Hamming Distance And Lee Distance Of Linear Negacyclic And Cyclic Codes Over Z_2α

Cyclic codes have good algebra structure and are one of the most important class of linear block codes. They have important applications by virtue of their high speed of encoding and decoding and strong capacity of error correction and detection.Prange first began to study cyclic codes over finite field in 1957 and Berkelamp first began to investigate negacyclic codes over finite field in 1968,but the codes over finite rings didn't received much attention until the 1990s due to the works of Nechaev in 1991 and Hammons in 1994,which showed that well-known nonlinear codes such as Kerdock and Preparata codes are constructed from linear codes over finite rings.Since then,many researchers have studied the codes over rings from different aspects.Hamming distance has a close relationship with the ability of error correction and detection of codes,meanwhile due to Lee distance there exists a isome-try,Gray mapping, between (Z₄^N, Lee measurement) and (Z₂^2N, Hamming measurement),henceforth Hamming distance and Lee distance are of important research value.In 2004,Taher Abualrub, Ali Ghrayeb and Robert H.Oehmke obtained the bound on the Hamming distance and Lee distance of cyclic codes of length 2^s over ring Z₄.In 2005,Hai Q.Dinh studied the Hamming distance of negacyclic codes of lerigth 2^s over ring Z_2a and proposed a conjecture about this problem.The thesis is devoted to the distance of negacyclic and cyclic codes over ring Z_2a. After some definitions and properties on cyclic codes, Hamming distance and Lee distance introduced, the Hamming distance and Lee distance of negacyclic codes of length 2^s over rings Z_2a and subsequently the Hamming weight distribution of these codes are investigated.Finally,the Hamming distance and Lee distance of cyclic codes of length 2^s over ring Z₄ are studied.Main results of this work are as follow:(1)We prove the conjecture proposed by Q.Dinh about the Hamming distance of negacyclic codes of length 2^s over ring Z_2a are correct,and thereby obtain the exact value of Hamming distance of these codes.Moreover,we gain the exact value of Lee distance of these codes and the Hamming weight distribution of codes 〈(x + 1)^2s(a-k)-1〉(1≤ k ≤ a - 1).(2)We obtain the exact value of Hamming distance of cyclic codes of length...