Filters








4,047 Hits in 3.9 sec

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

Arunwan Boripan, Somphong Jitman, Patanee Udomkavanich
2018 arXiv   pre-print
The class of 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.  ... 
arXiv:1801.08842v2 fatcat:2q3zbcfaubfipcrgwvm3ghavlm

Enumeration of self-reciprocal irreducible monic polynomials with prescribed leading coefficients over a finite field [article]

Zhicheng Gao
2021 arXiv   pre-print
In this paper we enumerate 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  ... 
arXiv:2109.09006v3 fatcat:tvm575uvhzgdblpsbfi7sxysqi

Improved error bounds for the number of irreducible polynomials and self-reciprocal irreducible monic polynomials with prescribed coefficients over a finite field [article]

Zhicheng Gao
2021 arXiv   pre-print
In this paper we obtain improved error bounds for the number of 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  ... 
arXiv:2109.14154v4 fatcat:mbnii4uhxzcrxevqaxfwb2i4mu

On the parity of the number of irreducible factors of self-reciprocal polynomials over finite fields

Omran Ahmadi, Gerardo Vega
2008 Finite Fields and Their Applications  
Using the Stickelberger-Swan theorem, the parity of the number of 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.  ... 
doi:10.1016/j.ffa.2006.09.004 fatcat:g7fw4355izaexp2dgbziygow7a

On the construction of irreducible self-reciprocal polynomials over finite fields

Helmut Meyn
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.  ... 
doi:10.1007/bf01810846 fatcat:g3mfzt3xcbcgdcf32jaidfb3pa

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

Theodoulos Garefalakis, Giorgos Kapetanakis
2015 Finite Fields and Their Applications  
In this note, we complete the work in [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.  ... 
doi:10.1016/j.ffa.2015.03.005 fatcat:v2u3m2n2unb2lmszhxeyubcrty

Some Properties of Generalized Self-reciprocal Polynomials over Finite Fields [article]

Ryul Kim, Ok-Hyon Song, Hyon-Chol Ri
2014 arXiv   pre-print
We consider some properties of the divisibility of a-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.  ... 
arXiv:1302.3051v3 fatcat:sdneu4ww3vhqjgaryd3ywmat2a

Self-reciprocal and self-conjugate-reciprocal irreducible factors of x^n-λ and their applications [article]

Yansheng Wu, Qin Yue, Shuqin Fan
2020 arXiv   pre-print
In this paper, we present some necessary and sufficient conditions under which an 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.  ... 
arXiv:2001.04766v1 fatcat:app45yauebdhdklq2jq2sihvba

Divisibility of Trinomials by Irreducible Polynomials over F2 [article]

Ryul Kim, Wolfram Koepf
2014 arXiv   pre-print
A condition for divisibility of 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.  ... 
arXiv:1311.1366v2 fatcat:fxusbgnocbcjhise4v2e5phf4u

Factors of Dickson polynomials over finite fields

Robert W. Fitzgerald, Joseph L. Yucas
2005 Finite Fields and Their Applications  
We give new descriptions of the factors of Dickson 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.  ... 
doi:10.1016/j.ffa.2004.12.002 fatcat:rfpern7245gz7ntcawldio2wba

On an iterated construction of irreducible polynomials over finite fields of even characteristic by Kyuregyan

Simone Ugolini
2016 Czechoslovak Mathematical Journal  
In this paper we deal with the construction of sequences of 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  ... 
doi:10.1007/s10587-016-0253-2 fatcat:d3esf4eogjc7bo4sjb6f5vgn4i

Construction of self-reciprocal normal polynomials over finite fields of even characteristic

Mahmood ALIZADEH, Saeid MEHRABI
2015 Turkish Journal of Mathematics  
The construction of N -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 .  ... 
doi:10.3906/mat-1407-32 fatcat:qyz4fnrwqngz7l3kqr224lfyia

Self-Conjugate-Reciprocal Irreducible Monic Factors of x^n-1 over Finite Fields and Their Applications [article]

Arunwan Boripan, Somphong Jitman, Patanee Udomkavanich
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.  ... 
arXiv:1804.06138v1 fatcat:thh3hmnbivb53n4fc3sm2dyxha

Generalization of a Theorem of Carlitz [article]

Omran Ahmadi
2010 arXiv   pre-print
We generalize Carlitz' result on the number of 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.  ... 
arXiv:1003.5856v1 fatcat:hg3fuoealrehbcbu7nop6inr3i

Self-reciprocal irreducible polynomials with prescribed coefficients

Theodoulos Garefalakis
2011 Finite Fields and Their Applications  
We prove estimates for the number of 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.  ... 
doi:10.1016/j.ffa.2010.11.004 fatcat:kctsf2afbzgpnh6xmzd7dp55xe
« Previous Showing results 1 — 15 out of 4,047 results