Research Of The 2-ary Constant Dimension Code C(n,4,3)based On Grassmannian Spaces

Sub space codes are particular class of error-correcting codes in network coding.Subspace codes are great different from the traditional error-correcting codes because that every code word of a subspace code is a whole space.Besides,the subspace distance is a metric for the capacity in error-checking and error-correcting.Constant dimension codes are different from other error-correcting codes in that the dimensions of all code words are equal and the distance of any two different code words is always even.What makes people pay more attention to constant dimension codes is that the constant dimension codes correspond to the vertex set of Grassman graph.How to get the maximum of code words in constant dimension codes when the four parameters n,k,d and q are fixed is the key point.The main problem of constant dimension codes C(n,4,3)is investigating performance bound in Grassmannian spaces,and then showing the necessary conditions of the optimal constant dimension codes C(n,4,3)and C(n,4,4).First of all,the Singleton type bound,Wang-Xing-Safavi-Naini bound and Johnson type bound Ⅱ are all the upper bound of constant dimension codes.While,the code words of constant dimension codes C(n,2,k)and C(n,2(k-1),k)are constructed by finite geometry and finite fields theory.Compared with other lower bounds,the lower bound from this paper is more optimal.Secondly,the existence conditions of Steiner structure are used to discuss the existence of constant dimension codes.The necessary conditions of the existence of q-ary constant dimension codes C(n,4,3),C(n,4,4)and the necessary and sufficient condition of existence of C(n,2k,k)are important to study sub space codes.At last,the sub space codes with fixed parameters n,d are stated by analyzing the dimension distribution in projective space such as C(4,3),C(4,2),C(5,2).