Some $k$-Valued Pseudo-random Sequences and Nearly Equidistant Codes

We consider pseudorandom sequences $\alpha$ of length $n$ in which the elements are the $k$-th order roots of unity. We show that for any $k$ and $n=q-1$, $q\equiv 1(\operatorname{mod}k)$ ($q$ is a power of a prime $p$), there exist pseudorandom sequences $\alpha$ with autocorrelation function $T(m)$, whose modulus does not exceed 4. In addition we consider nearly equidistant codes which can be obtained from the pseudorandom sequences considered.