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Self-Conjugate-Reciprocal Irreducible Monic Polynomials over Finite Fields
[article]

2018
*
arXiv
*
pre-print

The class of

arXiv:1801.08842v2
fatcat:2q3zbcfaubfipcrgwvm3ghavlm
*self*-conjugate-*reciprocal**irreducible*monic (SCRIM)*polynomials**over**finite**fields*are studied. ... Necessary and sufficient conditions for monic*irreducible**polynomials*to be SCRIM are given. The number of SCRIM*polynomials*of a given degree are also determined. ... Lemma 2.4 ([5,*Self*-Conjugate-*Reciprocal**Irreducible**Polynomials*In this section, we study*self*-conjugate-*reciprocal**irreducible*monic (SCRIM)*polynomials**over**finite**fields*. ...##
###
Enumeration of self-reciprocal irreducible monic polynomials with prescribed leading coefficients over a finite field
[article]

2021
*
arXiv
*
pre-print

In this paper we enumerate

arXiv:2109.09006v3
fatcat:tvm575uvhzgdblpsbfi7sxysqi
*self*-*reciprocal**irreducible*monic*polynomials**over*a*finite**field*with prescribed leading coefficients. ... A*polynomial*is called*self*-*reciprocal*(or palindromic) if the sequence of its coefficients is palindromic. ...*Polynomials*in P q are called*self*-*reciprocal*or palindromic. • I q ⊆ M q denotes the set of*irreducible*monic*polynomials*. • S q = I q ∩ P q denotes the set of*self*-*reciprocal**irreducible*monic*polynomials*...##
###
Improved error bounds for the number of irreducible polynomials and self-reciprocal irreducible monic polynomials with prescribed coefficients over a finite field
[article]

2021
*
arXiv
*
pre-print

In this paper we obtain improved error bounds for the number of

arXiv:2109.14154v4
fatcat:mbnii4uhxzcrxevqaxfwb2i4mu
*irreducible**polynomials*and*self*-*reciprocal**irreducible*monic*polynomials*with prescribed coefficients*over*a*finite**field*. ... A*polynomial*is called*self*-*reciprocal*(or palindromic) if the sequence of its coefficients is palindromic. ...*Polynomials*in P are called palindromic or*self*-*reciprocal*. • I ⊆ M denotes the set of*irreducible*monic*polynomials*. • S = I ∩ P denotes the set of*self*-*reciprocal**irreducible*monic*polynomials**over*F ...##
###
On the parity of the number of irreducible factors of self-reciprocal polynomials over finite fields

2008
*
Finite Fields and Their Applications
*

Using the Stickelberger-Swan theorem, the parity of the number of

doi:10.1016/j.ffa.2006.09.004
fatcat:g7fw4355izaexp2dgbziygow7a
*irreducible*factors of a*self*-*reciprocal*even-degree*polynomial**over*a*finite**field*will be hereby characterized. ... It will be shown that in the case of binary*fields*such a characterization can be presented in terms of the exponents of the monomials of the*self*-*reciprocal**polynomial*. ...*Self*-*reciprocal**irreducible**polynomials**over**finite**fields*have been studied by many authors. ...##
###
On the construction of irreducible self-reciprocal polynomials over finite fields

1990
*
Applicable Algebra in Engineering, Communication and Computing
*

*polynomials*

*over*the binary

*field*" (forthcoming). ... Simple criteria are given for the case that the

*irreducibility*of f is inherited by the

*self*-

*reciprocal*

*polynomial*fQ. ...

*Self*-

*reciprocal*

*polynomials*

*over*

*finite*

*fields*are used to generate reversible codes with a read-backward property (J. L. Massey [13] , S. J. Hong and D. C. Bossen [10] , A. M. Patel and S. J. ...

##
###
A note on the Hansen–Mullen conjecture for self-reciprocal irreducible polynomials

2015
*
Finite Fields and Their Applications
*

In this note, we complete the work in [

doi:10.1016/j.ffa.2015.03.005
fatcat:v2u3m2n2unb2lmszhxeyubcrty
*Finite**Fields*Appl., 18(4):832-841, 2012] by using computer calculations to prove that for odd q, there exists a monic*self*-*reciprocal**irreducible**polynomial*of degree ... 2n*over*F q , with any of its first (hence any of its last) n/2 coefficients arbitrarily prescribed, with a couple of genuine exceptions. ... In [4] , the existence of*self*-*reciprocal**irreducible*monic*polynomials*with prescribed coeffecients,*over*F q for odd q, was considered. ...##
###
Some Properties of Generalized Self-reciprocal Polynomials over Finite Fields
[article]

2014
*
arXiv
*
pre-print

We consider some properties of the divisibility of a-

arXiv:1302.3051v3
fatcat:sdneu4ww3vhqjgaryd3ywmat2a
*reciprocal**polynomials*and characterize the parity of the number of*irreducible*factors for a-*self**reciprocal**polynomials**over**finite**fields*of odd ... Numerous results on*self*-*reciprocal**polynomials**over**finite**fields*have been studied. In this paper we generalize some of these to a-*self**reciprocal**polynomials*defined in [4]. ... Carlitz [3] proposed a formula on the number of*self*-*reciprocal**irreducible*monic (srim)*polynomials**over**finite**fields*and Meyn [7] gave a simpler proof of it. ...##
###
Self-reciprocal and self-conjugate-reciprocal irreducible factors of x^n-λ and their applications
[article]

