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On Self-dual Bases of the Extensions of the Binary Field

Mika Hirvensalo, Jyrki Lahtonen, On Self-dual Bases of the Extensions of the Binary Field. In: H. Maurer G. Paun G. Rozenberg J. Karhumäki (Ed.), Theory Is Forever: Essays Dedicated to Arto Salomaa on the Occasion of His 70th Birthday, LNCS 3113, 102–111, Springer, 2004.

Abstract:

There are at least two points of view when representing elements of <B>F</B><SUB>2<SUP>n</SUP></SUB>, the field of
2<SUP>n</SUP> elements. We could represent the (nonzero) elements as powers of a generating element, the exponent ranging from 0 to 2<SUP>n</SUP>-2. On the other hand, we could represent the elements as strings of <I>n</I> bits. In the former representation, multiplication becomes a very easy task, whereas in the latter one, addition is obvious. In this note, we focus on representing <B>F</B><SUB>2<SUP>n</SUP></SUB>
as strings on <I>n</I> bits in such a way that the natural basis (1,0,...,0), (0,1,...,0),...,(0,0,...,1)$ becomes self-dual, and outline a very simple algorithm for finding a self-dual basis. We also study multiplication tables for the natural basis and present necessary and sufficient conditions for a multiplication table to give
<B>F</B><SUB>2<SUP>n</SUP></SUB> a field structure in such a way that the natural basis is self-dual.

BibTeX entry:

@INBOOK{cHiLa04a,
  title = {On Self-dual Bases of the Extensions of the Binary Field},
  booktitle = {Theory Is Forever: Essays Dedicated to Arto Salomaa on the Occasion of His 70th Birthday},
  author = {Hirvensalo, Mika and Lahtonen, Jyrki},
  volume = {3113},
  series = {LNCS},
  editor = {J. Karhumäki, H. Maurer G. Paun G. Rozenberg},
  publisher = {Springer},
  pages = {102–111},
  year = {2004},
}

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