New Lower Bounds for Minimum Distance of Linear Quasi-Cyclic and Almost Linear Cyclic Codes

New lower bounds are derived for the minimum distance of linear $(np,kp)$ quasi-cyclic codes over arbitrary fields $GF(q)$ for transmission rate $R=k/n$ and for almost linear cyclic codes of length $p$ over nonprime fields $GF(q^n)$ for transmission rate $R=k/n$, where $p$ is any prime number. It is not assumed that $q$ is a primitive root modulo $p$. This result makes it possible to establish the asymptotic attainability of the corresponding Gilbert–Varshamov bounds by quasi-cyclic and almost linear cyclic codes with the indicated characteristics for almost all prime numbers $p$.