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<p style='text-indent:20px;'>One of the best methods for constructing maximum distance separable (<inline-formula><tex-math id="M1">\begin{document}$ \operatorname{MDS} $\end{document}</tex-math></inline-formula>) matrices is based on making use of Cauchy matrices. In this paper, by using some extensions of Cauchy matrices, we introduce several new forms of <inline-formula><tex-math id="M2">\begin{document}$ \operatorname{MDS} $\end{document}</tex-math></inline-formula> matrices over finite fields of characteristic 2. A known extension of a Cauchy matrix, called the Cauchy-like matrix, with application in coding theory was introduced in 1985. One of the main contributions of this paper is to apply Cauchy-like matrices to introduce <b><inline-formula><tex-math id="M3">\begin{document}$ 2n \times 2n $\end{document}</tex-math></inline-formula> involutory <inline-formula><tex-math id="M4">\begin{document}$ \operatorname{MDS} $\end{document}</tex-math></inline-formula> matrices</b> in the semi-Hadamard form which is a generalization of the previously known methods. We make use of Cauchy-like matrices to construct <b>multiple <inline-formula><tex-math id="M5">\begin{document}$ \operatorname{MDS} $\end{document}</tex-math></inline-formula> matrices</b> which can be used in the Feistel structures. We also introduce a new extension of Cauchy matrices to be referred to as <i>Cauchy-light matrices</i>. The introduced Cauchy-light matrices are applied to construct <inline-formula><tex-math id="M6">\begin{document}$ n \times n $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M7">\begin{document}$ \operatorname{MDS} $\end{document}</tex-math></inline-formula> matrices having at least <inline-formula><tex-math id="M8">\begin{document}$ 3n-3 $\end{document}</tex-math></inline-formula> entries equal to the unit element <inline-formula><tex-math id="M9">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>; such a matrix is called a <b>lightweight <inline-formula><tex-math id="M10">\begin{document}$ \operatorname{MDS} $\end{document}</tex-math></inline-formula> matrix</b> and can be used in the lightweight cryptography. A simple closed-form expression is given for the determinant of Cauchy-light matrices.</p>