The generator matrix

 1  0  1  1  1 3X+2  1  1 2X  1  1 X+2  1  2  1  1  X  1 2X+2 3X  1  1  1  1  1  1  1  1  X  1  2  0  1  2
 0  1 X+1 3X+2 2X+3  1 2X+1 2X  1 3X+3 X+2  1 2X+2  1  3 3X  1 X+1  1  1  2 X+3  X  1  0  0 3X+1 2X+2 3X 3X  1 2X+2 2X+1  1
 0  0 2X+2  0 2X  0 2X 2X+2 2X+2  2  2  2 2X+2  0  2 2X 2X+2  0 2X+2 2X  0 2X 2X+2  2  0 2X+2 2X+2 2X  2 2X 2X  2 2X+2 2X
 0  0  0 2X 2X 2X  0 2X  0 2X  0 2X  0 2X  0  0  0 2X 2X  0 2X  0 2X 2X 2X  0 2X  0  0 2X  0 2X  0 2X

generates a code of length 34 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 31.

Homogenous weight enumerator: w(x)=1x^0+234x^31+292x^32+368x^33+353x^34+340x^35+200x^36+168x^37+46x^38+34x^39+1x^40+8x^41+1x^42+2x^44

The gray image is a code over GF(2) with n=272, k=11 and d=124.
This code was found by Heurico 1.16 in 134 seconds.