Two classes of p-ary linear codes and their duals
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Let F_p^m be the finite field of order p^m, where p is an odd prime and m is a positive integer. In this paper, we investigate a class of subfield codes of linear codes and obtain the weight distribution of C_k={(( Tr_1^m(ax^p^k+1+bx)+c)_x ∈F_p^m, Tr_1^m(a)) : a,b ∈F_p^m, c ∈F_p}, where k is a nonnegative integer. Our results generalize the results of the subfield codes of the conic codes in <cit.>. Among other results, we study the punctured code of C_k, which is defined as C̅_k={( Tr_1^m(a x^p^k+1+bx)+c)_x ∈F_p^m : a,b ∈F_p^m, c ∈F_p}. The parameters of these linear codes are new in some cases. Some of the presented codes are optimal or almost optimal. Moreover, let v_2(·) denote the 2-adic order function and v_2(0)=∞, the duals of C_k and C̅_k are optimal with respect to the Sphere Packing bound if p>3, and the dual of C̅_k is an optimal ternary linear code for the case v_2(m)≤ v_2(k) if p=3 and m>1.
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