On the third relative greedy weight of 4-dimensional codes

Motivated by the concepts of the relative generalized Hamming weight and the greedy weight, the relative greedy weight is introduced, and then it is shown that the codes achieving the upper bounds on the relative greedy weight are optimal on the security of the transmitted data symbols in the wire-tap channel. Based on such applications, the finite geometry method is generalized, and by using the generalized finite geometry method, certain upper bounds on the third relative greedy weight of 4-dimensional codes with respect to 1-dimensional subcodes are first determined, and then optimal codes are constructed with respect to these obtained upper bounds.