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  1  1  1  1  1  1  1  X  1  1  1  1  X  1  1  1  X  X  X  1
 0 X^3  0  0  0  0  0  0  0  0  0  0  0  0  0 X^3 X^3 X^3  0  0 X^3 X^3  0 X^3 X^3 X^3  0 X^3  0 X^3  0 X^3 X^3  0  0 X^3  0  0 X^3 X^3  0 X^3  0 X^3 X^3  0 X^3 X^3  0 X^3  0 X^3 X^3 X^3 X^3  0 X^3 X^3  0  0  0 X^3  0 X^3 X^3  0  0  0  0 X^3 X^3  0  0  0 X^3 X^3  0  0  0 X^3 X^3 X^3  0
 0  0 X^3  0  0  0  0  0  0  0  0  0  0  0 X^3 X^3 X^3  0  0 X^3 X^3  0 X^3  0 X^3  0 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 X^3  0 X^3  0  0  0 X^3 X^3 X^3  0 X^3 X^3 X^3  0  0  0 X^3  0 X^3  0 X^3  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0  0 X^3  0 X^3  0  0 X^3
 0  0  0 X^3  0  0  0  0  0  0  0  0  0 X^3  0 X^3 X^3  0 X^3 X^3  0 X^3 X^3 X^3 X^3  0  0  0 X^3 X^3 X^3 X^3  0 X^3  0 X^3 X^3  0  0 X^3 X^3  0 X^3  0 X^3  0 X^3  0  0  0  0  0 X^3 X^3 X^3 X^3  0  0 X^3 X^3  0 X^3 X^3  0  0 X^3  0  0 X^3  0 X^3  0 X^3 X^3  0  0  0  0 X^3  0  0 X^3 X^3
 0  0  0  0 X^3  0  0  0  0  0  0  0 X^3  0  0 X^3 X^3  0 X^3  0 X^3  0 X^3 X^3  0 X^3 X^3  0 X^3  0 X^3  0 X^3  0 X^3 X^3 X^3  0  0  0  0 X^3 X^3  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0  0  0 X^3 X^3  0 X^3 X^3 X^3 X^3  0  0  0  0  0 X^3  0 X^3 X^3  0 X^3  0  0 X^3  0  0 X^3
 0  0  0  0  0 X^3  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3  0 X^3 X^3  0  0 X^3  0 X^3  0 X^3  0  0 X^3  0  0  0  0  0 X^3 X^3  0 X^3  0  0  0 X^3 X^3  0 X^3  0 X^3 X^3 X^3  0  0  0 X^3 X^3 X^3  0  0 X^3  0 X^3  0  0 X^3  0 X^3  0 X^3 X^3  0  0 X^3 X^3  0 X^3 X^3  0  0 X^3  0  0 X^3 X^3  0  0
 0  0  0  0  0  0 X^3  0 X^3 X^3 X^3  0  0  0  0  0  0  0  0 X^3 X^3  0 X^3  0 X^3 X^3  0 X^3  0 X^3  0 X^3  0 X^3  0 X^3  0 X^3  0  0 X^3 X^3 X^3 X^3  0  0 X^3  0  0 X^3 X^3  0 X^3  0 X^3 X^3 X^3  0  0  0 X^3  0 X^3  0  0 X^3 X^3 X^3  0  0 X^3 X^3  0  0 X^3 X^3 X^3  0  0 X^3 X^3 X^3 X^3
 0  0  0  0  0  0  0 X^3 X^3  0 X^3 X^3  0  0  0  0  0  0 X^3 X^3  0 X^3 X^3 X^3 X^3  0 X^3 X^3  0  0  0  0  0 X^3 X^3  0 X^3 X^3  0 X^3  0 X^3  0  0  0  0 X^3  0 X^3  0 X^3 X^3  0 X^3  0  0  0 X^3 X^3 X^3  0 X^3 X^3  0 X^3  0 X^3  0  0  0 X^3  0 X^3  0 X^3 X^3  0 X^3 X^3 X^3  0  0 X^3

generates a code of length 83 over Z2[X]/(X^4) who´s minimum homogenous weight is 76.

Homogenous weight enumerator: w(x)=1x^0+65x^76+8x^78+102x^80+80x^82+1536x^83+127x^84+40x^86+48x^88+30x^92+9x^96+1x^100+1x^156

The gray image is a linear code over GF(2) with n=664, k=11 and d=304.
This code was found by Heurico 1.16 in 0.594 seconds.