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  X  1  1  X  X  1  1  1  1  1  X  1  1  X  1  1  1  X  1  1  1  1
 0 X^3  0  0  0  0  0  0  0  0 X^3 X^3  0  0  0 X^3  0  0  0  0 X^3 X^3 X^3 X^3  0 X^3 X^3 X^3 X^3 X^3 X^3  0 X^3 X^3 X^3  0  0  0  0  0 X^3  0  0 X^3 X^3  0  0  0
 0  0 X^3  0  0  0  0  0  0 X^3 X^3 X^3  0 X^3 X^3  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3  0 X^3 X^3  0  0  0  0  0  0 X^3  0 X^3 X^3 X^3  0  0 X^3  0  0 X^3  0  0 X^3 X^3
 0  0  0 X^3  0  0  0  0  0 X^3  0 X^3 X^3 X^3  0 X^3  0 X^3 X^3  0  0  0  0  0 X^3  0 X^3 X^3  0  0  0 X^3 X^3 X^3  0 X^3 X^3 X^3 X^3  0 X^3 X^3  0  0 X^3 X^3 X^3 X^3
 0  0  0  0 X^3  0  0  0 X^3  0  0 X^3 X^3  0 X^3 X^3  0  0 X^3 X^3  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0  0  0 X^3  0  0 X^3 X^3  0 X^3 X^3 X^3 X^3  0  0  0  0  0
 0  0  0  0  0 X^3  0  0 X^3  0 X^3  0  0 X^3 X^3 X^3  0 X^3 X^3  0  0  0  0 X^3 X^3  0  0  0  0 X^3 X^3  0 X^3 X^3  0 X^3  0 X^3  0  0 X^3 X^3 X^3 X^3 X^3 X^3  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  0  0 X^3  0  0 X^3  0 X^3  0 X^3  0 X^3  0  0 X^3  0  0  0 X^3 X^3 X^3  0  0  0 X^3  0 X^3 X^3 X^3  0  0
 0  0  0  0  0  0  0 X^3 X^3 X^3  0  0 X^3 X^3 X^3  0 X^3  0 X^3  0  0 X^3 X^3  0  0  0 X^3  0  0 X^3  0  0 X^3 X^3 X^3 X^3  0 X^3 X^3 X^3  0 X^3  0  0  0  0  0  0

generates a code of length 48 over Z2[X]/(X^4) who´s minimum homogenous weight is 40.

Homogenous weight enumerator: w(x)=1x^0+37x^40+106x^44+72x^46+1663x^48+48x^50+59x^52+8x^54+39x^56+10x^60+4x^64+1x^84

The gray image is a linear code over GF(2) with n=384, k=11 and d=160.
This code was found by Heurico 1.16 in 0.157 seconds.