# Why are folded Reed Solomon Codes considered non linear?

This is for my understanding. What am I missing?

Suppose we consider $s$-folded Reed-Solomon codes that are based on polynomials over a field $\mathbb{F} = \mathrm{GF}(p^t)$. Then the alphabet of those codes is of size $p^{t \cdot s}$. Hence, in order to be linear, those codes should be closed under multiplication by scalar from the field $\mathbb{F}' = \mathrm{GF}(p^{t \cdot s})$. There is no apparent reason why they should be closed under such operation.
However, it is true that they are closed under multiplication by scalar from $\mathbb{F}$. In other words, the codes are $\mathbb{F}$-linear.