Let $q$ be a prime greater than 4. In this paper, we determine the coefficients of the discrete Fourier transform over the finite field $\mathbb {F}_q$ of two classes of quaternary sequences of even length with optimal autocorrelation. They are quaternary sequence with period $2p$ derived from binary Legendre sequences and quaternary sequence with period $2p(p+2)$ derived from twin-prime sequences pair. As applications, the linear complexities over the finite field $\mathbb {F}_q$ of both of the quaternary sequences are determined.
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