The generator matrix

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

generates a code of length 44 over Z2[X]/(X^4) who´s minimum homogenous weight is 37.

Homogenous weight enumerator: w(x)=1x^0+98x^37+946x^38+2408x^39+5583x^40+9964x^41+14989x^42+19904x^43+22415x^44+21178x^45+15425x^46+9514x^47+5106x^48+2276x^49+919x^50+172x^51+141x^52+16x^53+9x^54+2x^55+2x^56+4x^57

The gray image is a linear code over GF(2) with n=352, k=17 and d=148.
This code was found by Heurico 1.16 in 84.1 seconds.