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.

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.

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.

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.

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.

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 } .

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.

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.

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.

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.

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.

Abstract: Publication date: January 2018 Source:Finite Fields and Their Applications, Volume 49 Author(s): F.E. Brochero Martínez, Lucas Reis Let f ( x ) ∈ F q [ x ] be an irreducible polynomial of degree m and exponent e. For each positive integer n, such that ν p ( q − 1 ) ≥ ν p ( e ) + ν p ( n ) for all prime divisors p of n, we show a fast algorithm to determine the irreducible factors of f ( x n ) . Using this algorithm, we give the complete factorization of x n − 1 into irreducible factors in the case where n = d p t , p is an odd prime, q is a generator of the group Z p 2 ⁎ and either d = 2 m with m ≤ ν 2 ( q − 1 ) or d = r a , where r is a prime dividing q − 1 but not p − 1 .

Abstract: Publication date: January 2018 Source:Finite Fields and Their Applications, Volume 49 Author(s): Baokun Ding, Tao Zhang, Gennian Ge Recently, Yaakobi et al. introduced codes for b-symbol read channels, where the read operation is performed as a consecutive sequence of b > 2 symbols. In this paper, we establish a Singleton-type bound for b-symbol codes. Codes meeting the Singleton-type bound are called maximum distance separable (MDS) codes, and they are optimal in the sense they attain the maximal minimum b-distance. We introduce a construction method using projective geometry, and then construct several infinite families of linear MDS b-symbol codes over finite fields. The lengths of these codes have a large range. And in some sense, we completely determine the existence of linear MDS b-symbol codes over finite fields for certain parameters.

Abstract: Publication date: January 2018 Source:Finite Fields and Their Applications, Volume 49 Author(s): Dalma Görcsös, Gábor Horváth, Anett Mészáros Let PPol ( R ) denote the group of permutation polynomial functions over the finite, commutative, unital ring R under composition. We generalize numerous results about permutation polynomials over Z p n to local rings by treating them under a unified manner. In particular, we provide a natural wreath product decomposition of permutation polynomial functions over the maximal ideal M and over the finite field R / M . We characterize the group of permutation polynomial functions over M whenever the condition M R / M = { 0 } applies. Then we derive the size of PPol ( R ) , thereby generalizing the same size formulas for Z p n . Finally, we completely characterize when the group PPol ( R ) is solvable, nilpotent, or abelian.

Abstract: Publication date: January 2018 Source:Finite Fields and Their Applications, Volume 49 Author(s): Guangkui Xu, Xiwang Cao, Jingshui Ping Six classes of permutation pentanomials are constructed from fractional polynomials which permute the set of ( 2 m + 1 ) -th roots of unity. Based on an approach which is a generalization of the work of Zha, Hu and Fan, some permutation pentanomials and trinomials are also obtained from known permutations of the set of ( 2 m + 1 ) -th roots of unity.

Abstract: Publication date: January 2018 Source:Finite Fields and Their Applications, Volume 49 Author(s): Xiusheng Liu, Yun Fan, Hualu Liu In this paper, we generalize the linear complementary dual codes (LCD codes for short) to k-Galois LCD codes, and study them by a uniform method. A necessary and sufficient condition for linear codes to be k-Galois LCD codes is obtained, two classes of k-Galois LCD MDS codes are exhibited. Then, necessary and sufficient conditions for λ-constacyclic codes being k-Galois LCD codes are characterized. Some classes of k-Galois LCD λ-constacyclic MDS codes are constructed. Finally, we study Hermitian LCD λ-constacyclic codes, and present a class of Hermitian LCD λ-constacyclic MDS codes.

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): 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): 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 ] .