环上的循环码理论及其应用

发布时间：2018-06-13 00:02

本文选题：DNA循环码 + WCC配对　；参考：《山东理工大学》2017年硕士论文

【摘要】：随着编码理论的发展,有限环上的编码理论在理论研究和实际应用中具有越来越重要的研究意义.近十多年来,国内外的很多学者致力于有限环上的编码理论研究,成为编码领域的研究热点之一.本文通过研究环上的循环码理论,建立了环上的循环码序列与DNA碱基序列之间的关系,得到了性能优异的DNA循环码.在有限环上DNA循环码理论已有研究成果的基础上,进一步深入研究了有限链环F_4[u]/〈u~2+1〉和有限非链环Z_4+νZ_4上的DNA循环码的结构理论和构造方法.具体内容如下:第二章介绍循环码的基本理论,给出文章中所涉及的代数学及相关的编码知识.第三章,我们主要研究有限链环F_4[u]/〈u~2+1〉上的循环码理论,建立了环F_4[u]/〈u~2+1〉中的元素与DNA碱基对之间的一一对应关系φ,构造出此环上的DNA循环码.由于DNA分子具有无序性、互补性,所以我们研究了此环上DNA循环码自反互补的结构与性质.同时给出了此环上DNA循环码的生成元.最后,利用上述结果,我们构造了环F_4[u]/〈u~2+1〉上一个码长为6具有自反互补特性的DNA循环码.第四章,我们主要研究有限非链环R=Z_4+νZ_4上的循环码理论,通过定义环R与4Z之间的Gray映射,建立了环Z_4上的循环码与环R上任意码长的循环码之间的关系,得到了环R上循环码的生成元.利用拓展后的Gray映射,建立了环R上的元素与DNA碱基对之间的关系,构造出环R上的DNA循环码.另外,还研究了环R上DNA循环码的可逆约束、可逆互补约束以及GC重量约束条件,同时也给出了环R上DNA循环码的生成元.最后,利用上述结果,构造了环R上码长分别为8与10的DNA循环码.
[Abstract]:With the development of coding theory, coding theory on finite ring has more and more important research significance in theory research and practical application. In recent ten years, many scholars at home and abroad have devoted themselves to the study of coding theory over finite rings, and become one of the research hotspots in the field of coding. In this paper, the relationship between cyclic code sequences and DNA base sequences over rings is established by studying cyclic codes over rings, and DNA cyclic codes with excellent performance are obtained. Based on the existing research results of DNA cyclic codes over finite rings, the structure theory and construction method of DNA cyclic codes on finite chain rings FSZ 4 [u] / < U 21 > and finite nonchained rings Z4 v Z4 are further studied. The main contents are as follows: in chapter 2, the basic theory of cyclic code is introduced, and the algebra and coding knowledge involved in this paper are given. In chapter 3, we mainly study the cyclic code theory on the finite chain F4 [u] / < uf21 >, and establish the one-to-one correspondence relationship between the elements in the ring F4 [u] / < UU 21 > and the base pairs of DNA, and construct the cyclic code of DNA on this ring. Because DNA molecules are disordered and complementary, we study the structure and properties of reflexive complementarity of DNA cyclic codes on this ring. At the same time, the generator of DNA cyclic code on this ring is given. Finally, using the above results, we construct a DNA cyclic code with the reflexive complementarity property of code length 6 on the ring F4 [u] / < UF 21 >. In chapter 4, we mainly study the cyclic code theory on the finite non-chained ring RX Z4 v Z4. By defining the Gray mapping between the ring R and 4Z, we establish the relation between the cyclic code on the ring Zs 4 and the cyclic code of any code length on the ring R. The generator of cyclic code over ring R is obtained. Based on the extended Gray mapping, the relationship between elements on ring R and DNA base pairs is established, and the cyclic code of DNA on ring R is constructed. In addition, the reversible constraints, reversible complementary constraints and GC weight constraints for DNA cyclic codes over ring R are studied. The generator of DNA cyclic codes over ring R is also given. Finally, using the above results, the cyclic codes with code lengths of 8 and 10 on ring R are constructed.
【学位授予单位】：山东理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O157.4

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