Two Classes Of Binary Cyclic Codes And Their Weight Distributions

In coding theory,cyclic code is a subclass of linear block codes.Due to their beauti-ful algebraic structure and efficient coding and decoding algorithms,cyclic codes have been widely used in communication system,data storage system and security.The weight distribu-tion of cyclic codes can determine the minimum distance of the codes and the error detection and correction ability of codes.In addition,the probability of decoding error can be estimated by the weight distribution of the employed cyclic codes with respect to some decodng algo-rithms.Therefore,it is of great value to study the weight distribution of cyclic codes from both theoretical and practical viewpoints.Searching cyclic codes with excellent parameters and determining the weight distribution of cyclic codes have been interesting research topics in coding theory.One objective of this thesis is to present two kinds of binary cyclic codes with two nonzeros.Another objective of this theis is to determine the weight distribution of the prposed cyclic codes and study their dual codes.The proposed cyclic codes are optimal in some cases with respect to certain bounds of linear codes,and has new dimension and minimal distance compared with known ones with the same code length.The thesis are organized as folllows.In Chapter one and Chapter two,we introduce respectively the background and known works on cyclic codes,and some necessary preliminaries.In Chapter three,we preset two classes of cylcic codes and determine their weight distri-butions.Specifically,let m be a positive integer and?a primitive element of the finite field F_2^m.The first class of binary cyclic codes has parity-check polynomial m_?-1m_?^(-2m-1)/5,where m_?ⁱ is the minimal polynomial of?ⁱ over F₂.We show that this class of cyclic codes is a sev-en weight cyclic codes with parameters[2^m-1,m+4]and determine its weight distribution based on Gaussian period.The second one has parity-check polynomial m_?-1m_?^(-2m-1)/9.We also determine the parameters and the weight distribution of this class of cyclic codes.In Chapter four,we study the properties of dual codes of the proposed cyclic codes.It turns out that the dual code of the first class cyclic code has parameters[2^m-1,2^m-m-5,d],where 4?d?6,and the second dual code has parameters[2^m-1,2^m-m-7,3].