The generator matrix

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

generates a code of length 34 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 89.

Homogenous weight enumerator: w(x)=1x^0+456x^89+348x^90+516x^91+1149x^92+2196x^93+1404x^94+1536x^95+2256x^96+3720x^97+2484x^98+3252x^99+3414x^100+6060x^101+4404x^102+4716x^103+4680x^104+6492x^105+4032x^106+3192x^107+2757x^108+3600x^109+1152x^110+612x^111+561x^112+516x^113+24x^116+6x^120

The gray image is a linear code over GF(4) with n=136, k=8 and d=89.
This code was found by Heurico 1.16 in 8.19 seconds.