【作者】 胡学琴；

【导师】 刘宏伟；

【作者基本信息】 华中师范大学 ， 基础数学， 2009， 硕士

【摘要】 有限环上纠错码的研究始于上世纪70年代,Blake和Speigel等学者首先研究了整数剩余类环Z_m上的码。接着,Calderbank和Sloane研究了p-进制整数环上的码,进一步,Doughertyl Liu和Park定义了一类类似Z_p^e的环R_i同时对这类环和形式幂级数环上的码进行了研究。本文继续对Dougherty,Liu和Park定义的这类环R_i和形式幂级数环上的码展开了研究,得到了以下主要结果:在第三章,定义了R_i上的Ⅱ型码,给出了这些码的一些性质和R_i上的自对偶码提升成R_i+1上的自对偶码的一个必要条件。在第四章,我们给出了形式幂级数环上的自对偶码的一种构造方法,并且给出了形式幂级数环上的自对偶码存在的充分必要条件。更多还原

【Abstract】 The research of error-correcting codes over the finite rings began in 1970s.Blake and Speigel and other researchers had discussed codes over the rings Z_m.Later Calderbank and Sloane discussed codes over the rings p-adic.Further,Dougherty, Liu and Park defined a class of ring,R_i which was similar with the rings Z_p^e and the ring of formal power series.Here,R_i= {a₀ + a₁r + a₂r² +…+ a_i-1r^i-1| a_s∈F,（?） 0≤s≤i - 1}.In this definition,F was a finite field and r^i-1 was equivalent to zero,but rⁱ was not equivalent to zero.At the same time,the researchers discussed codes over these two class of rings.In this thesis,we shall continue the stduy on codes over the rings R_i and the ring of formal power series.We obtain the following results.In chapter 3,we define a typeⅡcode over the rings R_i.We give some properties of this code and a necessary condition for the self-dual codes over R_i to lift to the self-dual codes over R_i+1.In chapter 4,we give a method of constructing the self-dual codes over the ring of formal power series,and the sufficient and necessary condition for the exist of the self-dual codes over the ring of formal power series.更多还原