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Revisiting products of the form X times a linearized polynomial L(X)

DESIGNS CODES AND CRYPTOGRAPHY 2025年第1期93卷 39-50页

作者： Beierle, Christof Ruhr Univ Bochum Fac Comp Sci Bochum Germany

For a q-polynomial L over a finite field F-q(n), we characterize the differential spectrum of the function f(L):F-q(n)-> F-q(n),xbar right arrowx center dot L(x) and show that, for n <= 5, it is completely determined by the image of the rational function r(L):F-q(n)*-> F-q(n),xbar right arrowL(x)/x. This result follows from the classification of the pairs (L, M) of q-polynomials in F-q(n)[X], n <= 5, for which r(L) and r(M) have the same image, obtained in Csajbok et al. (Ars Math Contemp 16(2):585-608, 2019). For the case of n>5, we pose an open question on the dimensions of the kernels of xbar right arrowL(x)-ax for a is an element of Fqn. We further present a link between functions f(L) of differential uniformity bounded above by q and scattered q-polynomials and show that, for odd values of q, we can construct CCZ-inequivalent functions f(M) with bounded differential uniformity from a given function f(L) fulfilling certain properties.

关键词： linearized polynomial Differential spectrum Differential uniformity Linear set Scattered polynomial

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From r-linearized polynomial equations to r^m-linearized polynomial equations

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FINITE FIELDS AND THEIR APPLICATIONS 2016年第Jan.期37卷 14-27页

作者： Fernando, Neranga Hou, Xiang-dong Northeastern Univ Dept Math Boston MA 02115 USA Univ S Florida Dept Math & Stat Tampa FL 33620 USA

Let r be a prime power and q = r(m). For 0 <= i <= m-1, let f(1) is an element of F-r [X] be q-linearized and a(i) is an element of F-q. Assume that z is an element of (F) over bar (r) satisfies the equation Sigma(m-1)(i=0) a(i)f(i)(z)(r1) = 0, where Sigma(m-1)(i=0) a(i)f(i)(z)(r2) is an element of F-q [X] is an r-linearized polynomial. It is shown that z satisfies a q-linearized polynomial equation with coefficients in F-r. This result provides an explanation for numerous permutation polynomials previously obtained through computer search. (C) 2015 Elsevier Inc. All rights reserved.

关键词： Determinant Finite field linearized polynomial Permutation polynomial

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Constructions of complete permutations in multiplication

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DESIGNS CODES AND CRYPTOGRAPHY 2025年 1-24页

作者： Li, Kangquan Natl Univ Def Technol Coll Sci Changsha 410073 Peoples R China

Complete permutations in addition over finite fields have attracted many scholars' attention due to their wide applications in combinatorics, cryptography, sequences, and so on. In 2020, Tu et al. introduced the concept of the complete permutation in the sense of multiplication (CPM for short). In this paper, we further study the constructions and applications of CPMs. We mainly construct many classes of CPMs through three different approaches, i.e., index, self-inverse binomial, which is a new concept proposed in this paper, and linearized polynomial. Particularly, we provide a modular algorithm to produce all CPMs with a given index and determine all CPMs with index 3. Many infinite classes of complete self-inverse binomials are proposed, which explain most of the experimental results about complete self-inverse binomials over F2n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {F}}_{2<^>n}$$\end{document} with n <= 10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\le 10$$\end{document}. Six classes of linearized CPMs are given by using standard arguments from fast symbolic computations and a general method is proposed by the AGW criterion. Finally, two applications of CPMs in cryptography are discussed.

关键词： Complete permutation in multiplication Index Self-inverse polynomial linearized polynomial

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Vector multispaces and multispace codes

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APPLIED MATHEMATICS AND COMPUTATION 2025年 486卷

作者： Kovacevic, Mladen Univ Novi Sad Fac Tech Sci Novi Sad Serbia

Basic algebraic and combinatorial properties of finite vector spaces in which individual vectors are allowed to have multiplicities larger than 1 are derived. An application in coding theory is illustrated by showing that multispace codes that are introduced here may be used in random linear network coding scenarios, and that they generalize standard subspace codes (defined in the set of all subspaces of F-q(n)) and extend them to an infinitely larger set of parameters. In particular, in contrast to subspace codes, multispace codes of arbitrarily large cardinality and minimum distance exist for any fixed n and q.

关键词： Multiset Multispace Projective space Grassmannian linearized polynomial Modular lattice Subspace code Random linear network coding

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Using multi-orbit cyclic subspace codes for constructing optical orthogonal codes

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CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN FUNCTIONS AND SEQUENCES 2025年 1-14页

作者： Ozbudak, Ferruh Santonastaso, Paolo Zullo, Ferdinando Sabanci Univ FENS Istanbul Turkiye Univ Campania Luigi Vanvitelli Dipartimento Matemat & Fis Caserta Italy

We present a new application of multi-orbit cyclic subspace codes to construct large optical orthogonal codes, with the aid of the multiplicative structure of finite fields extensions. This approach is different from earlier approaches using combinatorial and additive (character sum) structures of finite fields. Consequently, we immediately obtain new classes of optical orthogonal codes with different parameters.

