( m i , q ) = 1 for 1 ≤ i ≤ ℓ , and let n = m 1 + m 2 + ⋯ + m ℓ . Let Λ = ( λ 1 , λ 2 , ⋯ , λ ℓ ) be fixed, where λ 1 , λ 2 , ⋯ , λ ℓ are non-zero elements of F q . In this paper, we study the algebraic structure of Λ-multi-twisted codes of length n over F q and their dual codes with respect to the standard inner product on F q n . We provide necessary and sufficient conditions for the existence of a self-dual Λ-multi-twisted code of length n over F q , and obtain enumeration formulae for all self-dual and self-orthogonal Λ-multi-twisted codes of length n over F q . We also derive some sufficient conditions under which a Λ-multi-twisted code is linear with complementary dual (LCD). We determine the parity-check polynomial of all Λ-multi-twisted codes of length n over F q and obtain a BCH type bound on their minimum Hamming distances. We also determine generating sets of dual codes of some Λ-multi-twisted codes of length n over F q from the generating sets of the codes. Besides this, we provide a trace description for all Λ-multi-twisted codes of length n over F q by viewing these codes as direct sums of certain concatenated codes, which leads to a method to construct these codes. We also obtain a lower bound on their minimum Hamming distances using their multilevel concatenated structure.

Authors:Jun Zhang; Keqin Feng Pages: 338 - 355 Abstract: Publication date: March 2018 Source:Finite Fields and Their Applications, Volume 50 Author(s): Jun Zhang, Keqin Feng Relative generalized Hamming weights (RGHWs) of a linear code with respect to a linear subcode determine the security of the linear ramp secret sharing scheme based on the linear codes. They can be used to express the information leakage of the secret when some keepers of shares are corrupted. Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems. In this paper, we investigate the RGHWs of cyclic codes of two nonzeros with respect to its irreducible cyclic subcodes. We give two formulae for RGHWs of the cyclic codes. As applications of the formulae, explicit examples are computed. Moreover, RGHWs of cyclic codes in the examples are very large, comparing with the generalized Plotkin bound of RGHWs. So it guarantees very high security for the secret sharing scheme based on the dual codes.

Authors:Lin Tan; Hong Xu; Wen-Feng Qi Pages: 356 - 365 Abstract: Publication date: March 2018 Source:Finite Fields and Their Applications, Volume 50 Author(s): Lin Tan, Hong Xu, Wen-Feng Qi Modified de Bruijn sequences are created by removing a single zero from the longest run of zeros of de Bruijn sequences. There are few theoretical results on the minimal polynomial and linear complexity of modified de Bruijn sequences. Some preliminary results are presented in this paper. It shows that for the minimal polynomial of a modified de Bruijn sequence of order n there exists at least one irreducible factor of degree n. An equivalent condition on which a polynomial is the minimal polynomial of some modified de Bruijn sequence is derived, using the tool of rational fraction representation of periodic sequences. Based on the equivalent condition, the impossible linear complexity of modified de Bruijn sequences can be discussed.

Authors:Alexander Borisov Pages: 366 - 371 Abstract: Publication date: March 2018 Source:Finite Fields and Their Applications, Volume 50 Author(s): Alexander Borisov We construct some examples of polynomial maps over finite fields that admit subvarieties with a peculiar property: every geometric point is mapped to a fixed point by some iteration of the map, while the whole subvariety is not. Several related open questions are stated and discussed.

Authors:Dieter Jungnickel; Vladimir D. Tonchev Pages: 372 - 381 Abstract: Publication date: March 2018 Source:Finite Fields and Their Applications, Volume 50 Author(s): Dieter Jungnickel, Vladimir D. Tonchev By a classical result of Bonisoli, the equidistant linear codes over GF ( q ) are, up to monomial equivalence, just the replications of some q-ary simplex code, possibly with added 0-coordinates. We prove an analogous result for the antipodal two-weight linear codes over GF ( q ) (that is, one of the two weights is the length of the code): up to monomial equivalence, any such code is ether a replication of a first order generalized Reed–Muller code, or a replication of a 3-dimensional projective code associated with some non-trivial maximal arc in the classical projective plane PG ( 2 , 2 s ) , s ≥ 1 . An equivalent geometric formulation of this result reads as follows: a multiset S of points spanning Π = PG ( m , q ) , m ≥ 2 , for which every hyperplane intersecting S does so in a constant number of points (counted with multiplicity), is either a multiple of a maximal arc in Π (for m = 2 ) or a multiple of the complement of some hyperplane of Π.

