The generator matrix

 1  0  0  1  1  1  1  1  1  1  0  1  1  1  1  1  1 (a+1)X (a+1)X  1 aX  1 aX  1  1  1  1  1  1  1  1  1  1  1  X  1  1  0  1  1 aX  1 aX  1
 0  1  0  1  a a+1 (a+1)X (a+1)X a+1 (a+1)X+a  1  a  0 (a+1)X+1 (a+1)X+a+1 (a+1)X+a (a+1)X+a+1  1  1 (a+1)X+1  X (a+1)X+1  1  1 X+1 X+a+1 (a+1)X+a (a+1)X+a aX+a+1 aX+a aX+a+1 X+1 aX+a+1 X+a  1 (a+1)X+a+1 a+1  1 a+1 X+a  1 (a+1)X  1  0
 0  0  1 a+1  a  1 a+1  1 X+a+1  1  a  0 X+a  X  X a+1  a  1 a+1 aX+a  1 aX+a+1 X+a aX+1  0  a aX+1  X  1 aX+1  X aX+a  0 aX X+a+1 aX+a+1  1  X X+1 (a+1)X+1 a+1 (a+1)X (a+1)X+1 (a+1)X
 0  0  0  X  0  X  0  0 (a+1)X aX aX (a+1)X (a+1)X  X  X (a+1)X  0 aX (a+1)X aX aX  X  0 (a+1)X (a+1)X (a+1)X  X  0 aX  X  0 (a+1)X  0 (a+1)X  X  X  0 aX  X (a+1)X  0  0  X aX
 0  0  0  0  X (a+1)X aX (a+1)X (a+1)X  X  X  0 (a+1)X  X aX  X  0  0 aX  X aX aX aX  X (a+1)X  X  0  0 aX (a+1)X (a+1)X  0 aX (a+1)X aX (a+1)X  0  0 aX  0  0  X  0 aX

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

Homogenous weight enumerator: w(x)=1x^0+147x^116+132x^117+168x^118+492x^119+1092x^120+1260x^121+216x^122+1596x^123+2829x^124+2748x^125+408x^126+2532x^127+4638x^128+4548x^129+612x^130+3288x^131+7002x^132+5868x^133+792x^134+4404x^135+6366x^136+5028x^137+576x^138+2508x^139+3009x^140+1812x^141+264x^142+540x^143+390x^144+108x^145+36x^146+48x^148+42x^152+18x^156+15x^160+3x^164

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 12.5 seconds.