Properties Of Structure And Period Distribution Of Cyclic Codes Over F₂+uF₂

Cyclic codes are one of the most important classes of linear codes. Cyclic codes have strict algebra structure, whose performance is easy to analyze. They also have the character of cyclic which ensures the encoding and decoding of the codes is easy. Thus the cyclic codes have attracted many attentions.Recently, more and more scholars have taken interest in cyclic codes over finite rings. The study of cyclic codes over finite rings means a lot not only in theory but also in practice. On one hand, we can obtain good binary codes from some cyclic codes over finite rings using the Gray map. On the other hand, the studies of the structure, the property, and the estimating of parameters of cyclic codes over finite rings also play a key role in theory of cyclic codes.The period distribution of an error-correcting code is of great use, which is defined in 1992. Many scholars have made good efforts to solve the problem since then. The study of the period distribution of an error-correcting code does help to make out codeword which has good hamming correlation property to design, to acknowledge the deep structure of an error-correcting code, and to realize the lossless compression of information.The major jobs of this thesis are as follows:(1) We studied the structure of cyclic codes over F₂+uF₂ with oddly even length as well as their dual codes, and give their generators. We also discussed the Gray map over F₂+uF₂.(2) Considering the parameters estimation of a cyclic code, we studied the period distribution of cyclic codes over F₂+uF₂, and gave the formula for computing the number of codewords with the same period for a cyclic code with odd length and its dual code.