On the construction of certain odd degree irreducible polynomials over finite fields
For an odd prime power q , let F q 2 = F q ( α ) , α 2 = t ∈ F q be the quadratic extension of the finite field F q . In this paper, we consider the irreducible polynomials F ( x ) = x k - c 1 x k - 1 + c 2 x k - 2 - ⋯ - c 2 q x 2 + c 1 q x - 1 over F q 2 , where k is an odd integer and the coeffici...
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| Veröffentlicht in: | Designs, codes, and cryptography Jg. 92; H. 12; S. 4085 - 4097 |
|---|---|
| Hauptverfasser: | , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
New York
Springer US
01.12.2024
Springer Nature B.V |
| Schlagworte: | |
| ISSN: | 0925-1022, 1573-7586 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | For an odd prime power
q
, let
F
q
2
=
F
q
(
α
)
,
α
2
=
t
∈
F
q
be the quadratic extension of the finite field
F
q
. In this paper, we consider the irreducible polynomials
F
(
x
)
=
x
k
-
c
1
x
k
-
1
+
c
2
x
k
-
2
-
⋯
-
c
2
q
x
2
+
c
1
q
x
-
1
over
F
q
2
, where
k
is an odd integer and the coefficients
c
i
are in the form
c
i
=
a
i
+
b
i
α
with at least one
b
i
≠
0
. For a given such irreducible polynomial
F
(
x
) over
F
q
2
, we provide an algorithm to construct an irreducible polynomial
G
(
x
)
=
x
k
-
A
1
x
k
-
1
+
A
2
x
k
-
2
-
⋯
-
A
k
-
2
x
2
+
A
k
-
1
x
-
A
k
over
F
q
, where the
A
i
’s are explicitly given in terms of the
c
i
’s. This gives a bijective correspondence between irreducible polynomials over
F
q
2
and
F
q
. This fact generalizes many recent results on this subject in the literature. |
|---|---|
| Bibliographie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0925-1022 1573-7586 |
| DOI: | 10.1007/s10623-024-01479-7 |