The weight spectrum of certain affine Grassmann codes

Abstract

We consider the linear code corresponding to a special affine part of the Grassmannian ${G_{2,m}}$, which we denote by ${C^{\mathcal {A}}(2, m)}$. This affine part is the complement of the Schubert divisor of ${G_{2,m}}$. In view of this, we show that there is a projection of Grassmann code onto the affine Grassmann code which is also a linear isomorphism. This implies that the dimensions of Grassmann codes and affine Grassmann codes are equal. The projection gives a 1–1 correspondence between codewords of Grassmann codes and affine Grassmann codes. Using this isomorphism and the correspondence between codewords, we give a skew–symmetric matrix in some standard block form corresponding to every codeword of ${C^{\mathcal {A}}(2, m)}$. The weight of a codeword is given in terms of the rank of some blocks of this form and it is shown that the weight of every codeword is divisible by some power of q. We also count the number of skew–symmetric matrices in the block form to compute the weight spectrum of the affine Grassmann code ${C^{\mathcal {A}}(2, m)}$.