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

generates a code of length 78 over Z2[X]/(X^4) who´s minimum homogenous weight is 72.

Homogenous weight enumerator: w(x)=1x^0+61x^72+119x^74+80x^75+242x^76+304x^77+490x^78+304x^79+208x^80+80x^81+68x^82+55x^84+22x^86+6x^88+5x^90+2x^92+1x^140

The gray image is a linear code over GF(2) with n=624, k=11 and d=288.
This code was found by Heurico 1.16 in 0.625 seconds.