The generator matrix

 1  0  1  1  1 X^3+X^2+X  1  X  1 X^3+X^2  1  1  1  1 X^3  1 X^3+X^2+X  1  1  1 X^2+X  1 X^2  1 X^3+X  1  1  1  1  1 X^2  1  1  1  1  1  1  1  1  1 X^3+X  1  1  1 X^2+X  1  X  0
 0  1 X+1 X^2+X X^3+X^2+1  1 X^3+X^2  1 X^2+X+1  1 X^3+X^2+X X^2+1  X X^3+1  1  0  1 X^3+X+1 X^3+X  1  1 X^3  1 X^2+1  1 X^2 X^3+1 X+1 X^3+X^2+X+1 X^3+X  1  1 X^3+1 X^2+X+1 X^3+X+1 X^3+X^2+X+1 X^3+X+1 X^2+X+1 X^3+X+1 X^3  1 X^3+1 X^3+1  0  1 X^2+X+1 X^2+X  0
 0  0 X^2  0 X^3+X^2 X^2  0 X^2 X^3+X^2 X^3 X^2  0 X^3+X^2 X^3 X^3+X^2 X^3 X^3+X^2 X^3+X^2 X^3 X^2  0 X^2 X^2  0 X^3 X^3+X^2 X^3  0  0  0 X^3+X^2 X^2 X^2 X^2 X^3 X^3 X^3 X^3 X^2 X^2  0 X^3+X^2 X^3+X^2 X^2  0 X^2 X^2 X^2
 0  0  0 X^3  0  0  0  0 X^3  0  0 X^3  0 X^3 X^3 X^3 X^3 X^3  0 X^3 X^3 X^3  0  0 X^3 X^3  0  0 X^3  0  0  0 X^3 X^3  0  0 X^3 X^3  0  0  0 X^3 X^3  0  0  0 X^3 X^3
 0  0  0  0 X^3  0 X^3 X^3  0 X^3 X^3 X^3  0  0 X^3 X^3  0 X^3 X^3 X^3  0  0  0 X^3 X^3 X^3  0  0 X^3  0 X^3  0  0 X^3 X^3  0  0 X^3 X^3  0  0 X^3  0 X^3 X^3  0  0 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+393x^44+104x^45+740x^46+480x^47+865x^48+272x^49+696x^50+160x^51+324x^52+8x^53+36x^54+6x^56+9x^60+2x^68

The gray image is a linear code over GF(2) with n=384, k=12 and d=176.
This code was found by Heurico 1.16 in 28 seconds.