【作者】 冯倩倩；

【导师】 刘宏伟；

【作者基本信息】 华中师范大学 ， 基础数学， 2008， 硕士

【摘要】 上世纪70年代起，Blake[1]和Speigel[2]等学者开始将纠错码的研究从有限域上转移到整数剩余类环Zm上．90年代初，Forney等学者在[3]，Hammons等学者在[4]中证明了Kerdock码，Preparata码，Delsarte-Goethals码比同样长度，同样距离的线性码有更多的码字，这些非线性码实际上就是一些Z4上的线性码在Gray映射下的像．1998年，Carlet在文献[6]中，通过Boolean函数在Z2k上定义了Gray映射，通过Gray映射将Z2k上的线性码映射成Z2上的非线性码，得到了广义的Kerdock码和广义的Goethals码．Ling在文献[9]中进一步将Gray映射推广到环Zpk+1上，给出了（1-pk）-循环码的Gray像是Zp上的准循环码，且通过建立（1-pk）-循环码与一般循环码的一一对应，得到了环Zpk+1上的循环码的Gray像等价于准循环码，且给出了它们的像为线性的充分条件．本文继续对环Zpk+1上的码展开研究，得到了以下主要结果．在第二章，通过利用环Zpk+1中的元素可以唯一写成p进制的形式，以及从Zpk+1n到Zppkn的Gray映射，我们给出了环Zpk+1上的（1+pk）-循环码的Gray像和一般循环码的Gray像以及负循环码的Gray像。在第三章，我们给出了环Zpk+1上长为n，（n，p）=1的常循环码的生成元．通过建立Zpk+1上的循环码与（1+pk）-循环码的一一对应给出了（1+pk）-循环码的生成元．在第四章，我们考虑环Zp2上的码．由于本文所说的循环码，常循环码，准循环码不一定是线性的．最后本文给出了环Zp2上（1+p）-循环码和循环码的Gray像是线性的充分条件．更多还原

【Abstract】 Since 1970s, some researchers, such as Blake[1] and Speigel[2], began to discuss error-correcting codes over the rings Zm of integers modulo m instead of studying codes over finite fields. In the beginning of 1990s, Forney etc[3] and Hammous etc[4] proved that some nonlinear codes, such as Kerdock code, Preparata code and Delsarte-Goethals code are the images of some linear codes over Z4, where the map is the Gray map. Since these non-linear codes have more codewords than those linear codes with the same length and the same Hamming distance, the research interest in codes over finite rings has grown rapidly. In 1998, Carlet[6] defined Gray map over Z2k by using Boolean function, then mapped the linear codes over Z2k to nonlinear codes over Z2 by the Gray map in this paper, and finally obtained generalized Kerdock code and generalized Goethals code. Ling[9] extended the definition of Gray map to Zpk+1 and proved that the Gray image of （1-pk）- cyclic codes are quasi-cyclic codes over Zp. In [9], they also proved that cyclic codes over the rings Zpk+1 are equivalent to quasi-cyclic codes by formulating a one-to-one correspondence between （1-pk）-cyclic codes and general cyclic codes, they also provided a sufficient condition for the images of codes over Zpk+1 to be linear.In this thesis, we shall continue the study on codes over Zpk+1. We obtain the following results.In chapter 2, by using the Gray map （see [9]） from Zpk+1n to Zppkn, and the unique p-adic expression of each element in Zpk+1, the Gray images of （1+pk）- cyclic codes of length n over the rings Zpk+1, general cyclic codes and negacyclic codes are obtained, where （n,p） = 1.In chapter 3, when （n,p） = 1, the generator of constacyclic codes with length n over Zpk+1 are obtained. By using the one-to-one correspondence between cyclic codes over Zpk+1 and （1+pk）- cyclic codes over Zpk+1, the generator of （1+pk）-cyclic codes are obtained.In chapter 4, we focus on codes over Zp2. First we note that the three class of codes-cyclic codes, constacyclic codes and quasi-cyclic codes, which are discussed in this thesis, may be nonlinear. In this chapter, the sufficient conditions for the Gray image of （1+p）- cyclic codes and the cyclic codes over Zp2 to be linear axe also given.更多还原