极大—加代数上形式多项式的除法运算与编码的线性码

发布时间：2018-03-28 16:04

本文选题：极大　切入点：加代数　出处：《河北师范大学》2016年硕士论文

【摘要】：极大加代数是一个具有重要理论意义和应用价值的代数系统.多项式是代数学中基本的研究对象之一.极大加线性系统理论不断发展和完善,而极大加代数上多项式理论鲜见研究.带余除法是数论和多项式论中的一个重要方法,在Euclid除法、因式分解、求解多项式方程、有理函数分解中扮演着重要角色.本文将研究极大加代数上多项式的除法运算和带余除法.在此基础上,本文还将首次研究极大加代数上编码的线性码.首先,我们研究极大加代数上形式多项式与多项式函数之间的关系,证明在形式多项式幂等代数与多项式函数幂等代数之间存在一个赋值同态.其次,我们研究凹形式多项式的性质,发现任一凹形式多项式具有全支集.然后,我们引入极大加代数上形式多项式可除、商式和余式的概念,并给出可除的一些性质.我们利用凹形式多项式具有全支集的特性及其相邻项系数之差的单调性,研究2次凹形式多项式与次数小于2的形式多项式的带余除法,分别给出2次凹形式多项式除以1次形式多项式的商式和余式存在的充分必要条件,商式和余式唯一的充分必要条件,以及商式和余式的求法.凹形式多项式的带余除法对于研究极大加代数上多项式函数的带余除法有着特别的意义.利用赋值同态的性质,我们证明:如果两个凹形式多项式可除,那么它们所对应的多项式函数可除.利用2次凹形式多项式除以1次形式多项式的商式和余式的计算公式,我们可计算任一2次多项式函数除以1次多项式函数的商式和余式.另一方面,我们还举例说明,当两个凹形式多项式不可除时,它们所对应的多项式函数也不可除.最后,我们研究极大加代数上编码的线性码,并引入极大加代数上线性码和循环码的概念,给出极大加代数上线性码的多项式描述,即极大加码多项式,且用数值例子加以说明.极大加代数上多项式的带余除法可用于计算极大加循环码的循环移位.
[Abstract]:Maximal plus algebra is an algebraic system with important theoretical significance and application value. Polynomial is one of the basic research objects in algebra. However, the theory of polynomials on maximal addition algebra is seldom studied. The method with coaddition division is an important method in number theory and polynomial theory. In Euclid division, factorization of factorization, solving polynomial equations, Rational function decomposition plays an important role. In this paper, we will study the division of polynomials and the division with coaddition of polynomials on maximal plus algebras. On this basis, we will also study the linear codes encoded on maximal plus algebras for the first time. We study the relationship between formal polynomials and polynomial functions on maximal additive algebras, and prove that there exists an assignment homomorphism between formal polynomial idempotent algebras and polynomial function idempotent algebras. We study the properties of concave polynomials and find that any concave polynomial has a set of full branches. Then we introduce the concepts of divisible quotient and covariance of formal polynomials on maximal additive algebras. Some properties of divisibility are given. By using the monotonicity of the concave polynomial with full support set and the difference between the coefficients of adjacent terms, we study the division method with remainder between the concave polynomial of order 2 and the formal polynomial of degree less than 2. The necessary and sufficient conditions for the existence and uniqueness of quotient and covariance of quadratic concave polynomial divided by first-order formal polynomial are given respectively, and the necessary and sufficient conditions for the uniqueness of quotient and covariance are obtained. In addition, the method of quotient and covariance is of special significance for the study of the division of polynomial functions with coaddition on maximal addition algebras, using the property of assignment homomorphism, the method with codivision of concave polynomials is of special significance in studying the division of polynomial functions on maximal addition algebras. We prove that if two concave polynomials are divisible, then their corresponding polynomial functions can be divisible. We can calculate the quotient and covariance of any polynomial function of degree 2 divided by polynomial function of degree 1. On the other hand, we also show that when two concave polynomials are not divisible, the corresponding polynomial functions are also not divisible. In this paper, we study the linear codes coded on maximal additive algebras, and introduce the concepts of linear codes and cyclic codes on maximal additive algebras, and give the polynomial description of linear codes on maximal plus algebras, that is, the polynomial of maximal additive codes. A numerical example is given to illustrate that the division of polynomials on maximal addition algebras can be used to calculate the cyclic shift of maximal plus cyclic codes.
【学位授予单位】：河北师范大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O157.4

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