Abstract
In this chapter, we study several aspects of matrices over finite fields. We begin with some basic results on the rank and order of matrices and then apply some of these results to discuss an interesting theoretical topic, namely matrix representations of an extension field \({\text{GF}}(q^{n} )\) over \({\text{GF}}(q)\). Following this, we provide results on the numbers of matrices over \({\text{GF}}(q)\) of various important special types – such as symmetric, orthogonal and circulant matrices – that will be needed later to study basis representations of extension fields. Last but not least, we also consider the Discrete Fourier Transform, which is not only a very useful tool for some results given in the present chapter, but has many interesting further applications.
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