On Self-dual Codes Over The Ring Of Formal Power Series

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 typeâ…¡code 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.