Research On LCD Codes And Constacyclic Codes Over Finite Fields And Their Theory Application

With the continuous development and application of classical communication technology,the theory of classical error-correcting codes over finite fields is becoming more complete.As we all known,classical error-correcting codes are an important guarantee for traditional communications.Similarly,quantum error-correcting codes are a necessary prerequisite for reliable quantum computing and quantum communication.Quantum error-correcting codes and classical error-correcting codes are essentially different,but they are also inextricably linked.With the in-depth study of quantum error-correcting codes over finite fields,quantum synchronizable codes and entanglement-assisted quantum codes have also attracted much attention.Firstly,the equivalence,existence and enumeration of LCD(Linear complementar-y dual)codes over finite fields are studied.Secondly,we investigate a class of cyclic codes over finite fields.Thirdly,a series of dual-containing cyclic codes with parameters reaching the Code Tables are constructed,and a large number of new quantum codes and new quantum synchronizable codes are obtained by them.Finally,three classes of constacyclic MDS(Maximum distance separable)codes are obtained and we construct three families of entanglement-assisted quantum MDS codes.The specific research content is as follows:(1)We show that the LCD codes are not equivalent to non-LCD codes over finite fields(GF(2),GF(3)and GF(4)).Based on the Griesmer bound of linear codes,the bounds of LCD codes with given dimension and lengths over finite fields are presented.We study the existence of LCD codes with achieving the bound and the enumeration of optimal LCD codes in the equivalent sense.The maximum value of the minimum distance of the LCD codes for given length and dimension over GF(3)and GF(4)is determined.(2)We study the cyclic codes of length n=2p~e over finite fields.The enumeration of the cyclic codes of length n is given by classifying the q-cyclotomic coset.We study the properties of its dual codes and the dimension of the intersection between the cyclic codes and its dual codes.The enumeration of the cyclic codes is obtained with hull of given dimension.A new class of generalized cyclotomic classes of order two is given,then we construct some optimal cyclic codes of length n=2pe.(3)Using the cyclotomic classes of order r,a large number of optimal or almost optimal dual-containing cyclic codes are obtained,whose length is prime and n?r+1(mod 2r).Some new quantum codes and new quantum synchronizable codes are constructed by them.Because the cyclic codes used are generally optimal or almost optimal,the quantum synchronizable codes obtained generally have good error correction ability.(4)We construct three classes of constacyclic MDS codes by studying its defining sets.We decompose the defining set,and determine the number of entangled state c.Three classes of entanglement-assisted quantum MDS codes are constructed.Compared with known entanglement-assisted quantum codes,the entanglement-assisted quantum MDS codes obtained have more flexible parameters.