2020
*
arXiv
*
pre-print

In this paper, we present some necessary and sufficient conditions under which an

arXiv:2001.04766v1
fatcat:app45yauebdhdklq2jq2sihvba
*irreducible**polynomial*is*self*-*reciprocal*(SR) or*self*-conjugate-*reciprocal*(SCR). ... By these characterizations, we obtain some enumeration formulas of SR and SCR*irreducible*factors of x^n-λ, λ∈ F_q^*,*over*F_q, which are just open questions posed by Boripan et al (2019). ...*Self*-conjugate-*reciprocal**polynomials*In this section, we deal with*self*-conjugate-*reciprocal**polynomials**over**finite**fields*. 3.1. ...##
###
Divisibility of Trinomials by Irreducible Polynomials over F2
[article]

2014
*
arXiv
*
pre-print

A condition for divisibility of

arXiv:1311.1366v2
fatcat:fxusbgnocbcjhise4v2e5phf4u
*self*-*reciprocal*trinomials by*irreducible**polynomials**over*F_2 is established. ... In this paper we consider some conditions under which*irreducible**polynomials*divide trinomials*over*F_2. ... Numerous results are known concerning*self*-*reciprocal**irreducible**polynomials**over**finite**fields*. ...##
###
Factors of Dickson polynomials over finite fields

2005
*
Finite Fields and Their Applications
*

We give new descriptions of the factors of Dickson

doi:10.1016/j.ffa.2004.12.002
fatcat:rfpern7245gz7ntcawldio2wba
*polynomials**over**finite**fields*. ... The*irreducible*factors of H n (x) are the (b(x)),*over*all*irreducible*,*self*-*reciprocal**polynomials*of degree 2v and order n.(b) Suppose −1 / ∈ q n . ... Let P n be the collection of all*polynomials**over*F q of degree n and let S n denote the family of all*self*-*reciprocal**polynomials**over*F q of degree n. ...##
###
On an iterated construction of irreducible polynomials over finite fields of even characteristic by Kyuregyan

2016
*
Czechoslovak Mathematical Journal
*

In this paper we deal with the construction of sequences of

doi:10.1007/s10587-016-0253-2
fatcat:d3esf4eogjc7bo4sjb6f5vgn4i
*irreducible**polynomials*with coefficients in*finite**fields*of even characteristic. ... of*irreducible**polynomials*of non-decreasing degree starting from any*irreducible**polynomial*. ... Theorem 1.1. [3, Theorem 9] The Q-transform of a*self*-*reciprocal**irreducible*monic*polynomial*f (x) = x n + a 1 x n−1 + · · · + a 1 x + 1 ∈ F 2 k [x] with Tr n (a 1 ) = 1 is a*self*-*reciprocal**irreducible*...##
###
Construction of self-reciprocal normal polynomials over finite fields of even characteristic

2015
*
Turkish Journal of Mathematics
*

The construction of N -

doi:10.3906/mat-1407-32
fatcat:qyz4fnrwqngz7l3kqr224lfyia
*polynomials**over*any*finite**field*is a challenging mathematical problem. ... , are N -*polynomials*and the*polynomials*F k (x + 1) are*self*-*reciprocal*normal*polynomials**over*F 2 s . ...##
###
Self-Conjugate-Reciprocal Irreducible Monic Factors of x^n-1 over Finite Fields and Their Applications
[article]

2018
*
arXiv
*
pre-print

*Self*-

*reciprocal*and

*self*-conjugate-

*reciprocal*

*polynomials*

*over*

*finite*

*fields*have been of interest due to their rich algebraic structures and wide applications. ...

*Self*-

*reciprocal*

*irreducible*monic factors of x^n-1

*over*

*finite*

*fields*and their applications have been quite well studied. ... A

*polynomial*is said to be

*self*-

*reciprocal*

*irreducible*monic (SRIM) if it is

*self*-

*reciprocal*,

*irreducible*and monic. ...

##
###
Generalization of a Theorem of Carlitz
[article]

2010
*
arXiv
*
pre-print

We generalize Carlitz' result on the number of

arXiv:1003.5856v1
fatcat:hg3fuoealrehbcbu7nop6inr3i
*self**reciprocal*monic*irreducible**polynomials**over**finite**fields*by showing that similar explicit formula hold for the number of*irreducible**polynomials*obtained ...*Self*-*reciprocal**irreducible**polynomials**over**finite**fields*have been studied by many authors. Carlitz [3] counted the number of srim*polynomials*of degree 2n*over*a*finite**field*for every n. ... A*polynomial*f (x) is called*self*-*reciprocal*if f * (x) = f (x). The*reciprocal*of an*irreducible**polynomial*is also*irreducible*. ...##
###
Self-reciprocal irreducible polynomials with prescribed coefficients

2011
*
Finite Fields and Their Applications
*

We prove estimates for the number of

doi:10.1016/j.ffa.2010.11.004
fatcat:kctsf2afbzgpnh6xmzd7dp55xe
*self*-*reciprocal*monic*irreducible**polynomials**over*a*finite**field*of odd characteristic, that have the t lower degree coefficients fixed to given values. ... Our estimates imply that one may specify up to m/2 − log q (2m) − 1 values in the*field*and a*self*-*reciprocal*monic*irreducible**polynomial*of degree 2m exists with its low degree coefficients fixed to ...*self*-*reciprocal**polynomials*of degree 2m*over*a*finite**field*of odd characteristic, that have up to m/2 − log q (2m) − 1 low degree coefficients prescribed. ...
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