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Classes and equivalence of linear sets in PG(1, qⁿ)

JOURNAL OF COMBINATORIAL THEORY SERIES A 2018年 157卷 402-426页

作者： Csajbok, Bence Marino, Giuseppe Polverino, Olga Eotvos Lorand Univ ELTE MTA ELTE Geometr & Algebra Combinator Res Grp Dept Geometry Pazmany P Stny 1-C H-1117 Budapest Hungary Univ Campania Luigi Vanvitelli Dipartimento Matemat & Fis Viale Lincoln 5 I-81100 Caserta Italy

The equivalence problem of F-q-linear sets of rank n of PG(1, q(n)) is investigated, also in terms of the associated variety, projecting configurations,]Fq-linear blocking sets of Redei type and MRD-codes. We call an F-q-linear set L-U of rank n in PG(W,F-qn) = PG(1, q(n)) simple if for any n-dimensional F-q-subspace V of W, L-v is P Gamma L(2, q(n))-equivalent to L-U only when U and V lie on the same orbit of Gamma L(2, q(n)). We prove that U = {(x,Tr q(n)/q (x)): x is an element of F-qn defines a simple]Fq-linear set for each n. We provide examples of non-simple linear sets not of pseudoregulus type for n > 4 and we prove that all F-q-linear sets of rank 4 are simple in PG(1, q(4)). (C) 2018 Elsevier Inc. All rights reserved.

关键词： linearized polynomial Linear set Blocking set MRD-code

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On maximum additive Hermitian rank-metric codes

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JOURNAL OF ALGEBRAIC COMBINATORICS 2021年第1期54卷 151-171页

作者： Trombetti, Rocco Zullo, Ferdinando Univ Napoli Federico II Dipartimento Matemat & Applicaz Renato Caccioppod Via Cintia I-80126 Naples Italy Univ Campania Luigi Vanvitelli Dipartimento Matemat & Fis Viale Lincoln 5 I-81100 Caserta Italy

Inspired by the work of Zhou (Des Codes Cryptogr 88:841-850, 2020) based on the paper of Schmidt (J Algebraic Combin 42(2):635-670, 2015), we investigate the equivalence issue of maximum d-codes of Hermitian matrices. More precisely, in the space H-n(q(2)) of Hermitian matrices over F-q2 we have two possible equivalences: the classical one coming from the maps that preserve the rank in F-q2(nxn) , and the one that comes from restricting to those maps preserving both the rank and the space H-n(q(2)). We prove that when d < n and the codes considered are maximum additive d-codes and (n - d)-designs, these two equivalence relations coincide. As a consequence, we get that the idealisers of such codes are not distinguishers, unlike what usually happens for rank metric codes. Finally, we deal with the combinatorial properties of known maximum Hermitian codes and, by means of this investigation, we present a new family of maximum Hermitian 2-code, extending the construction presented by Longobardi et al. (Discrete Math 343(7):111871, 2020).

关键词： Hermitian matrix Rank metric code linearized polynomial

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Generalized twisted Gabidulin codes

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JOURNAL OF COMBINATORIAL THEORY SERIES A 2018年 159卷 79-106页

作者： Lunardon, Guglielmo Trombetti, Rocco Zhou, Yue Univ Napoli Federico II Dipartimento Math & Applicaz R Caccioppoli I-80126 Naples Italy Guangzhou Univ Sch Math & Informat Sci Guangzhou 510006 Guangdong Peoples R China Natl Univ Def Technol Dept Math Changsha 410073 Hunan Peoples R China

Let C be a set of m by n matrices over F-q such that the rank of A - B is at least d for all distinct A, B is an element of C. Suppose that m <= n. If #C = q(n.(m-d+1)), then C is a maximum rank distance (MRD for short) code. Until 2016, there were only two known constructions of MRD codes for arbitrary 1 < d < m - 1. One was found by Delsarte (1978) [8] and Gabidulin (1985) [10] independently, and it was later generalized by Kshevetskiy and Gabidulin (2005) [16]. We often call them (generalized) Gabidulin codes. Another family was recently obtained by Sheekey (2016) [22], and its elements are called twisted Gabidulin codes. In the same paper, Sheekey also proposed a generalization of the twisted Gabidulin codes. However the equivalence problem for it is not considered, whence it is not clear whether there exist new MRD codes in this generalization. We call the members of this putative larger family generalized twisted Gabidulin codes. In this paper, we first compute the Delsarte duals and adjoint codes of them, then we completely determine the equivalence between different generalized twisted Gabidulin codes. In particular, it can be proven that, up to equivalence, generalized Gabidulin codes and twisted Gabidulin codes are both proper subsets of this family (C) 2018 Elsevier Inc. All rights reserved.

