Projective systems and perfect codes with a poset metric

The notion of a projective system, defined as a set X of n-points in a projective space over a finite field, was introduced by Tsfasman and Vl dut. By this notion, the weight distribution of a nondegenerate linear code can be computed by the configuration of X and hyperplanes in the dual projective space of C. In this paper, we construct a projective system X defined as a set of n-linear subspaces on the dual projective space of a poset code. This gives a natural one-to-one correspondence between the set of equivalence classes of nondegenerate poset projective systems and the set of nondegenerate poset codes. By this correspondence, the weight distribution of a nondegenerate poset code can be also computed by the configuration of X and hyperplanes in the dual projective space of C. Finally, we provide an algorithm for finding perfect poset codes.