Complete weight enumerators of a class of linear codes

Let F q be the finite field with q = p m elements, where p is an odd prime and m is a positive integer. For a positive integer t , let D ⊂ F q t and let Tr m be the trace function from F q onto F p . In this paper, let D = { ( x 1 , x 2 , … , x t ) ∈ F q t \ { ( 0 , 0 , … , 0 ) } : Tr m ( x 1 + x 2 + ⋯ + x t ) = 0 } , we define a p -ary linear code C D by C D = { c ( a 1 , a 2 , … , a t ) : ( a 1 , a 2 , … , a t ) ∈ F q t } , where c ( a 1 , a 2 , … , a t ) = ( Tr m ( a 1 x 1 2 + a 2 x 2 2 + ⋯ + a t x t 2 ) ) ( x 1 , x 2 , … , x t ) ∈ D . We shall present the complete weight enumerators of the linear codes C D and give several classes of linear codes with a few weights. This paper generalizes the results of Yang and Yao (Des Codes Cryptogr, 2016 ).