Research On Error-correcting Code And Its Application

The theory of error-correcting codes is not only the theoretical basis for information security, but also the theoretical basis for quantum information. Since 1994, Hammons, Calderbank, Sloane and others in IEEE Trans. Inform. Theory published in the well-known award-winning paper "The Z4-linearity of Kerdock,Preparata, Goethals, and related codes", codes over finite rings have received much attention since then. In 1998, Calderbank, Rains, Shor and others established the mathematical theory of quantum error-correcting codes, and gave a effective way about the quantum error-correcting codes constructed by error-correcting codes. Their research has greatly accelerated the theory of error-correcting codes in quantum information applications.In this dissertation, we take the theory of error-correcting codes over finite ring as the foundation, and take the quantum error-correcting codes construction as the application. The contributions of the dissertation are outlined as follows:1. Theory of error-correcting codes over finite rings(1) Constacyclic codes and cyclic codes over polynomial residue class rings are introduced. A new Gray map between codes over polynomial residue class rings and codes over F2 is defined. As a result, some good propositions are obtained.(2) Repeated-root cyclic codes over polynomial residue class rings are studied. Descriptions are given in terms of discrete Fourier transform. An isomorphism between repeated-root cyclic codes and a direct sum is defined. This shows that any repeated-root cyclic codes over polynomial residue class rings can be described by a direct sum of ideals using this decomposition.(3) MacDonald codes over finite ring Fq+uFq are constructed. Based on the MacDonald codes, we present an access structure of a secret sharing scheme.(4) Constacyclic codes and cyclic codes over the finite chain ring R are introduced, where R is a finite chain ring with nilpotency index e= 2, and residue field Fk.We prove that the Gray image of a linear constacyclic code over R of length n is a distance invariant quasi-cyclic code over Fpk. We also prove that if (n,p)= 1, then every code over Fpk which is the Gray image of a cyclic code over R of length n is equivalent to a quasi-cyclic code.(5) The problem of the Gray image of constacyclic codes over finite chain ring R is considered, where R is a finite chain ring with nilpotency index e, and residue field Fp. A Gray map between codes over finite chain ring R and codes over finite field Fp is defined. As a consequence, some good propositions are obtained. 2. Error-correcting codes in the application of quantum information(1) A new method to obtain self-orthogonal codes over finite field F2 is presented. Based on this method, we provide a construction for quantum error-correcting codes starting from cyclic codes over finite ring F2+uF2. As an example, we present infinite families of quantum error-correcting codes which are derived from cyclic codes over the ring F2+uF2.(2) A new method of constructing nonbinary nonadditive quantum codes starting from linear codes over finite ring Zp2 (the ring of integers modulo p2) is provided, where p is any prime. Furthermore, infinite families of nonbinary nonadditive quantum codes with parameters ((pm+1, p2pm-4m-4,3(p-1))) are presented.(3) Families of asymmetric quantum codes from classical quadratic residue codes and Reed-Muller codes are presented. Furthermore, families of asymmetric quantum codes from classical generalized Reed-Solomon codes are constructed.(4) A method of constructing quantum error-correcting codes on the basis of classical quasi-cyclic codes is provided. We give conditions for quasi-cyclic codes that contain their duals, and obtain a large number of new quantum qausi-cyclic codes by CSS construction.(5) A unified approach to construct quantum error-correcting codes is presented by using quasi-twisted codes. A sufficient and necessary condition for quasi-twisted contained its dual codes and a new method for constructing quasi-twisted codes are given. Moreover, new quantum quasi-twisted codes are obtained by using quasi-twisted codes.(6) Generally, it is difficult to construct the generator matrices of stabilizer for large length N. We employ the recursive relations of matrices to solve this problem so that quantum error correcting codes with large length N can be constructed easily.