本文選題：線性碼 + 常循環(huán)碼��； 參考：《合肥工業(yè)大學(xué)》2017年博士論文

【摘要】：糾錯編碼理論作為現(xiàn)代數(shù)學(xué)和計(jì)算機(jī)科學(xué)的一個交叉研究領(lǐng)域,無論是對于數(shù)學(xué)本身還是信息安全領(lǐng)域都起著日益重要的作用。經(jīng)過將近70年的發(fā)展,有限域上的經(jīng)典糾錯碼在理論上獲得系統(tǒng)而全面的研究,同時也在工程實(shí)踐中得到廣泛應(yīng)用。隨著糾錯碼理論的深入發(fā)展,有限環(huán)上糾錯碼的極其重要的理論意義和應(yīng)用價值也逐漸被人們認(rèn)識。有限環(huán)上的糾錯編碼理論成為近年來糾錯碼理論研究的熱點(diǎn)問題之一。有限環(huán)上常循環(huán)碼與自對偶碼的研究是有限環(huán)上糾錯碼研究的重點(diǎn)。20世紀(jì)末,量子計(jì)算與量子通信被廣泛關(guān)注。與數(shù)字通信情況一樣,量子糾錯碼理論是量子信息傳輸?shù)靡詫?shí)現(xiàn)的必要保障之一。1998年,Calderbank等人建立了量子糾錯碼的數(shù)學(xué)表達(dá)形式,并且給出了利用經(jīng)典糾錯碼來構(gòu)造量子糾錯碼的第一種系統(tǒng)有效的數(shù)學(xué)方法,這極大推動了量子糾錯碼構(gòu)造的研究。本文在前人對編碼理論研究工作的基礎(chǔ)上,進(jìn)一步深入研究有限環(huán)上線性碼特別是常循環(huán)碼理論研究以及利用有限域上的常循環(huán)糾錯碼來構(gòu)造參數(shù)好的量子糾錯碼。具體研究內(nèi)容如下:第一,研究了有限鏈環(huán)R上任意長度的(l + wγ)-常循環(huán)碼的距離分布與深度譜等重要性質(zhì),其中w是R中的單位,γ是R的極大理想的一個生成元。首先,利用環(huán)R上(1 + wγ)-常循環(huán)碼的生成多項(xiàng)式,給出這類常循環(huán)碼的各階撓碼的生成多項(xiàng)式,確定了所有這類常循環(huán)碼的最小漢明距離。研究了有限鏈環(huán)上(1+ wγ)-常循環(huán)碼的最小齊次距離。給出了最小齊次距離的上界和下界,并得到在某些特殊情況下,該類常循環(huán)碼的精確最小齊次距離。其次,根據(jù)各階撓碼的代數(shù)結(jié)構(gòu),確定了這類常循環(huán)碼中任一碼字的深度值的一個下界。利用這個下界,完全給出了有限鏈環(huán)R上任意長度的每個(1 + wγ)-常循環(huán)碼的深度譜。最后,利用最高階撓碼的生成多項(xiàng)式,構(gòu)造了 Galois環(huán)GR(pt,a)上的(1 + wp)-常循環(huán)MDR碼,其中w是GR(pt,a)中的任一單位。第二,研究了有限環(huán)上自對偶碼。一方面,利用中國剩余定理,給出了有限鏈環(huán)上的自對偶循環(huán)碼的生成多項(xiàng)式。利用生成多項(xiàng)式,得到了有限鏈環(huán)上(非平凡)單根自對偶循環(huán)碼存在的充分必要條件。利用撓碼和有限域上經(jīng)典循環(huán)MDS碼,構(gòu)造了 Galois環(huán)GR(pt,m)上長度為n的循環(huán)自對偶MDR碼,其中n≥2是pm-1的正因數(shù)。另一方面,研究了 16元素環(huán)Z4+vZ4=Z4[v]/v2-1上的線性碼與自對偶碼。得到了環(huán)Z4+vZ4上的自對偶碼的一些重要性質(zhì),給出了(Z4+vZ4)n到Z42n的一個Z4 -線性保距Gray映射,證明了 Z4 + vZ4上的長度為n的自對偶碼的Gray像是Z4上長度為2n的自對偶碼,由此構(gòu)造了 Z4上的一些極優(yōu)類型Ⅰ與類型Ⅱ自對偶碼。第三,利用有限域Fq2上長度為n =(q2m- 1)/(q+1)的ωq1 -常循環(huán)碼構(gòu)造了Fq2上長為n的厄米特對偶包含碼�；诖�,利用量子碼的厄米特構(gòu)造方法,得到了幾類參數(shù)好的q元量子糾錯碼,其中ω是Fq2的一個本原元。與已知的量子BCH碼相比,這類量子常循環(huán)碼具有更好的參數(shù)。
[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.
【學(xué)位授予單位】：合肥工業(yè)大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2017
【分類號】：O157.4

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