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

generates a code of length 81 over Z2[X]/(X^4) who´s minimum homogenous weight is 77.

Homogenous weight enumerator: w(x)=1x^0+20x^77+99x^78+138x^79+163x^80+234x^81+157x^82+76x^83+64x^84+32x^85+23x^86+10x^87+4x^88+2x^89+1x^130

The gray image is a linear code over GF(2) with n=648, k=10 and d=308.
This code was found by Heurico 1.16 in 0.453 seconds.