On The Depth Spectrums Of Cyclic Codes On Ring Z_2^k And The Width Of Sequence

We have five parts in this thesis.The first part: We introduce not only the important results in the depth spectrums of linear cyclic codes on ring Z_2^k and the width of sequence but also the main contributions in this thesis.The second part: We give some preliminaries about codes,including some concepts about linear codes and cyclic codes.The third part:The depth spectrums of linear cyclic codes on ring Z_2^k are studied,and it is proved that the depth spectrum of linear code of type (2^k) (2^k-1)^k₁…(2)^k_k-1 has at least k₀ + k₁+…+k_k-1 nonzero values,and the depth spectrum of linear cyclic code of type (2^k)^k₀ is {n , n-1,…,n-k₀ +1}.The fourth part:A number of properties on the width of codewords are presented,and two recursive algorithms for computing the width of acodeword on the finite ring Z kare given.The fiveth part:The relationship of the period and width of infinite sequence over GF (p^m)are studied. The following statements are the main proven results: If x = ( x₀ , x₁,...)is infinite sequence over GF (p^m),the width of x is limited and above zero,then the minimal positive period of x is 2p^{[log_p w]}, p is a odd prime.