On the Second Relative Greedy Weight of 4-Dimensional Codes

Abstract

Based on applications to the wire-tap channel of type II with the coset coding scheme, we will define optimal codes which are described as the ones achieving upper bounds on $e_{2}-M_{2}$ , where $e_{2}$ and $M_{2}$ are the second relative greedy weight and the second relative generalized Hamming weight, respectively. By using the finite geometry method, we will first classify all the 4-dimensional codes with respect to their $k_{1}$ -dimensional subcodes for $k_{1}=1$ and $k_{1}=2$ and determine the upper bounds on $e_{2}-M_{2}$ for each class of codes, and then obtain optimal codes by explicitly constructing the value assignments which achieve upper bounds on $e_{2}-M_{2}$ .