The generator matrix

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

generates a code of length 44 over Z2[X]/(X^4) who´s minimum homogenous weight is 38.

Homogenous weight enumerator: w(x)=1x^0+35x^38+154x^39+166x^40+366x^41+383x^42+662x^43+617x^44+618x^45+444x^46+350x^47+91x^48+130x^49+30x^50+10x^51+15x^52+6x^53+2x^54+8x^55+4x^56+1x^58+2x^60+1x^62

The gray image is a linear code over GF(2) with n=352, k=12 and d=152.
This code was found by Heurico 1.16 in 0.282 seconds.