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

generates a code of length 62 over Z2[X]/(X^4) who´s minimum homogenous weight is 57.

Homogenous weight enumerator: w(x)=1x^0+190x^57+205x^58+348x^59+383x^60+694x^61+688x^62+618x^63+318x^64+254x^65+126x^66+92x^67+49x^68+70x^69+19x^70+30x^71+1x^72+8x^73+1x^74+1x^102

The gray image is a linear code over GF(2) with n=496, k=12 and d=228.
This code was found by Heurico 1.16 in 3.55 seconds.