Authors:Cícero Carvalho; Victor G.L. Neumann Pages: 382 - 390 Abstract: Publication date: March 2018 Source:Finite Fields and Their Applications, Volume 50 Author(s): Cícero Carvalho, Victor G.L. Neumann In this paper we present several values for the next-to-minimal weights of projective Reed–Muller codes. We work over F q with q ≥ 3 since in [3] we have determined the complete values for the next-to-minimal weights of binary projective Reed–Muller codes. As in [3] here we also find examples of codewords with next-to-minimal weight whose set of zeros is not in a hyperplane arrangement.

Authors:Simone Costa; Tao Feng; Xiaomiao Wang Pages: 391 - 405 Abstract: Publication date: March 2018 Source:Finite Fields and Their Applications, Volume 50 Author(s): Simone Costa, Tao Feng, Xiaomiao Wang Strong difference families will be used to get: seven 2-designs whose parameter sets were previously in doubt; a strong indication about the existence of 2- ( p q , p , 1 ) designs with p ∈ { 13 , 17 } and q a prime power; a better asymptotic bound on prime powers q for which there exists an ( F p × F q , F p × { 0 } , k , λ ) relative difference family with p a prime power and k ∈ { p , p + 1 } .

Authors:Wei Zhao; Xilin Tang; Ze Gu Pages: 1 - 16 Abstract: Publication date: March 2018 Source:Finite Fields and Their Applications, Volume 50 Author(s): Wei Zhao, Xilin Tang, Ze Gu Let F p m be a finite field with cardinality p m and R = F p m + u F p m with u 2 = 0 . We aim to determine all α + u β -constacyclic codes of length n p s over R, where α , β ∈ F p m ⁎ , n , s ∈ N + and gcd ( n , p ) = 1 . Let α 0 ∈ F p m ⁎ and α 0 p s = α . The residue ring R [ x ] / 〈 x n p s − α − u β 〉 is a chain ring with the maximal ideal 〈 x n − α 0 〉 in the case that x n − α 0 is irreducible in F p m [ x ] . If x n − α 0 is reducible in F p m [ x ] , we give the explicit expressions of the ideals of R [ x ] / 〈 x n p s − α − u β 〉 . Besides, the number of codewords and the dual code of every α + u β -constacyclic code are provided.

Authors:Martino Borello; Javier de la Cruz Pages: 17 - 34 Abstract: Publication date: March 2018 Source:Finite Fields and Their Applications, Volume 50 Author(s): Martino Borello, Javier de la Cruz The existence of an extremal self-dual binary linear code of length 120 is a long-standing open problem. We continue the investigation of its automorphism group, proving that automorphisms of order 30 and 57 cannot occur. Supposing the involutions acting fixed point freely, we show that also automorphisms of order 8 cannot occur and the automorphism group is of order at most 120, with further restrictions. Finally, we present some necessary conditions for the existence of the code, based on shadow and design theory.

Authors:Domingo Gómez-Pérez; Alina Ostafe; Min Sha Pages: 35 - 65 Abstract: Publication date: March 2018 Source:Finite Fields and Their Applications, Volume 50 Author(s): Domingo Gómez-Pérez, Alina Ostafe, Min Sha Motivated by a question of van der Poorten about the existence of an infinite chain of prime numbers (with respect to some base), in this paper we advance the study of sequences of consecutive polynomials whose coefficients are chosen consecutively from a sequence in a finite field of odd prime characteristic. We study the arithmetic of such sequences, including bounds for the largest degree of irreducible factors, the number of irreducible factors, as well as for the number of such sequences of fixed length in which all the polynomials are irreducible.

Authors:José Gómez-Torrecillas; F.J. Lobillo; Gabriel Navarro; Alessando Neri Pages: 84 - 112 Abstract: Publication date: March 2018 Source:Finite Fields and Their Applications, Volume 50 Author(s): José Gómez-Torrecillas, F.J. Lobillo, Gabriel Navarro, Alessando Neri The use of skew polynomial rings allows to endow linear codes with cyclic structures which are not cyclic in the classical (commutative) sense. Whenever these skew cyclic structures are carefully chosen, some control over the Hamming distance is gained, and it is possible to design efficient decoding algorithms. In this paper, we give a version of the Hartmann–Tzeng bound that works for a wide class of skew cyclic codes. We also provide a practical method for constructing them with designed distance. For skew BCH codes, which are covered by our constructions, we discuss decoding algorithms. Detailed examples illustrate both the theory as the constructive methods it supports.

