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

generates a code of length 55 over Z2[X]/(X^4) who´s minimum homogenous weight is 48.

Homogenous weight enumerator: w(x)=1x^0+67x^48+94x^52+96x^54+1536x^55+139x^56+32x^58+40x^60+32x^64+10x^68+1x^104

The gray image is a linear code over GF(2) with n=440, k=11 and d=192.
This code was found by Heurico 1.16 in 0.203 seconds.