The setting FunctionOfCoefficients[g] sets the input form and output form for the element GF[p, ilist][{e1, e2, ...}] to be g[e1, e2, ...].
The setting FunctionOfCode[g] sets the input form and output form for the element GF[p, ilist][{e1, e2, ...}] to be g[i], where i is the element code e1+e2p+e3p2+....