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

generates a code of length 87 over Z2[X]/(X^4) who´s minimum homogenous weight is 84.

Homogenous weight enumerator: w(x)=1x^0+121x^84+224x^86+256x^87+378x^88+43x^92+1x^168

The gray image is a linear code over GF(2) with n=696, k=10 and d=336.
This code was found by Heurico 1.16 in 96.7 seconds.