The Constructions Of Subspace Codes Based On Rank Distance Codes And Unitary Spaces

Subspace codes have became a hot research topic in the field of network coding because they have good error detection and error correction capability.Rank distance codes are a special class of error-correcting codes,and good subspace codes can be obtained by lifting rank distance codes.At the same time,various spaces in the geometry of classical groups over finite fields have been widely used in coding theory because of their good algebraic combinatorial structures.In this paper,we mainly study how to construct subspace codes using rank distance codes and unitary spaces over finite fields,with the aim of achieving the full application of group theory and finite geometry in coding and obtaining subspace codes with good performance,and we obtain two research results.Firstly,we construct cyclic unitary groups and abelian noncyclic unitary group by using the companion matrix of a primitive polynomial over finite fields and give several constructions of rank-distance cyclic orbit codes and rank-distance abelian noncyclic orbit code based on the constructed unitary groups,two maximum rank distance codes are obtained by the merge operation and the sum operation of rank distance codes.The constructed maximum rank distance codes are lifted in two different ways to construct new subspace codes,finally a spread code is obtained and a concrete example of a quadratic spread code is given.Secondly,one class of cyclic orbit code is constructed by using the set of subspaces of type(n,0)in unitary spaces Fq2(2n),calculating their relevant parameters and giving a new quadratic orbit code[6,63,4,3]4.Based on this code,we obtain orbit codes with large size using the external direct product of unitary groups acting on the direct sum of subspaces.Moreover,the constructed codes are compared with the existing constructions and have certain advantages.