As a branch of applied mathematics, coding theory plays an important role. Among them, cyclic codes have attracted much attention because of their good algebraic structure and easy analysis performance. In this paper,...As a branch of applied mathematics, coding theory plays an important role. Among them, cyclic codes have attracted much attention because of their good algebraic structure and easy analysis performance. In this paper, we will study one class of cyclic codes over F<sub>3</sub>. Given the length and dimension, we show that it is optimal by proving its minimum distance is equal to 4, according to the Sphere Packing bound.展开更多
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={((T r(a f(x)+b x)+c)...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={((T r(a f(x)+b x)+c)_(x∈F_(p)^(m)),T r(a)):a,b∈F_(p)^(m),c∈F_(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.展开更多
文摘As a branch of applied mathematics, coding theory plays an important role. Among them, cyclic codes have attracted much attention because of their good algebraic structure and easy analysis performance. In this paper, we will study one class of cyclic codes over F<sub>3</sub>. Given the length and dimension, we show that it is optimal by proving its minimum distance is equal to 4, according to the Sphere Packing bound.
基金This work was supported in part by the National Natural Science Foundation of China(NSFC)under Grants 11971156 and 12001175.
文摘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={((T r(a f(x)+b x)+c)_(x∈F_(p)^(m)),T r(a)):a,b∈F_(p)^(m),c∈F_(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.
