Self-dual bases in \(\mathbb{F}_{q^ n}\)
type: Item
instance of: scholarly article
title: Self-dual bases in \(\mathbb{F}_{q^ n}\)
zbMATH Open document ID: 0788.11058
DOI: 10.1007/BF01418530
author: Willi Geiselmann
author: Dieter Gollmann
published in: Designs, Codes and Cryptography
publication date: 1994-06-05T00:00:00Z
review text: Let \(F_{q^ n}\) denote the finite field of order \(q^ n\) so that \(F_{q^ n}\) may be viewed as a vector space of dimension \(n\) over \(F_ q\). Given a basis \(A=\{\alpha_ 0,\dots, \alpha_{n-1}\}\) of \(F_{q^ n}\) over \(F_ q\), the dual basis is given by \(\{\beta_ 0,\dots,\beta_{n-1}\}\) where \(\text{tr}(\alpha_ i\beta_ j)= \delta_{ij}\) for \(0\leq i,j<n\), where \(\text{tr}(\;)\) denotes the trace function. The basis \(A\) is self-dual if \(\beta_ i= \alpha_ i\), \(0\leq i<n\) and it is said to be weakly self-dual if there exists a \(\gamma\in F_{q^ n}\) and a permutation \(\pi\) of the indices \(\{0,1,\dots,n-1\}\) so that \(\beta_ i= \gamma\alpha_{\pi(i)}\) for all \(0\leq i<n\). A polynomial basis is a basis of the form \(\{1,\alpha,\dots, \alpha^{n- 1}\}\) where \(\alpha\) is a root of an irreducible of degree \(n\) over \(F_ q\). Weakly self-dual polynomial bases are studied and the following important and useful characterization is proved: The field \(F_{q^ n}\) has a weakly self-dual polynomial basis if and only if there exists an irreducible trinomial \(p(x)= x^ n- cx^ k -1\) or an irreducible binomial \(p(x)= x^ n -d\). The permutation \(\pi\) yielding the dual basis is given by \(\pi(i)= k-1-i\pmod n\) (if \(p(x)\) is a binomial, set \(k=0\)). Weakly self-dual normal bases are also considered and applications of weakly self-dual bases to dual basis multipliers are discussed.
reviewed by: Gary L. Mullen
Mathematics Subject Classification ID: 11T30
Mathematics Subject Classification ID: 11T71
zbMATH DE Number: 474717
zbMATH Keywords: weakly self-dual polynomial bases
zbMATH Keywords: weakly self-dual normal bases
zbMATH Keywords: finite field
zbMATH Keywords: dual basis multipliers
MaRDI profile type: Publication
cites work: Bit-serial Reed - Solomon encoders
cites work: Q2949568
cites work: Selfdual normal bases on \(\mathrm{GF}(q)\)
cites work: Q3731518
cites work: Q3827943
cites work: Q3203109
cites work: Factorization of Symmetric Matrices and Trace-Orthogonal Bases in Finite Fields
cites work: Q3765876
cites work: Q4146667
cites work: Efficient bit-serial multiplication and the discrete-time Wiener-Hopf equation over finite fields
cites work: Optimal normal bases in \(GF(p^ n)\)
cites work: Bit Serial Multiplication in Finite Fields
full work available at URL: https://doi.org/10.1007/bf01418530
OpenAlex ID: W103938206
Recommended article: Selfdual normal bases on \(\mathrm{GF}(q)\)
Recommended article: On the Number of Self-Dual Bases of GF(q m ) Over GF(q)
Recommended article: Q3203109
Recommended article: On the self-duality of \(\mathbb{Q}_ p\)
Recommended article: Self-dual bases for varieties of orthomodular lattices
Recommended article: Q4349314
Recommended article: Almost weakly self-dual bases for finite fields
Recommended article: Self-conjugate normal bases of finite fields
Recommended article: Theory Is Forever
Recommended article: Q5320179
description: scientific article; zbMATH DE number 474717