(untitled) (Communications in Mathematical Research, 2022)
Let F_p^m be a finite field with p^m elements, where p is an odd prime and m is a positive integer. Recently, [17] and [35] determined the weight distributions of subfield codes with the form C_f={((Tr(af(x)+bx)+c)_xF_p^m ,Tr(a)):a,bF_p^m,cF_p} for f(x) = x^2 and f(x) = x^p^k+1 , respectively, where Tr(·) is the trace function from F_p^m to F_p, and k is a nonnegative integer. In this paper, we further investigate the subfield code C_f for f(x) being a known perfect nonlinear function over F_p^m and generalize some results in [17, 35]. The weight distributions of the constructed codes are determined by applying the theory of quadratic forms and the properties of perfect nonlinear functions over finite fields. In addition, the parameters of the duals of these codes are also determined. Several examples show that some of our codes and their duals have the best known parameters according to the code tables in [16]. The duals of some proposed codes are optimal according to the Sphere Packing bound if p≥ 5.