The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
 0  X  0  0  0  X  0 a*X a*X  X a^2*X a^2*X  X a*X  X a^2*X  0  0  0
 0  0  X  0  0 a^2*X  X  0 a*X a*X a*X  X a*X  0 a^2*X  0 a^2*X  X  0
 0  0  0  X  0  X a^2*X a^2*X a*X  0 a*X  X  0  X a*X  X a*X  X a*X
 0  0  0  0  X a^2*X a*X  X a^2*X  X  0  X a*X  X a*X a^2*X a*X a*X a*X

generates a code of length 19 over F4[X]/(X^2) who´s minimum homogenous weight is 44.

Homogenous weight enumerator: w(x)=1x^0+54x^44+156x^48+174x^52+177x^56+3072x^57+177x^60+141x^64+120x^68+21x^72+3x^76

The gray image is a linear code over GF(4) with n=76, k=6 and d=44.
This code was found by Heurico 1.16 in 0.0383 seconds.