Several Classes of $p$-ary Linear Codes with Few Weights

Few weights codes have been an important topic of coding theory for decades. In this paper, a class of two-weight and three-weight codes for the homogeneous metric over the chain ring R = Fp + uFp + · · ·+ uFp are constructed. These codes are defined as trace codes. They are shown to be abelian. Their homogeneous weight distributions are computed by using exponential sums. In particular, we give a necessary and sufficient condition of the optimality for their Gray images by using the Griesmer bound in the two-weight case, and the information about the dual homogeneous distance is also given. In addition, the codewords of these codes have ∗This research is supported by National Natural Science Foundation of China (61672036), Technology Foundation for Selected Overseas Chinese Scholar, Ministry of Personnel of China (05015133) and the Open Research Fund of National Mobile Communications Research Laboratory, Southeast University (2015D11) and Key projects of support program for outstanding young talents in Colleges and Universities (gxyqZD2016008).