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

generates a code of length 78 over Z2[X]/(X^4) who´s minimum homogenous weight is 74.

Homogenous weight enumerator: w(x)=1x^0+204x^74+32x^75+402x^76+192x^77+464x^78+288x^79+214x^80+156x^82+86x^84+8x^86+1x^144

The gray image is a linear code over GF(2) with n=624, k=11 and d=296.
This code was found by Heurico 1.16 in 18 seconds.