Subfield Codes And Their Dual Codes Of A Class Of Linear Codes

Ding and Heng in[12]proposed the concept of subfield code,gave their trace representation and studied a kind of special geometric codes,the subfield code of ovoid code.Assume that F_p^m={x₁,x₂,···,x_pm}be a finite field,f（x） is a polynomial over F_p^m.Let C be a[p^m+1,3]code with generator matrix Then the subfield code of C,remember it as C_f,can be represented by the following trace form:In particular,Heng and Ding in[18]studied the weight distribution of subfield code C_f for f（x）=x².Wang et.al.[34]extended their work and studied the weight distri-bution of subfield code C_ffor f（x）=x^pk+1,where k is a nonnegative integer.Along the line of the work in[18,34],in this paper,we further study the weight distribution of C_fand the parameters of its dual for f（x）being a known perfect nonlinear function.The weight distribution of C_fis determined by applying the theory of quadratic forms and the properties of perfect nonlinear functions over finite fields.In addition,the pa-rameters 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 with respect to the code tables in[15].The duals of some proposed codes are optimal with respect to the Sphere Packing bound if p≥5.