On automorphisms of linear codes over a prime field

We discuss linearity of code automorphisms for codes ina space over a finite field. We introduce a concept of minimal supportsand minimal codewords, which in some cases are turned out useful toprove that an automorphism of a linear code is linear. Also we constructa graph on the set of minimal supports of a code as a vertex set. In thispaper for a linear code in a space over a prime field it is shown that allits autotopies fixing the zero vector are linear if and only if the graph ofminimal supports of the code does not contain any isolated vertices. Wealso characterize the autotopy group of a linear code over a prime field.