On the Second Relative Greedy Weight of 4-Dimensional Codes

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
.