Constructions Of Cyclic Subspace Codes Over Finite Fields

Let q be a prime power and N a positive integer.Let F be a finite field with q elementsand FqN an extension of Fq.Let FqN denote the vector space of dimension N over Fqwhere N=nd,n and d are positive integers.In this thesis,firstly,by using linearized polynomials over finite fields,we construct some cyclic subspace codes.Then,we get more constructions of cyclic subspace codes via Sidon spaces.Detailed works and results as follows:From chapter three,it is obvious that:We consider two linearized polynomials(?),where ?1,?2,?1 and ?2?Fqn*andl,s are positive integers relatively prime to k and 1 ?s<k,1?l<k.Let U and V be the kernels of T1(x)and T2(x),respectively.Under some conditions,we can prove that(?)is a cyclic subspace code with size of 2qN-1/q-1 andminimum distance 2k-2.By the same methods,we will find that it is possible to add morecodewords into the cyclic subspace codes and increase the code size from 2qN-1/q-1 up to(r+t)qN-1/q-1 without compromising minimum distance 2k-2.In chapter four,Employing Sidon spaces we give second construction of cyclic subspace codes.Let(?),we prove that it is a Sidon space,where ??Fqn.Then,using these Sidon spaces,we construct cyclic subspace codes with size ?qN-1/q-1 andminimum distance 2k-2.The fact that two cyclic subspace codes can be joined into one subspace code without compromising minimum distance enable the construction of a cyclicsubspace code whose cardinality is 2?qN-1/q-1.