【摘要】：循環(huán)碼是一類非常重要的碼.常循環(huán)碼是循環(huán)碼的自然推廣,它保留了循環(huán)碼的幾乎所有良好性質(zhì).對偶性質(zhì)是編碼理論的重要研究對象,它在碼的重量結(jié)構(gòu)研究和代數(shù)結(jié)構(gòu)研究等方面都有重要作用.本文主要從三個(gè)不同角度來研究常循環(huán)碼(包括循環(huán)碼)與對偶相關(guān)的性質(zhì).1.推廣歐幾里得內(nèi)積和厄米特內(nèi)積,對有限域的任意自同構(gòu),我們引入了伽羅瓦內(nèi)積.我們用統(tǒng)一的方法研究最一般情形的常循環(huán)碼(包括循環(huán)碼,包括重根情形)的伽羅瓦自對偶性質(zhì).這個(gè)統(tǒng)一的方法包括：定義q-陪集函數(shù)用于刻畫和構(gòu)造常循環(huán)碼,它推廣了半單情形的用零點(diǎn)集刻畫常循環(huán)碼的方法；定義新的保距同構(gòu),它既適用于半單情形也適用于非半單情形；等等.我們得到一系列關(guān)于伽羅瓦對偶性與自對偶性的結(jié)果.特別地,對有限域的任意自同構(gòu),給出了伽羅瓦自對偶常循環(huán)碼以及伽羅瓦自對偶循環(huán)碼存在的充要條件.這些結(jié)果包括了常循環(huán)碼的歐幾里得對偶性,厄米特對偶性的有關(guān)結(jié)果作為特例.2.推廣duadic循環(huán)碼和Ⅱ-型duadic負(fù)循環(huán)碼的概念,我們引進(jìn)了even-like(也即,Ⅱ-型)和odd-like duadic常循環(huán)碼的概念,并研究它們的一系列性質(zhì)和存在條件.我們證明了even-like duadic常循環(huán)碼是保距自正交的,并且even-like duadic常循環(huán)碼的對偶碼是odd-like duadic常循環(huán)碼.另外,對常循環(huán)碼,我們證明了當(dāng)長度為n的Ⅰ-型duadic對不存在但Ⅱ-型duadic對存在時(shí),Ⅱ-型duadic對是最大保距自正交的.隨后我們給出了存在even-like duadic常循環(huán)碼的充要條件,并構(gòu)造了一類稱作交錯MDS-碼的even-like duadic常循環(huán)碼的例子.3.我們引進(jìn)了一類新的保距同構(gòu)來研究循環(huán)碼的自對偶性.現(xiàn)有文獻(xiàn)中的保距同構(gòu)都是用模n剩余類Zn上的乘法置換來構(gòu)造的,稱為乘子.這類保距同構(gòu)無法用于循環(huán)碼保距自對偶性的研究.我們在模n剩余類Zn上用加法置換定義了一類新的保距自同構(gòu),稱作平移算子.我們用這種新的方法來研究保距自對偶循環(huán)碼存在的充要條件,及相關(guān)的性質(zhì).特別地,我們給出了保距自對偶循環(huán)碼存在的幾個(gè)等價(jià)條件.另外,我們把這種方法推廣到常循環(huán)碼上,并與用乘子構(gòu)造Ⅰ-型duadic常循環(huán)碼的方法一起來研究了保距自對偶常循環(huán)碼的存在條件及相關(guān)性質(zhì).
[Abstract]:Cyclic codes are a kind of very important codes. Constant cyclic code is a natural generalization of cyclic code, which preserves almost all good properties of cyclic code. Dual property is an important research object of coding theory, which plays an important role in the study of weight structure and algebra structure of codes. In this paper, we mainly study the dual-related properties of constant cyclic codes (including cyclic codes) from three different angles. In this paper, we generalize Euclidean inner product and Hermitian inner product. For any automorphism of finite fields, we introduce Galava inner product. In this paper, we study the Galava self-duality property of the most general constant cyclic codes (including cyclic codes, including double root cases) by using a unified method. This unified method includes defining Q-coset functions to characterize and construct constant cyclic codes, which generalize the method of describing constant cyclic codes with zero sets in semi-simple cases. Define a new distance-preserving isomorphism, which is applicable to both semi-simple and non-semi-simple cases, and so on. We get a series of results on Galova duality and self-duality. In particular, for any automorphism of finite fields, the necessary and sufficient conditions for the existence of Galava self-dual constant cyclic codes and Galava self-dual cyclic codes are given. These results include the Euclidean duality of constant cyclic codes and the results of Hermitian duality as special cases. 2. In this paper, we generalize the concepts of duadic cyclic codes and type II duadic negative cyclic codes. We introduce the concepts of even-like (that is, type II) and odd-like duadic constant cyclic codes, and study a series of properties and existence conditions of them. We prove that even-like duadic constant cyclic codes are distance-preserving self-orthogonal, and the dual codes of even-like duadic constant cyclic codes are odd-like duadic constant cyclic codes. In addition, for constant cyclic codes, we prove that when the I-type duadic pair of length n does not exist but the 鈪，

本文編號：2492153