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

generates a code of length 68 over Z2[X]/(X^4) who´s minimum homogenous weight is 62.

Homogenous weight enumerator: w(x)=1x^0+103x^62+124x^63+278x^64+316x^65+398x^66+704x^67+356x^68+736x^69+310x^70+308x^71+208x^72+100x^73+70x^74+16x^75+34x^76+12x^78+13x^80+2x^82+3x^84+1x^86+2x^88+1x^108

The gray image is a linear code over GF(2) with n=544, k=12 and d=248.
This code was found by Heurico 1.16 in 0.734 seconds.