Authors:Tathagata Basak Pages: 113 - 121 Abstract: Publication date: March 2018 Source:Finite Fields and Their Applications, Volume 50 Author(s): Tathagata Basak We show that the octonions can be defined as the R -algebra with basis { e x : x ∈ F 8 } and multiplication given by e x e y = ( − 1 ) φ ( x , y ) e x + y , where φ ( x , y ) = tr ( y x 6 ) . While it is well known that the octonions can be described as a twisted group algebra, our purpose is to point out that this is a useful description. We show how the basic properties of the octonions follow easily from our definition. We give a uniform description of the sixteen orders of integral octonions containing the Gravesian integers, and a computation-free proof of their existence.

Authors:Zohreh Rajabi; Kazem Khashyarmanesh Pages: 122 - 137 Abstract: Publication date: March 2018 Source:Finite Fields and Their Applications, Volume 50 Author(s): Zohreh Rajabi, Kazem Khashyarmanesh Let p be an odd prime. The main purpose of this paper is to establish and exploit the algebraic structure of some repeated-root two dimensional constacyclic codes of length 2 p s . 2 k over the finite field F p m . Moreover, we study all self-dual repeated-root two-dimensional ( λ , 1 ) -constacyclic codes of length 2 p s . 2 or 2 p s . 2 2 over F p m for λ ∈ { − 1 , 1 } .

Authors:Lin Sok; Minjia Shi; Patrick Solé Pages: 138 - 153 Abstract: Publication date: March 2018 Source:Finite Fields and Their Applications, Volume 50 Author(s): Lin Sok, Minjia Shi, Patrick Solé In this paper 1 , we prove existence of optimal complementary dual codes (LCD codes) over large finite fields. We also give methods to generate orthogonal matrices over finite fields and then apply them to construct LCD codes. Construction methods include random sampling in the orthogonal group, code extension, matrix product codes and projection over a self-dual basis.

Authors:Ziran Tu; Xiangyong Zeng; Chunlei Li; Tor Helleseth Pages: 178 - 195 Abstract: Publication date: March 2018 Source:Finite Fields and Their Applications, Volume 50 Author(s): Ziran Tu, Xiangyong Zeng, Chunlei Li, Tor Helleseth In this paper, we characterize the coefficients of f ( x ) = x + a 1 x q ( q − 1 ) + 1 + a 2 x 2 ( q − 1 ) + 1 in F q 2 [ x ] for even q that lead f ( x ) to be a permutation of F q 2 . We transform the problem into studying some low-degree equations with variable in the unit circle, which are intensively investigated with some parameterization techniques. From the numerical results, the coefficients that lead f ( x ) to be a permutation appear to be completely characterized in this paper. It is also demonstrated that some permutations proposed in this paper are quasi-multiplicative (QM) inequivalent to the previously known permutation trinomials.

Authors:Rohit Gupta; R.K. Sharma Pages: 196 - 208 Abstract: Publication date: March 2018 Source:Finite Fields and Their Applications, Volume 50 Author(s): Rohit Gupta, R.K. Sharma Let F q denote the finite field of order q. In this paper, some new classes of permutation polynomials of the form ( x p m − x + δ ) s + x over F p 2 m are obtained by determining the number of solutions of certain equations.

Authors:Koji Momihara; Qing Xiang Pages: 222 - 250 Abstract: Publication date: March 2018 Source:Finite Fields and Their Applications, Volume 50 Author(s): Koji Momihara, Qing Xiang In this paper, we give a new lifting construction of “hyperbolic” type of strongly regular Cayley graphs. Also we give new constructions of strongly regular Cayley graphs over the additive groups of finite fields based on partitions of subdifference sets of the Singer difference sets. Our results unify some recent constructions of strongly regular Cayley graphs related to m-ovoids and i-tight sets in finite geometry. Furthermore, some of the strongly regular Cayley graphs obtained in this paper are new or nonisomorphic to known strongly regular graphs with the same parameters.

Authors:Shudi Yang; Chuangqiang Hu Pages: 251 - 271 Abstract: Publication date: March 2018 Source:Finite Fields and Their Applications, Volume 50 Author(s): Shudi Yang, Chuangqiang Hu In this paper, by employing some results on Kummer extensions, we give an arithmetic characterization of pure Weierstrass gaps at many totally ramified places on a quotient of the Hermitian curve, including the well-studied Hermitian curve as a special case. The cardinality of these pure gaps is explicitly investigated. In particular, the numbers of gaps and pure gaps at a pair of distinct places are determined precisely, which can be regarded as an extension of the previous work by Matthews (2001) considered Hermitian curves. Additionally, some concrete examples are provided to illustrate our results.

