On Self-Complementary Bases GF (qn) over GF (q)
Summary :
A self-complementary basis of a finite field corresponds to the orthonormal basis of a vector metric space. Seroussi and Lempel showed that a finite field GF (qn) has a self-complementary basis over GF (q) if and only if either q is even or both q and n are odd. In this paper, firstly we show that by using the complementary basis of a polynomial basis we can write a self-complementary basis explicitly. Since a polynomial basis and a normal basis are the most popular bases in finite fields, in this paper we consider whether a polynomial basis or a normal basis can be self-complementary. Secondly we show that any polynomial basis can not be self-complementary. Thirdly we tabulate the numbers of all the different self-complementary normal bases of GF (qn) over GF (q) for various q and n. From this table we present a conjecture concerning the existence of nonexistence of self-complementary normal bases.
- Publication
- IEICE TRANSACTIONS on transactions Vol.E66-E No.12 pp.717-721
- Publication Date
- 1983/12/25
- Publicized
- Online ISSN
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- Type of Manuscript
- PAPER
- Category
- Mathematics
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Kyoki IMAMURA, "On Self-Complementary Bases GF (qn) over GF (q)" in IEICE TRANSACTIONS on transactions,
vol. E66-E, no. 12, pp. 717-721, December 1983, doi: .
Abstract: A self-complementary basis of a finite field corresponds to the orthonormal basis of a vector metric space. Seroussi and Lempel showed that a finite field GF (qn) has a self-complementary basis over GF (q) if and only if either q is even or both q and n are odd. In this paper, firstly we show that by using the complementary basis of a polynomial basis we can write a self-complementary basis explicitly. Since a polynomial basis and a normal basis are the most popular bases in finite fields, in this paper we consider whether a polynomial basis or a normal basis can be self-complementary. Secondly we show that any polynomial basis can not be self-complementary. Thirdly we tabulate the numbers of all the different self-complementary normal bases of GF (qn) over GF (q) for various q and n. From this table we present a conjecture concerning the existence of nonexistence of self-complementary normal bases.
URL: https://global.ieice.org/en_transactions/transactions/10.1587/e66-e_12_717/_p
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@ARTICLE{e66-e_12_717,
author={Kyoki IMAMURA, },
journal={IEICE TRANSACTIONS on transactions},
title={On Self-Complementary Bases GF (qn) over GF (q)},
year={1983},
volume={E66-E},
number={12},
pages={717-721},
abstract={A self-complementary basis of a finite field corresponds to the orthonormal basis of a vector metric space. Seroussi and Lempel showed that a finite field GF (qn) has a self-complementary basis over GF (q) if and only if either q is even or both q and n are odd. In this paper, firstly we show that by using the complementary basis of a polynomial basis we can write a self-complementary basis explicitly. Since a polynomial basis and a normal basis are the most popular bases in finite fields, in this paper we consider whether a polynomial basis or a normal basis can be self-complementary. Secondly we show that any polynomial basis can not be self-complementary. Thirdly we tabulate the numbers of all the different self-complementary normal bases of GF (qn) over GF (q) for various q and n. From this table we present a conjecture concerning the existence of nonexistence of self-complementary normal bases.},
keywords={},
doi={},
ISSN={},
month={December},}
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TY - JOUR
TI - On Self-Complementary Bases GF (qn) over GF (q)
T2 - IEICE TRANSACTIONS on transactions
SP - 717
EP - 721
AU - Kyoki IMAMURA
PY - 1983
DO -
JO - IEICE TRANSACTIONS on transactions
SN -
VL - E66-E
IS - 12
JA - IEICE TRANSACTIONS on transactions
Y1 - December 1983
AB - A self-complementary basis of a finite field corresponds to the orthonormal basis of a vector metric space. Seroussi and Lempel showed that a finite field GF (qn) has a self-complementary basis over GF (q) if and only if either q is even or both q and n are odd. In this paper, firstly we show that by using the complementary basis of a polynomial basis we can write a self-complementary basis explicitly. Since a polynomial basis and a normal basis are the most popular bases in finite fields, in this paper we consider whether a polynomial basis or a normal basis can be self-complementary. Secondly we show that any polynomial basis can not be self-complementary. Thirdly we tabulate the numbers of all the different self-complementary normal bases of GF (qn) over GF (q) for various q and n. From this table we present a conjecture concerning the existence of nonexistence of self-complementary normal bases.
ER -
English
