The generator matrix

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

generates a code of length 21 over F4[X]/(X^2) who´s minimum homogenous weight is 44.

Homogenous weight enumerator: w(x)=1x^0+57x^44+300x^48+96x^51+1008x^52+192x^54+1152x^55+2964x^56+1728x^58+5184x^59+6426x^60+5184x^62+10368x^63+9447x^64+5184x^66+7776x^67+7632x^68+540x^72+213x^76+60x^80+24x^84

The gray image is a linear code over GF(4) with n=84, k=8 and d=44.
This code was found by Heurico 1.16 in 4.41 seconds.