The generator matrix

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

generates a code of length 44 over F4[X]/(X^2) who´s minimum homogenous weight is 121.

Homogenous weight enumerator: w(x)=1x^0+348x^121+336x^122+276x^123+297x^124+1176x^125+1140x^126+516x^127+315x^128+1596x^129+1200x^130+552x^131+540x^132+1536x^133+1320x^134+408x^135+399x^136+1380x^137+960x^138+420x^139+213x^140+744x^141+420x^142+132x^143+12x^144+132x^145+6x^148+3x^152+6x^160

The gray image is a linear code over GF(4) with n=176, k=7 and d=121.
This code was found by Heurico 1.16 in 1.27 seconds.