关键词： Maximum rank distance code linearized polynomial Sernifield

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A Large Family of Maximum Scattered Linear Sets of PG(1, qⁿ) and Their Associated MRD Codes

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COMBINATORICA 2023年第4期43卷 681-716页

作者： Longobardi, G. Marino, Giuseppe Trombetti, Rocco Zhou, Yue Univ Napoli Federico II Dipartimento Matemat & Applicazioni Renato Caccio Via Vicinale Cupa Cintia I-80126 Naples Italy Natl Univ Def Technol Dept Math Changsha 410073 Peoples R China

Linear sets in projective spaces over finite fields were introduced by Lunardon (GeomDedic 75(3):245-261, 1999) and they play a central role in the study of blocking sets, semifields, rank-metric codes, etc. A linear set with the largest possible cardinality and rank is called maximum scattered. Despite two decades of study, there are only a limited number of maximum scattered linear sets of a line PG(1,qn). In this paper, we provide a large family of new maximum scattered linear sets over PG(1,qn)for any evenn=6 and oddq. In particular, the relevant family contains at least [GAPHIC]inequivalent members for givenq=prandn=2t>8, where p=char(Fq).Thisis a great improvement of previous results: for givenqandn>8, the number of inequivalent maximum scattered linear sets of PG(1,qn)in all classes known so far, issmaller thanq2f(n)/2, wherefdenotes Euler's totient function. Moreover, we showt hat there are a large number of new maximum rank-distance codes arising from the constructed linear sets

关键词： Linear set linearized polynomial Rank-metric code

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On Mathon's construction of maximal arcs in Desarguesian planes II

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JOURNAL OF COMBINATORIAL THEORY SERIES A 2004年第1期108卷 99-122页

作者： Fiedler, F Leung, KH Xiang, Q Univ Delaware Dept Math Sci Newark DE 19716 USA Natl Univ Singapore Dept Math Singapore 117548 Singapore

In a recent paper, Mathon (J. Combin. Theory (A) 97 (2002) 353) gives a new construction of maximal arcs which generalizes the construction of Denniston. In relation to this construction, Mathon asks the question of determining the largest degree of a non-Denniston maximal arc arising from his new construction. In this paper, we give a nearly complete answer to this problem. Specifically, we prove that when m greater than or equal to 5 and m not equal 9, the largest d of a non-Denniston maximal arc of degree 2(d) in PG(2, 2(m)) generated by a {p,1}-map is ([m/2] + 1). This confirms our conjecture in (Fiedler et al. (Adv. Geom. (2003) (Suppl.) S119)). For {p, q}-maps, we prove that if m greater than or equal to 7 and m not equal 9, then the largest d of a non-Denniston maximal arc of degree 2(d) in PG(2, 2(m)) generated by a {p, q}-map is either [m/2] + 1 or [m/2] + 2. (C) 2004 Elsevier Inc. All rights reserved.

关键词： arc linearized polynomial maximal arc Moore determinant quadratic form

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Triple-Cycle Permutations Over Finite Fields of Characteristic Two

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INTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE 2019年第2期30卷 275-292页

作者： Liu, Xianping Chen, Yuan Xu, Yunge Sun, Zhimin Hubei Univ Fac Math & Stat Hubei Key Lab Appl Math Wuhan 430062 Hubei Peoples R China Chinese Acad Sci State Key Lab Informat Secur Inst Informat Engn Beijing 100093 Peoples R China

Triple-cycle permutations over finite fields of characteristic two are studied, and some classes of triple-cycle permutations are proposed in this paper. In addition, new triple-cycle permutations can be constructed b... 详细信息

关键词： Triple-cycle permutation Dickson polynomial linearized polynomial switching construction

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Constructions and equivalence of Sidon spaces

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JOURNAL OF ALGEBRAIC COMBINATORICS 2023年第4期58卷 1299-1329页

作者： Castello, Chiara Polverino, Olga Santonastaso, Paolo Zullo, Ferdinando Univ Campania Luigi Vanvitelli Dipartimento Matemat & Fis Viale Lincoln 5 I-81100 Caserta Italy

