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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the number of self-dual bases of $ {\rm GF}(q\sp m)$ over $ {\rm GF}(q)$

Authors: Dieter Jungnickel, Alfred J. Menezes and Scott A. Vanstone
Journal: Proc. Amer. Math. Soc. 109 (1990), 23-29
MSC: Primary 11T30
MathSciNet review: 1007501
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Abstract: Let $ E = GF({q^m})$ be the $ m$-dimensional extension of $ F = GF(q)$. We are concerned with the numbers $ sd(m,q)$ and $ sdn(m,q)$ of self-dual bases and self-dual normal bases of $ E$ over $ F$, respectively. We completely determine $ sd(m,q)$, en route giving a very simple proof for the Sempel-Seroussi theorem which states that $ sd(m,q) = 0$ iff $ q$ is odd and $ m$ is even. Using results of Lempel and Weinberger and MacWilliams, we can also determine $ sdn(m,p)$ for primes $ p$.

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Keywords: Self-dual basis, normal basis, finite field, orthogonal matrix, circulant matrix
Article copyright: © Copyright 1990 American Mathematical Society

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