Authors:Hua Huang; Shanmeng Han; Wei Cao Pages: 272 - 278 Abstract: Publication date: March 2018 Source:Finite Fields and Their Applications, Volume 50 Author(s): Hua Huang, Shanmeng Han, Wei Cao A normal basis of F q n over F q is a basis of the form { α , α q , … , α q n − 1 } . An irreducible polynomial in F q [ x ] is called an N-polynomial if its roots are linearly independent over F q . Let p be the characteristic of F q . Pelis et al. showed that every monic irreducible polynomial with degree n and nonzero trace is an N-polynomial provided that n is either a power of p or a prime different from p and q is a primitive root modulo n. Chang et al. proved that the converse is also true. By comparing the number of N-polynomials with that of irreducible polynomials with nonzero traces, we present an alternative treatment to this problem and show that all the results mentioned above can be easily deduced from our main theorem.

Authors:Lucas Reis Pages: 279 - 292 Abstract: Publication date: March 2018 Source:Finite Fields and Their Applications, Volume 50 Author(s): Lucas Reis Let q be a prime power and F q n be the finite field with q n elements, where n > 1 . We introduce the class of the linearized polynomials L ( X ) over F q n such that L ( t ) ( X ) : = L ∘ L ∘ ⋯ ∘ L ︸ t times ( X ) ≡ 0 ( mod X q n − X ) for some t ≥ 2 , called nilpotent linearized polynomials (NLP's). We discuss the existence and construction of NLP's and, as an application, we show how to obtain permutations of F q n from these polynomials. For some of those permutations, we can explicitly give the compositional inverse map and the cycle decomposition. This paper also contains a method for constructing involutions over binary fields with no fixed points, which are useful in block ciphers.

Authors:Ziran Tu; Xiangyong Zeng; Tor Helleseth Pages: 304 - 318 Abstract: Publication date: March 2018 Source:Finite Fields and Their Applications, Volume 50 Author(s): Ziran Tu, Xiangyong Zeng, Tor Helleseth In this paper, we propose a class of permutation polynomials over the finite field F 2 2 m for odd m. These permutations are generally quadrinomials, and some permutation trinomials can also be obtained.

Authors:Ioulia N. Baoulina Pages: 319 - 337 Abstract: Publication date: March 2018 Source:Finite Fields and Their Applications, Volume 50 Author(s): Ioulia N. Baoulina We explicitly determine the values of reduced cyclotomic periods of order 2 m , m ≥ 4 , for finite fields of characteristic p ≡ 3 or 5 ( mod 8 ) . These evaluations are applied to obtain explicit factorizations of the corresponding reduced period polynomials. As another application, the weight distributions of certain irreducible cyclic codes are described.

Authors:Sudhir R. Ghorpade; Prasant Singh Pages: 1 - 28 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.

Authors:Hao Zhang Pages: 29 - 48 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 .

Authors:Yoshinori Hamahata Pages: 49 - 61 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.

Authors:Hiroaki Taniguchi Pages: 62 - 79 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.

Authors:Masaaki Homma; Seon Jeong Kim Pages: 80 - 93 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 .

Authors:Mireille Fouquet; Josep M. Miret; Javier Valera Pages: 108 - 125 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.

Authors:Daniele Bartoli; Giovanni Zini Pages: 126 - 131 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 .

Authors:Roswitha Hofer Pages: 23 - 29 Abstract: Publication date: March 2018 Source:Finite Fields and Their Applications, Volume 50 Author(s): Roswitha Hofer In this paper we investigate the distribution properties of hybrid sequences which are made by combining Halton sequences in the ring of polynomials and digital Kronecker sequences. We give a full criterion for the uniform distribution and prove results on the discrepancy of such hybrid sequences.

Authors:Kamil Otal; Ferruh Abstract: Publication date: March 2018 Source:Finite Fields and Their Applications, Volume 50 Author(s): Kamil Otal, Ferruh Özbudak In this paper, a construction of maximum rank distance (MRD) codes as a generalization of generalized Gabidulin codes is given. The family of the resulting codes is not covered properly by additive generalized twisted Gabidulin codes, and does not cover all twisted Gabidulin codes. When the basis field has more than two elements, this family includes also non-affine MRD codes, and such codes exist for all parameters. Therefore, these codes are the first non-additive MRD codes for most of the parameters.

Authors:I.F. 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.