本文選題：高斯和 + 雅可比和��； 參考：《合肥工業(yè)大學(xué)》2014年博士論文

【摘要】：有限域上高斯和與雅可比和,作為指數(shù)和的特殊情況,不僅是數(shù)論中研究的重要對象,而且在通信中已有廣泛的應(yīng)用,伽羅華環(huán)上指數(shù)和是有限域上指數(shù)和的推廣,隨著有限域上糾錯碼理論的迅速發(fā)展,伽羅華環(huán)及其上的指數(shù)和也受到眾多學(xué)者的關(guān)注和重視,被用于構(gòu)造好參數(shù)的糾錯碼以及相關(guān)性好的序列等。 本文主要研究了伽羅華環(huán)GR(p2,r)上高斯和與雅可比和及其在通信中的應(yīng)用。首先,給出了伽羅華環(huán)GR(p2,r)上加法特征及乘法特征,定義了伽羅華環(huán)GR(p2,r)上的高斯和與雅可比和,計(jì)算了伽羅華環(huán)GR(p2,r)上高斯和與雅可比和在平凡和非平凡情況下的值,并證明了在所有非平凡的情況下,可將伽羅華環(huán)GR(p2,r)上高斯和與雅可比和的計(jì)算化簡到有限域Fpr的情況,給出了伽羅華環(huán)GR(p2,r)上高斯和與雅可比和的關(guān)系式,得到了伽羅華環(huán)GR(p2,r)與環(huán)R(l)=GR(p2,rl)(l≥1)上高斯和及雅可比和的關(guān)系。其次,將伽羅華環(huán)GR(p2,r)上高斯和與雅可比和應(yīng)用于兩方面：一方面利用伽羅華環(huán)GR(p2,r)上的高斯和的值,得到環(huán)Zp2上的幾類線性碼的重量分布,在此基礎(chǔ)上,利用伽羅華環(huán)GR(p2,r)上高斯和的Davenport-Hasse提升,得到環(huán)Zp2上一系列線性碼C(l)={cβ(l)=(TR(l)(βx))x∈H(l)：β∈R(l)}的重量分布,其中H(l)是環(huán)R(l)的單位群R(l)*的子群,Tr(l)是環(huán)R(l)到環(huán)Zp2的跡映射,并通過Gray映射從環(huán)Zp2上線性碼得到了有限域Fp上好參數(shù)的線性碼與非線性碼；另一方面利用伽羅華環(huán)GR(p2,r)上高斯和、雅可比和以及張量方法給出新參數(shù)的近似相互無偏正交基的三種構(gòu)造方法。
[Abstract]:As a special case of exponential sum, Gao Si sum and Jacobian sum on finite field are not only important objects in number theory, but also widely used in communication. The exponential sum on Galois ring is a generalization of exponential sum on finite field. With the rapid development of error-correcting code theory over finite fields, many scholars pay attention to the Galois ring and its exponent sum, which are used to construct error-correcting codes with good parameters and sequences with good correlation, etc. In this paper, the Gao Si sum and Jacobian sum on Galova ring GRP2 / r and their applications in communication are studied. First of all, the additive and multiplicative characteristics of the Galova ring GRP2Or) are given. The sum of Gao Si and Jacobian on the Galova ring GRP2Or) is defined. The values of Gao Si sum and Jacobian sum in ordinary and nontrivial cases are calculated. It is proved that the Gao Si sum and Jacobian sum can be simplified to finite field Fpr in all nontrivial cases, and the relation between Gao Si sum and Jacobian sum on Galova ring GRp2G) is given. The relations between the Gao Si sum and Jacobian sum on the Galova ring GRP2N) and the ring RGV P2N rll 鈮，

本文編號：1900971