The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X  X  1  1  1  1  X  X  X  X  X
 0 2X  0  0  0  0  0 2X  0  0  0 2X 2X  0 2X 2X  0  0  0  0 2X 2X  0 2X  0  0 2X 2X 2X 2X 2X  0 2X  0
 0  0 2X  0  0  0  0 2X  0  0 2X 2X  0 2X 2X  0  0 2X  0 2X 2X 2X  0  0 2X 2X  0 2X 2X  0  0  0  0 2X
 0  0  0 2X  0  0  0 2X  0 2X 2X  0  0 2X  0 2X  0  0  0  0 2X 2X 2X 2X  0 2X 2X  0  0 2X  0 2X  0 2X
 0  0  0  0 2X  0  0 2X  0 2X  0  0 2X 2X 2X  0 2X  0 2X  0 2X 2X 2X  0 2X 2X 2X  0 2X  0 2X  0  0  0
 0  0  0  0  0 2X  0 2X 2X  0  0  0 2X 2X  0 2X 2X  0  0 2X  0 2X  0  0 2X  0 2X 2X 2X 2X  0 2X  0 2X
 0  0  0  0  0  0 2X 2X 2X  0 2X  0  0  0 2X 2X  0 2X 2X  0 2X  0 2X 2X 2X 2X  0  0  0  0 2X  0 2X  0

generates a code of length 34 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 29.

Homogenous weight enumerator: w(x)=1x^0+14x^29+38x^30+46x^32+70x^33+768x^34+42x^37+21x^38+14x^40+2x^41+4x^46+3x^48+1x^54

The gray image is a code over GF(2) with n=272, k=10 and d=116.
This code was found by Heurico 1.16 in 8.36 seconds.