信息安全中常循环纠错码的代数理论研究与应用

发布时间：2018-06-15 09:17

本文选题：线性码 + 常循环码　；参考：《合肥工业大学》2017年博士论文

【摘要】：纠错编码理论作为现代数学和计算机科学的一个交叉研究领域,无论是对于数学本身还是信息安全领域都起着日益重要的作用。经过将近70年的发展,有限域上的经典纠错码在理论上获得系统而全面的研究,同时也在工程实践中得到广泛应用。随着纠错码理论的深入发展,有限环上纠错码的极其重要的理论意义和应用价值也逐渐被人们认识。有限环上的纠错编码理论成为近年来纠错码理论研究的热点问题之一。有限环上常循环码与自对偶码的研究是有限环上纠错码研究的重点。20世纪末,量子计算与量子通信被广泛关注。与数字通信情况一样,量子纠错码理论是量子信息传输得以实现的必要保障之一。1998年,Calderbank等人建立了量子纠错码的数学表达形式,并且给出了利用经典纠错码来构造量子纠错码的第一种系统有效的数学方法,这极大推动了量子纠错码构造的研究。本文在前人对编码理论研究工作的基础上,进一步深入研究有限环上线性码特别是常循环码理论研究以及利用有限域上的常循环纠错码来构造参数好的量子纠错码。具体研究内容如下:第一,研究了有限链环R上任意长度的(l + wγ)-常循环码的距离分布与深度谱等重要性质,其中w是R中的单位,γ是R的极大理想的一个生成元。首先,利用环R上(1 + wγ)-常循环码的生成多项式,给出这类常循环码的各阶挠码的生成多项式,确定了所有这类常循环码的最小汉明距离。研究了有限链环上(1+ wγ)-常循环码的最小齐次距离。给出了最小齐次距离的上界和下界,并得到在某些特殊情况下,该类常循环码的精确最小齐次距离。其次,根据各阶挠码的代数结构,确定了这类常循环码中任一码字的深度值的一个下界。利用这个下界,完全给出了有限链环R上任意长度的每个(1 + wγ)-常循环码的深度谱。最后,利用最高阶挠码的生成多项式,构造了 Galois环GR(pt,a)上的(1 + wp)-常循环MDR码,其中w是GR(pt,a)中的任一单位。第二,研究了有限环上自对偶码。一方面,利用中国剩余定理,给出了有限链环上的自对偶循环码的生成多项式。利用生成多项式,得到了有限链环上(非平凡)单根自对偶循环码存在的充分必要条件。利用挠码和有限域上经典循环MDS码,构造了 Galois环GR(pt,m)上长度为n的循环自对偶MDR码,其中n≥2是pm-1的正因数。另一方面,研究了 16元素环Z4+vZ4=Z4[v]/v2-1上的线性码与自对偶码。得到了环Z4+vZ4上的自对偶码的一些重要性质,给出了(Z4+vZ4)n到Z42n的一个Z4 -线性保距Gray映射,证明了 Z4 + vZ4上的长度为n的自对偶码的Gray像是Z4上长度为2n的自对偶码,由此构造了 Z4上的一些极优类型Ⅰ与类型Ⅱ自对偶码。第三,利用有限域Fq2上长度为n =(q2m- 1)/(q+1)的ωq1 -常循环码构造了Fq2上长为n的厄米特对偶包含码。基于此,利用量子码的厄米特构造方法,得到了几类参数好的q元量子纠错码,其中ω是Fq2的一个本原元。与已知的量子BCH码相比,这类量子常循环码具有更好的参数。
[Abstract]:As a cross research field of modern mathematics and computer science, the theory of error correction coding has played an increasingly important role in both mathematics and information security. After nearly 70 years of development, the classical error correcting codes on the finite field have been studied systematically and fully in theory, and also in engineering practice. With the development of the theory of error correcting codes, the extremely important theoretical significance and application value of the error correcting codes on the finite ring are gradually recognized. The theory of error correcting coding on the finite ring has become one of the hot issues in the research of the theory of error correcting codes in recent years. The study of the constant cyclic code and the self dual code on the finite ring is a finite ring correction. At the end of the.20 century, quantum computing and quantum communication are widely concerned. As with digital communication, the quantum error correction code theory is one of the necessary guarantees for the realization of quantum information transmission. Calderbank et al. Has established the mathematical expression form of the quantum error correction code, and gives the use of the classical error correcting code to construct the quantity. The first system effective mathematical method of the error correcting codes has greatly promoted the study of the construction of quantum error correcting codes. Based on the previous work on the research of the coding theory, this paper further studies the theory of linear codes on finite rings, especially the theory of constant cyclic codes and the use of constant cyclic error correcting codes on the finite field to construct the good parameters. Quantum error correction code. The main contents are as follows: first, we study the important properties of the distance distribution and depth spectrum of any length (L + W gamma) - constant cyclic codes over a finite chain R, in which w is a unit in R, and gamma is a generating element of the maximal ideal of R. First, this kind of regular cycle is given by using the generating polynomial of (1 + W gamma) - constant cyclic codes over the ring R. The minimum Hamming distance of all such codes is determined. The minimum homogeneous distance of the (1+ w) - constant cyclic code on the finite chain is studied. The upper and lower bounds of the minimum homogeneous distance are given, and the exact minimum homogeneous distance of the constant cyclic code is obtained in some special cases. Secondly, it is based on the minimum homogeneous distance of the constant cyclic code. The lower bound of the depth value of any code in this kind of constant cyclic code is determined by the algebraic structure of each order. Using this lower bounds, the depth spectrum of every (1 + W) - constant cyclic code of any length on the finite chain R is completely given. Finally, the (1 + WP) of the Galois ring GR (PT, a) is constructed by using the generation of the highest order torsion code. The ring MDR code, in which w is any unit in GR (PT, a). Second, the self dual code on a finite ring is studied. On the one hand, using the Chinese remainder theorem, the generating polynomial of the self dual cyclic code on a finite chain is given. Using the torsion code and the classical cyclic MDS codes on the finite field, a cyclic self dual MDR code with the length of N on the Galois ring GR (PT, m) is constructed. In which n > 2 is a positive factor of PM-1. On the other hand, the linear code and the self dual code on the 16 element ring Z4+vZ4=Z4[v]/v2-1 are studied. Some important properties of the self dual code on the loop Z4+vZ4 are obtained. To a Z4 linear distance preserving Gray mapping of Z42n, it is proved that the Gray image of a self dual code with a length of N on Z4 + vZ4 is a self dual code with a length of 2n on Z4, and thus constructs some extremely excellent types I and type II self dual codes on Z4. Third, using the Omega constant cyclic codes of n = = (1) / (), the length of the Z4 is constructed on the limited domain Fq2. The Omidt dual inclusion code is n. Based on this, several kinds of Q element quantum error correction codes with good parameters are obtained by using the Hermite construction method of quantum code, of which Omega is a primitive of Fq2. Compared with the known quantum BCH codes, this kind of quantum normal cyclic code has better parameters.
【学位授予单位】：合肥工业大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O157.4

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