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  X  1  1  1  1  1  1  X  1  1  1 X^2  X  1  1 X^2  1  1  1  X  1 X^3  1 X^2
 0 X^3+X^2  0  0  0 X^2 X^3+X^2 X^2  0 X^3 X^3+X^2 X^3+X^2  0 X^3 X^2 X^2  0 X^3 X^3+X^2 X^3+X^2 X^3+X^2 X^3+X^2 X^3  0 X^2 X^3  0 X^3+X^2 X^2 X^3 X^3+X^2  0 X^2 X^3 X^3 X^3+X^2 X^2 X^2 X^3+X^2  0 X^3  0 X^2 X^3+X^2 X^3+X^2  0  0 X^3
 0  0 X^3+X^2  0 X^2 X^2 X^2 X^3  0 X^3 X^2 X^3+X^2 X^2 X^2  0  0  0 X^3+X^2 X^3 X^3 X^2 X^3+X^2  0 X^2 X^2  0 X^2 X^3 X^3+X^2 X^3 X^3+X^2 X^3+X^2 X^3 X^3  0 X^2 X^2  0 X^3 X^2 X^2 X^3  0 X^2  0 X^2  0 X^2
 0  0  0 X^3+X^2 X^2 X^3 X^3+X^2 X^3+X^2  0 X^3+X^2 X^3 X^3+X^2 X^3+X^2  0 X^2  0 X^3 X^3+X^2 X^2 X^3 X^3 X^2 X^3+X^2 X^3  0 X^3+X^2 X^3 X^3+X^2 X^3  0 X^3+X^2  0  0 X^3  0 X^3+X^2  0 X^3 X^2 X^3+X^2 X^2 X^2  0 X^2 X^3+X^2 X^3+X^2 X^3+X^2  0
 0  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3  0  0  0 X^3  0 X^3 X^3 X^3  0  0 X^3  0  0 X^3 X^3 X^3  0 X^3  0  0 X^3  0 X^3 X^3  0 X^3  0 X^3 X^3 X^3  0  0  0  0  0  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+246x^44+32x^45+256x^46+224x^47+573x^48+224x^49+256x^50+32x^51+172x^52+16x^56+14x^60+1x^64+1x^80

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 97.5 seconds.