Partially scattered linearized polynomials and rank metric codes
A linearized polynomial f(x)∈Fqn[x] is called scattered if for any y,z∈Fqn, the condition zf(y)−yf(z)=0 implies that y and z are Fq-linearly dependent. In this paper two generalizations of the notion of a scattered linearized polynomial are provided and investigated. Let t be a nontrivial positive d...
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| Published in | Finite fields and their applications Vol. 76; p. 101914 |
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| Language | English |
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01.12.2021
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| ISSN | 1071-5797 1090-2465 |
| DOI | 10.1016/j.ffa.2021.101914 |
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| Abstract | A linearized polynomial f(x)∈Fqn[x] is called scattered if for any y,z∈Fqn, the condition zf(y)−yf(z)=0 implies that y and z are Fq-linearly dependent. In this paper two generalizations of the notion of a scattered linearized polynomial are provided and investigated. Let t be a nontrivial positive divisor of n. By weakening the property defining a scattered linearized polynomial, L-qt-partially scattered and R-qt-partially scattered linearized polynomials are introduced in such a way that the scattered linearized polynomials are precisely those which are both L-qt- and R-qt-partially scattered. Also, connections between partially scattered polynomials, linear sets and rank metric codes are exhibited. |
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| AbstractList | A linearized polynomial f(x)∈Fqn[x] is called scattered if for any y,z∈Fqn, the condition zf(y)−yf(z)=0 implies that y and z are Fq-linearly dependent. In this paper two generalizations of the notion of a scattered linearized polynomial are provided and investigated. Let t be a nontrivial positive divisor of n. By weakening the property defining a scattered linearized polynomial, L-qt-partially scattered and R-qt-partially scattered linearized polynomials are introduced in such a way that the scattered linearized polynomials are precisely those which are both L-qt- and R-qt-partially scattered. Also, connections between partially scattered polynomials, linear sets and rank metric codes are exhibited. |
| ArticleNumber | 101914 |
| Author | Longobardi, Giovanni Zanella, Corrado |
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| Cites_doi | 10.1016/j.ffa.2018.05.003 10.1007/s00493-016-3531-6 10.1007/s10801-017-0755-5 10.1016/j.ffa.2013.03.003 10.1016/j.jalgebra.2018.03.010 10.3934/amc.2016019 10.26493/1855-3974.2137.7fa 10.1016/j.disc.2009.04.007 10.1016/j.ffa.2018.08.001 10.1515/form.2004.029 10.1016/j.laa.2020.05.009 10.1016/0097-3165(78)90015-8 10.1016/j.jcta.2018.03.007 10.1016/j.laa.2020.01.004 10.1016/j.jcta.2017.01.002 10.1007/PL00012530 10.1023/A:1005283806897 10.1016/j.laa.2018.02.027 10.1007/s10801-020-01011-9 10.1016/j.disc.2020.111985 10.1007/s00013-016-0949-4 |
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| Snippet | A linearized polynomial f(x)∈Fqn[x] is called scattered if for any y,z∈Fqn, the condition zf(y)−yf(z)=0 implies that y and z are Fq-linearly dependent. In this... |
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| SubjectTerms | Finite field Finite projective space Linear set Linearized polynomial MRD-code Rank metric code Subgeometry |
| Title | Partially scattered linearized polynomials and rank metric codes |
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