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深洞在广义Reed-Solomon码的译码中发挥重要的作用.最近,Wu和Hong通过循环码对于标准Reed-Solomon码发现了一类新的深洞.本文给出一个简洁的方法,对于一般广义Reed-Solomon码给出新的一类深洞.特别地,对于标准Reed-Solomon码,我们得到了Wu和Hong给出的深洞.对于广义Reed-Solomon码GRSk(Fq,D),Li和Wan研究和刻画了k+1次多项式定义的深洞,并且指出这个问题归结为在有限域中的子集和问题.在偶特征的情形下,利用他们的方法,我们对于一些特殊的Reed-Solomon码得到了更多一类新的深洞.此外,我们研究扩展Reed-Solomon码(即赋值集合为D=Fq)k+2次多项式定义的深洞,并且证明没有k+2次多项式定义的深洞.
Deep holes play an important role in the decoding of generalized Reed-Solomon codes. Recently, Wu and Hong discovered a new class of deep holes for standard Reed-Solomon codes by using cyclic codes. This paper gives a concise method for general The generalized Reed-Solomon codes give a new class of deep holes. In particular, we obtain the deep holes given by Wu and Hong for the standard Reed-Solomon codes. For generalized Reed-Solomon codes GRSk (Fq, D), Li And Wan studied and characterized the deep-hole in the definition of k + 1 polynomials, and pointed out that the problem boils down to subsets and problems in finite fields.In the case of even features, we use their method for some special Reed - Solomon codes and so on. In addition, we study deep cavities that extend the definition of k + 2 polynomials of the Reed-Solomon code (ie, the set of assignment is D = Fq) and prove that there is no polynomial of degree k + 2 Defined deep hole.