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  1  1  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 X+a^2 a*X+a^2  0 X+1  1 X+1 X+1 a*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*X  X a*X+a a*X+a^2 a^2 a^2 X+1  a
 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  X a^2*X  0 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+408x^122+240x^123+285x^124+1152x^125+972x^126+576x^127+336x^128+1608x^129+1236x^130+504x^131+576x^132+1620x^133+1452x^134+504x^135+327x^136+1344x^137+900x^138+312x^139+231x^140+636x^141+408x^142+168x^143+18x^144+204x^145+6x^148+3x^152+6x^156+3x^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 3.59 seconds.