Recently, subfiled codes of linear code over GF$ (q) $ with good parameters were studied, and many optimal subfield codes were obtained. In this paper, Our mainly motivation is to generlize the results of the subfield codes of hyperoval in Ding and Heng (Finite Fields Their Appl. 56, 308-331 (2019)), and generlize the results of two families of subfield codes in Xiang and Yin (Cryptogr. Commun. 13(1), 117-127 (2021)) to $ p $-ary where $ p $ is odd. We get the parameters and weight distribution of these subfield codes. At the same time, the parameters of their dual codes are also studied. When $ m=1 $, The dual codes of these subfield codes are almost MDS code, when $ m>1 $ and $ p $ odd, these dual codes are dimension-optimal with respect to the sphere-backing bound.

翻译：最近,对超过(q)GF$的线性代码进行了次级提交,并获得了许多最佳的子字段代码。在本文件中,我们的主要动机是将Ding和Heng(Fite Fields their Appl. 56, 308-331 (2019))的超声波代码结果基因化,并将Chiang和Yin(Cryptogr. Commun. 13(1), 117-127 (2021))的两个子字段代码家庭的结果基因化为p-ary美元,其中P美元为奇数。我们得到了这些子字段代码的参数和重量分布。与此同时,还研究了其双层代码的参数。当 $m=1 美元时,这些子字段代码的双重代码几乎是MDS代码,当M>1美元和 p 美元为奇数时,这些双层代码的维度最优于域框约束。