Dissertation > Excellent graduate degree dissertation topics show

Research on the Chinese Product of Cyclic Codes and the MacWilliams Identities for Linear Codes over Finite Rings

Author: TangYongSheng
Tutor: ZhuShiXin
School: Hefei University of Technology
Course: Applied Mathematics
Keywords: Cyclic codes The Chinese Remainder Theorem Galois rings MacWilliams identities m - plyweight enumerator Lee weight enumerator Euclidean weight enumerator Support weight Homogenous weight Generalized Plotkin bound
CLC: TN911.22
Type: Master's thesis
Year: 2009
Downloads: 66
Quote: 0
Read: Download Dissertation

Abstract


A great deal of attention has been paid to codes over finite rings from the 1990s, because scholars have found that certain nonlinear codes (such as Kerdock codes, Preparata codes, Goethal codes) over finite fields can be constructed from linear codes over finite rings via Gray maps. Since then, studying coding theory over finite rings is a hot topic. In this thiese, we study the Chinese product of cyclic codes over Z k, ( k = ( i∏=s 1pi)m, with the condition that the code length n can not be divided by pi ). We also study the MacWilliams identities for Lee and Euclidean weights of linear codes over Galois rings; and we study the MacWilliams identities with respect to any partition and support weight for linear codes over Z_p~2( p is a prime). The details are given as follows:1. We describe the Chinese Remainder Theorem for studying cyclic and dual cyclic codes over Z k, (k = ( i∏=s 1pi)m, with the condition that the code length n can not be divided by pi ). A necessary and suffcient condition of the existence of nontrivial self-dual cyclic codes is given. The upper bound of minimum distance of cyclic codes is also obtained. Furthermore, we determine the minimum generators of such cyclic codes.2. The m ? ply Lee and Euclidean weight enumerators of linear codes over Z k are defned, and then we obtain a new kind of MacWilliams identities for the Lee and Euclidean weights of such codes. Furthermore, we regard Galois ring GR ( p e, m ) as a free Z pe module of rank m , then we discuss the relationships between the Lee and Euclidean weight enumerators of linear codes over GR ( p e, m ), which have generator matrices over Z peand the m ? ply Lee and Euclidean weight enumerators of linear codes over Z pe. At last, we get the Lee and Euclidean weights MacWilliams identities of linear codes over GR ( p e, m ).3. The generalized Lee weight enumerators for linear codes of length n over Z 4are defined. It is given that the generalized Lee weight MacWilliams identities for linear codes of length n over Z 4. The equivalent form of generalized Lee weight enumerators for linear codes of length n over Z 4are defined. Correspondingly, the equivalent form of generalized Lee weight MacWilliams identities for linear codes of length n over Z 4are obtained.4. When we interpret CU as restriction codes of C , we give an equation which connects the weight distribution of a code with those of its restrictions to sets of a given size; when we interpret C U as subcodes of C , we get a set of MacWilliams identities between any partition of the coordinate set of a Z_p~2-linear code with type ( p 2)k and that of its dual are established.5. We also study support weight and the generalized Plotkin bound of linear codes over Z_p~2, and give the generalized Plotkin bound for homogenous weight of linear codes over Z_p~2.

Related Dissertations

  1. The Design and Realization about the Secure Video Multicast Key Management System Based on Chinese Remainder Theorem,TP393.08
  2. The Design and Implementation of the Database Encryption System,TP311.13
  3. Research on Data Simulation and Phase Unwrapping for Multi-Baseline InSAR,TN958
  4. The Numerical Algorithm for the Linear Congruential Equations with Single Modulus / Multi-modulus and Its Application in Cryptography,TN918.1
  5. The Constructions of Difference Set Pairs and Frequency-Hopping Sequences Based on the Cyclotomy,O157.2
  6. Constructions and Analysis Some Low Correlation and High Linear Complexity Sequences,TN918.1
  7. Study of Multi-secret Sharing Scheme,TN918.1
  8. Research on Key Distribution Management in CAS,TN918.1
  9. New Constructions for Optimal Frequency-Hopping Sequences,TN914.41
  10. The Research on Static Balance and the Brain Electricity Signals of the Paralysis after Stroke,R743
  11. Dynamic multi- secret sharing scheme,TN918
  12. The Research of Secret Sharing among Weighted Participants,TN918
  13. The Study on Color and Distinctiveness Information Based Local Stereo Matching,TP391.41
  14. Research on Verifiable Secret Sharing Based on Chinese Remainder Theorem,TN918.1
  15. Based on the Chinese Remainder Theorem elliptic curve threshold group signature scheme,TN918.1
  16. The Structure and Construction of Generalized Quasi-Cyclic Codes,TN911.2
  17. Quadratic Residue Codes Over Z2K,O157.4
  18. A Class of Cyclic Codes Over ZPk+1,O157.4
  19. The Application Research of RSA Cryptographic Algorithm in Electricity Market Operation System,TN918.1
  20. FEC technology of high - speed backplane Ethernet,TP393.11

CLC: > Industrial Technology > Radio electronics, telecommunications technology > Communicate > Communication theory > Information Theory > Channel coding theory
© 2012 www.DissertationTopic.Net  Mobile