Research On Algebraic Theory Of Constacyclic Error-correcting Codes And Their Applications In Information Security

As an interdisciplinary research field between mathematics and compter science, the theory of error-correcting codes is playing more and more significant role in either mathematics itself or information security. After the development for nearly seventy years, classical error-correcting codes over finite fields have been subjected to systematic and comprehensive studies in theory, and have been widely applied in engineering practice. With the further development of the theory of error-correcting codes, the theoretical significance and applicable value of error-correcting codes over finite rings have been recognized and known gradually. In recent years, the studies on error-correcting codes over finite rings have been one of the hottest issues in the theory of error-correcting codes. Among these studies, constacyclic codes and self-dual codes over finite rings have been paid much attention. Meanwhile, quantum communication and quantum computation have attracted many interests at the end of the twentieth century. Just as with digital communication, the theory of quantum error-correcting codes is one of the important guarantees of transmitting quantum information. In 1998,Calderbank et al. proposed the mathematical expression for quantum error-correcting codes and gave the first effective method to construct quantum error-correcting codes by using classical codes, which has greatly motivated the development of the construction of quantum error-correcting codes.Based on the previous study on the theory of error-correcting codes, this dissertation presents a further research on linear codes over finite rings, especially constacyclic codes over certain finite rings, and constructs some good and new quantum error-correcting codes from constacyclic codes over finite fields. The details are as follows:1. The class of (1 + w?) -constacyclic codes of an arbitrary length over a finite chain ring R is studied, where w is a unit of R and ? is a fixed generator of the maximal ideal of R. Firstly, the generator polynomials of all torsion codes of each(1+w?)-constacyclic code are obtained. The minimum Hamming distance of each such constacyclic code is determined, and upper and lower bounds for the minimum homogeneous distance of every such constacyclic code are established. Especially, the exact homogeneous distance of some such constacyclic codes is obtained. Secondly, by using the generator polynomials of all torsion codes, a lower bound for each codeword in each (1 + w?) -constacyclic code of any length over R is obtained. The depth spectrum of all such constacyclic codes is completely determined by using the lower bound. Thirdly, by using the highest torsion code of each such constacyclic code,(1 + wp)-constacyclic MDR codes over the Galois ring GR(pt,a) of length n are constructed, where w is a unit of GR(pt,a) and n is a divisor of pm -1 and pm+1.2. The constructions of self-dual codes over finite rings are explored. On one hand,by using the Chinese remainder theorem, a generator polynomial of any self-dual cyclic code over a finite chain ring is obtained. Through the generator polynomial, necessary and sufficient conditions for the existence of self-dual cyclic codes are given. By employing torsion codes and classical cyclic MDS codes, self-dual cyclic MDR codes of length n over the Galois ring GR(pt,m) are constructed, where n?2 is a divisor of pm - 1. On the other hand, some important properties of self-dual codes over the ring Z4 + vZ4 = Z4 [v]/<v2 - 1> are given. A Z4 -linear Gray map from (Z4 + v24)n to Z42n is introduced. It is proved that the Gray image of a self-dual code of length n over Z4+vZ4 is a self-dual code of length 2n over Z4. Some extremal or optimal Type I and Type II self-dual codes over Z4 are constructed.3. The Hermitian dual-containing codes are derived from ?q-1-constacyclic codes of length n = (q2m-1)/(q+1) over the finite field Fq2,where ? is a primitive element of Fq2 . Based on this, several classes of q -ary quantum stabilizer codes are constructed by applying the Hermitian construction. The parameters of these quantum codes are better than quantum codes constructed from classical BCH codes.