Abstract
In this paper, several classes of three-weight codes and two-weight codes for the homogeneous metric over the chain ring \(R=\mathbb {F}_p+u\mathbb {F}_p+\cdots +u^{k-1}\mathbb {F}_{p},\) with \(u^k=0,\) are constructed that generalize the construction of Shi et al. (IEEE Commun. Lett. 20(12):2346–2349, 2016), which is the special case of \(p=k=2.\) These codes are defined as trace codes. In some cases of their defining sets, they are abelian. Their homogeneous weight distributions are computed by using exponential sums. In particular, in the two-weight case, we give some conditions of optimality of their Gray images by using the Griesmer bound. Their dual homogeneous distance is also given. The codewords of these codes are shown to be minimal for inclusion of supports, a fact favorable to an application to secret sharing schemes.
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Communicated by Miin Huey Ang.
This research of the first author 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). The third author is supported by the Project of Graduate Academic Innovation of Anhui University (No. yfc100015) and the fourth author is supported by China Postdoctoral Science Foundation funded project (2016M601991).
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Shi, M., Wu, R., Qian, L. et al. New Classes of p-Ary Few Weight Codes. Bull. Malays. Math. Sci. Soc. 42, 1393–1412 (2019). https://doi.org/10.1007/s40840-017-0553-1
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DOI: https://doi.org/10.1007/s40840-017-0553-1