The generator matrix

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

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+45x^116+108x^118+60x^119+261x^120+168x^121+348x^122+396x^123+450x^124+372x^125+864x^126+492x^127+711x^128+720x^129+1512x^130+600x^131+1137x^132+840x^133+1884x^134+948x^135+861x^136+744x^137+1212x^138+444x^139+405x^140+228x^141+216x^142+132x^143+99x^144+48x^148+42x^152+21x^156+9x^160+6x^164

The gray image is a linear code over GF(4) with n=176, k=7 and d=116.
This code was found by Heurico 1.16 in 0.901 seconds.