The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 X^2  1  0  0 X^3+X^2  1  1
 0  X X^3+X^2 X^2+X  0 X^2+X X^3+X^2 X^3+X X^2+X  0 X^3+X X^3+X^2 X^3 X^3+X X^2 X^3+X^2+X  0 X^2+X X^3+X^2 X^3+X  0 X^2+X X^3 X^2+X X^3+X X^3+X^2 X^3+X^2 X^3+X  0 X^3+X^2+X X^2  X  0 X^3 X^2+X X^3+X^2+X X^2 X^2  X X^3+X X^3+X X^3+X^2  X  X X^2  X X^3+X^2 X^3+X^2
 0  0 X^3  0  0  0 X^3  0 X^3  0 X^3  0  0 X^3  0 X^3 X^3  0 X^3  0 X^3 X^3  0  0 X^3  0 X^3 X^3 X^3 X^3 X^3  0  0 X^3  0  0 X^3  0 X^3  0 X^3  0 X^3  0  0 X^3 X^3 X^3
 0  0  0 X^3  0  0  0  0  0  0  0 X^3  0  0 X^3 X^3  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0 X^3 X^3 X^3  0 X^3 X^3  0 X^3 X^3  0  0 X^3 X^3  0 X^3 X^3  0  0  0  0  0  0  0
 0  0  0  0 X^3  0 X^3 X^3 X^3  0  0  0 X^3 X^3 X^3 X^3 X^3  0 X^3 X^3  0  0  0 X^3  0 X^3 X^3 X^3  0  0  0 X^3 X^3  0  0 X^3  0  0 X^3 X^3 X^3  0  0  0  0  0  0 X^3
 0  0  0  0  0 X^3  0 X^3 X^3 X^3 X^3 X^3 X^3  0 X^3  0 X^3 X^3 X^3 X^3 X^3 X^3  0  0  0  0  0 X^3 X^3  0  0  0 X^3  0  0  0 X^3  0 X^3  0  0  0  0  0 X^3  0 X^3 X^3

generates a code of length 48 over Z2[X]/(X^4) who´s minimum homogenous weight is 44.

Homogenous weight enumerator: w(x)=1x^0+211x^44+296x^46+256x^47+564x^48+256x^49+240x^50+201x^52+8x^54+14x^56+1x^88

The gray image is a linear code over GF(2) with n=384, k=11 and d=176.
This code was found by Heurico 1.16 in 25.6 seconds.