The generator matrix

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

generates a code of length 44 over F4[X]/(X^2) who´s minimum homogenous weight is 116.

Homogenous weight enumerator: w(x)=1x^0+162x^116+120x^117+12x^118+468x^119+1290x^120+1320x^121+288x^122+1284x^123+2829x^124+2712x^125+468x^126+2604x^127+4782x^128+4428x^129+624x^130+4104x^131+6462x^132+5976x^133+900x^134+3852x^135+6192x^136+5136x^137+624x^138+2580x^139+3099x^140+1656x^141+156x^142+468x^143+651x^144+156x^145+54x^148+48x^152+18x^156+12x^160

The gray image is a linear code over GF(4) with n=176, k=8 and d=116.
This code was found by Heurico 1.16 in 11.9 seconds.