关键词： Optical orthogonal code Cyclic subspace code linearized polynomial

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On generalized Sidon spaces

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LINEAR ALGEBRA AND ITS APPLICATIONS 2025年 704卷 270-308页

作者： Castello, Chiara Univ Campania Luigi Vanvitelli Dipartimento Matemat & Fis Viale Lincoln 5 I-81100 Caserta Italy

Sidon spaces have been introduced by Bachoc, Serra and Z & eacute;mor as the q-analogue of Sidon sets, classical combinatorial objects introduced by Simon Szidon. In 2018 Roth, Raviv and Tamo introduced the notion of r-Sidon spaces, as an extension of Sidon spaces, which may be seen as the q-analogue of Br-sets, a generalization of classical Sidon sets. Thanks to their work, the interest on Sidon spaces has increased quickly because of their connection with cyclic subspace codes they pointed out. This class of codes turned out to be of interest since they can be used in random linear network coding. In this work we focus on a particular class of them, the one-orbit cyclic subspace codes, through the investigation of some properties of Sidon spaces and r-Sidon spaces, providing some upper and lower bounds on the possible dimension of their r-span and showing explicit constructions in the case in which the upper bound is achieved. Moreover, we provide further constructions of r-Sidon spaces, arising from algebraic and combinatorial objects, and we show examples of Br-sets constructed by means of them. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://

关键词： Sidon space Cyclic subspace code linearized polynomial Generalized Sidon space

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A strengthening of McConnel's theorem on permutations over finite fields

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CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES 2025年第1期68卷 213-218页

作者： Yip, Chi Hoi Georgia Inst Technol Sch Math Atlanta GA 30332 USA

Let p be a prime, q = p(n), and D subset of F-q(& lowast;). A celebrated result of McConnel states that if D is a proper subgroup of F-q(& lowast;), and f : F-q -> F-q is a function such that (f(x)-f(y))/(x-y) is an element of D whenever x not equal y, then f(x) necessarily has the form ax(pj)+b. In this notes, we give a sufficient condition on D to obtain the same conclusion on f. In particular, we show that McConnel's theorem extends if D has small doubling.

关键词： Finite field linearized polynomial direction

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linearized polynomials over finite fields revisited

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FINITE FIELDS AND THEIR APPLICATIONS 2013年 22卷 79-100页

作者： Wu, Baofeng Liu, Zhuojun Chinese Acad Sci Acad Math & Syst Sci Key Lab Math Mech Beijing 100190 Peoples R China

We give new characterizations of the algebra L-n(F-qn) formed by all linearized polynomials reduced modulo (x(qn) - x) over the finite field F-qn after briefly surveying some known ones. One isomorphism we construct is between L-n(F-qn) and the composition algebra F-qn(V) circle times F-q F-qn. The other isomorphism we construct is between L-n(F-qn) and the so-called Dickson matrix algebra D-n(F-qn) We also further study the relations between a linearized polynomial and its associate Dickson matrix, generalizing a well-known criterion of Dickson on linearized permutation polynomials. Adjugate polynomial of a linearized polynomial is then introduced, and connections between them are discussed. Both of the new characterizations can bring us new approaches to establish some special forms of representations of linearized polynomials proposed recently by several authors. Structure of the subalgebra L-n(F-qm) which is formed by all linearized polynomials reduced modulo (x(qn) - x) over a subfield F-qm of F-qn where m vertical bar n is also described. (C) 2013 Elsevier Inc. All rights reserved.

关键词： linearized polynomial Composition algebra Dickson matrix Representation

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A new family of 2-scattered subspaces and related MRD codes

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FINITE FIELDS AND THEIR APPLICATIONS 2025年 103卷

作者： Bartoli, Daniele Ghiandoni, Francesco Giannoni, Alessandro Marino, Giuseppe Univ Perugia Dipartimento Matemat & Informat Perugia Italy Univ Firenze Dipartimento Matemat & Informat Ulisse Dini Florence Italy Univ Napoli Federico II Dipartimento Matemat & Applicazioni R Caccioppoli Naples Italy

Scattered subspaces and h-scattered subspaces have been extensively studied in recent decades for both theoretical purposes and their connections to various applications. While numerous constructions of scattered subspaces exist, relatively few are known about h-scattered subspaces with h >= 2. In this paper, we establish the existence of maximum 2-scattered Fqsubspaces in V (r, q(6)) whenever r >= 3, r not equal 5, and q is an odd power of 2. Additionally, we explore the corresponding MRD codes. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://***/licenses/by-nc-nd/4.0/).

关键词： linearized polynomial Scattered subspaces Rank metric code

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Planarity of products of two linearized polynomials

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FINITE FIELDS AND THEIR APPLICATIONS 2012年第6期18卷 1076-1088页

作者： Kyureghyan, Gohar Ozbudak, Ferruh Univ Magdeburg Dept Math D-39106 Magdeburg Germany Middle E Tech Univ Dept Math TR-06800 Ankara Turkey Middle E Tech Univ Inst Appl Math TR-06800 Ankara Turkey

Let L-1(x) and L-2(x) be linearized polynomials over F-qn. We give conditions when the product L-1(x) . L-2(x) defines a planar mapping on F-qn. For a polynomial L over F-qn, let M(L) = {alpha is an element of F-qn: L(x) + alpha . x is bijective on F-qn}. We show that the planarity of the product L-1(x) . L-2(x) is linked with the set M(L) of a suitable linearized polynomial L. We use this relation to describe families of such planar mappings as well as to obtain nonexistence results. (c) 2012 Elsevier Inc. All rights reserved.

关键词： Planar mapping Quadratic mapping Dembowski-Ostrom polynomial linearized polynomial Directions defined by linear functions Cubic irreducible polynomials

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