On automorphisms of linear codes over a prime field

We discuss linearity of code automorphisms for codes in a space over a finite field. We introduce a concept of minimal supports and minimal codewords, which in some cases are turned out useful to prove that an automorphism of a linear code is linear. Also we construct a graph on the set of minimal supports of a code as a vertex set. In this paper for a linear code in a space over a prime field it is shown that all its autotopies fixing the zero vector are linear if and only if the graph of minimal supports of the code does not contain any isolated vertices. We also characterize the autotopy group of a linear code over a prime field.