Sidon spaces have been introduced by Bachoc et al. (in: Mathematical Proceedings of the Cambridge Philosophical Society, Cambridge University Press, 2017) as the q-analogue of Sidon sets. The interest on Sidon spaces has increased quickly, especially after the work of Roth et al. (IEEE Trans Inform Theory 64(6):4412-4422, 2017), in which they highlighted the correspondence between Sidon spaces and cyclic subspace codes. Up to now, the known constructions of Sidon Spaces may be divided in three families: the ones contained in the sum of two multiplicative cosets of a fixed subfield of F-qn, the ones contained in the sum of more than two multiplicative cosets of a fixed subfield of F-qn and finally the ones obtained as the kernel of subspace polynomials. In this paper, we will mainly focus on the first class of examples, for which we provide characterization results and we will show some new examples, arising also from some well-known combinatorial objects. Moreover, we will give a quite natural definition of equivalence among Sidon spaces, which relies on the notion of equivalence of cyclic subspace codes and we will discuss about the equivalence of the known examples.

关键词： Sidon space Cyclic subspace code linearized polynomial q-analog

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Scalar q-subresultants and Dickson matrices

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JOURNAL OF ALGEBRA 2020年 547卷 116-128页

作者： Csajbok, Bence Eotvos Lorand Univ MTA ELTE Geometr & Algebra Combinator Res Grp ELT Budapest Hungary Dept Geometry Pazmany P Stny 1-C H-1117 Budapest Hungary

Following the ideas of Ore and Li we study q-analogues of scalar subresultants and show how these results can be applied to determine the rank of an F-q-linear transformation f of F-qn. As an application we show how certain minors of the Dickson matrix D(f), associated with f, determine the rank of D(f) and hence the rank of f. (C) 2019 Elsevier Inc. All rights reserved.

关键词： Dickson matrix Subresultant linearized polynomial

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New results on permutation polynomials over finite fields

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INTERNATIONAL JOURNAL OF NUMBER THEORY 2015年第2期11卷 437-449页

作者： Qin, Xiaoer Qian, Guoyou Hong, Shaofang Sichuan Univ Math Coll Chengdu 610064 Peoples R China Yangtze Normal Univ Coll Math & Comp Sci Chongqing 408100 Peoples R China

In this paper, we get several new results on permutation polynomials over finite fields. First, by using the linear translator, we construct permutation polynomials of the forms L(x) + Sigma(k)(j)(= 1) gamma(j)h(j)(f(j) (x)) and x + Sigma(k)(j)(= 1) gamma(j)f(j) (x). These forms generalize the results obtained by Kyureghyan in 2011. Consequently, we characterize permutation polynomials of the form L(x) + Sigma(l)(i)(= 1) gamma iTrFqm/F-q (h(i)(x)), which extends a theorem of Charpin and Kyureghyan obtained in 2009.

关键词： Permutation polynomial linearized polynomial linear translator matrix

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A new semifield of order 2¹⁰

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DISCRETE MATHEMATICS 2010年第22期310卷 3108-3113页

作者： Marino, Giuseppe Trombetti, Rocco Univ Naples 2 Dipartimento Matemat I-81100 Caserta Italy Univ Naples Federico II Dipartimento Matemat I-80126 Naples Italy

In [G. Lunardon, Semifields and linear sets of PG(1, q(t)), Quad. Mat., Dept. Math., Seconda Univ. Napoli, Caserta (in press)], G. Lunardon has exhibited a construction method yielding a theoretical family of semifields of order q(2n), n > 1 and n odd, with left nucleus F-qn, middle and right nuclei both F-q2 and center F-q. When n = 3 this method gives an alternative construction of a family of semifields described in [N.L.] Johnson, G. Marino, O. Polverino, R. Trombetti, On a generalization of cyclic semifields, J. Algebraic Combin. 26 (2009), 1-34], which generalizes the family of cyclic semifields obtained by Jha and Johnson in [V. Jha, N.L. Johnson, Translation planes of large dimension admitting non-solvable groups, J. Geom. 45 (1992), 87-104]. For n > 3, no example of a semifield belonging to this family is known. In this paper we first prove that, when n > 3, any semifield belonging to the family introduced in the second work cited above is not isotopic to any semifield of the family constructed in the former. Then we construct, with the aid of a computer, a semifield of order 2(10) belonging to the family introduced by Lunardon, which turns out to be non-isotopic to any other known semifield. (C) 2009 Elsevier B.V. All rights reserved.

关键词： Semifield linearized polynomial Linear set

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