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

generates a code of length 82 over Z2[X]/(X^4) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+43x^78+24x^79+188x^80+48x^81+446x^82+32x^83+171x^84+16x^85+27x^86+8x^87+15x^88+4x^90+1x^140

The gray image is a linear code over GF(2) with n=656, k=10 and d=312.
This code was found by Heurico 1.16 in 0.469 seconds.