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

generates a code of length 59 over Z2[X]/(X^4) who´s minimum homogenous weight is 54.

Homogenous weight enumerator: w(x)=1x^0+142x^54+46x^56+192x^57+362x^58+640x^59+256x^60+192x^61+130x^62+16x^64+70x^66+1x^112

The gray image is a linear code over GF(2) with n=472, k=11 and d=216.
This code was found by Heurico 1.16 in 17.8 seconds.