【作者】 罗玲；

【导师】 刘宏伟；

【作者基本信息】 华中师范大学 ， 基础数学， 2013， 硕士

【摘要】 拟循环码是循环码的推广,本文研究通过线性代数理论构造出的有限域上一类拟循环码的和与交.本文通过探讨矩阵多项式环中两个矩阵的右最大公因子与左最小公倍式的存在性问题,并利用矩阵多项式环中可逆多项式的性质以及两个矩阵的右最大公因子与左最小公倍式的存在性,证明了长度相同,指数相同的两个拟循环码的和与交仍然是拟循环码,并具体给出了它们的表达形式.本文还进一步研究了长度相同,指数不同的两个拟循环码的和与交,并证明了在满足一定的条件时,长度相同,指数不同的两个拟循环码的和与交仍然是拟循环码,并能变换成指数相同的两个拟循环码,利用指数相同的两个拟循环码的计算办法,得到指数不同的两个拟循环码的和与交的具体表达形式.更多还原

【Abstract】 Quasi-cyclic codes are generalizations of cyclic codes. In this paper, we study the sum and intersection of two quasi-cyclic codes, which are constructed through the theory of linear algebra over finite fields.Through exploring the existence of right greatest common divisor and left least common multiple between two matrices in the ring of matrix polynomials, by using invertible properties of polynomials over the ring of matrix polynomials, and exis-tence of right greatest common divisor and left least common multiple between two matrices in the ring of matrix polynomials, we prove the sum and intersection of two quasi-cyclic codes with same length and same index are still quasi-cyclic codes, and give their specific forms of expression.In this paper, we further study the sum and intersection of two quasi-cyclic codes with same length and different index, and we obtain that when certain condi-tions are given, the sum and intersection of two quasi-cyclic codes with same length and different index are still quasi-cyclic codes. And the two quasi-cyclic codes with same length and different index can be transformed into two quasi-cyclic codes with same length and same index. Then by using the method of calculation of the two quasi-cyclic codes with same length and same index, we obtain the specific forms of expression of the sum and intersection of two quasi-cyclic codes with same length and different index.更多还原