The generator matrix

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

generates a code of length 20 over F4[X]/(X^2) who´s minimum homogenous weight is 44.

Homogenous weight enumerator: w(x)=1x^0+144x^44+12x^45+462x^48+180x^49+192x^51+2382x^52+1080x^53+1728x^55+8049x^56+3240x^57+5184x^59+16536x^60+4860x^61+5184x^63+12462x^64+2916x^65+534x^68+303x^72+84x^76+3x^80

The gray image is a linear code over GF(4) with n=80, k=8 and d=44.
This code was found by Heurico 1.16 in 4.43 seconds.