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We construct two series of linear codes over finite ring$\mathbb{F}_{q}[x]/(x^2)$ and Galois ring $GR(p^2,m)$ respectively reaching theGriesmer bound. They derive two series of codes over finite field$\mathbb{F}_{q}$ by Gray map. The first series of codes over $\mathbb{F}_{q}$derived from $\mathbb{F}_{q}[x]/(x^2)$ are linear and also reach the Griesmerbound in some cases. Many of linear codes over finite field we constructed havetwo Hamming (non-zero) weights.

Linear Codes over $\mathbb{F}_{q}[x]/(x^2)$ and $GR(p^2,m)$ Reaching the Griesmer Bound