The generator matrix

 1  0  0  1  1  1  0  1  1  1  1  1  0  2  0  1  1  1  0  0  0  2  1  1  2  1  0  1  1  1  1  0  1  1  2  2  2  1  1  2  0  0  1  1
 0  1  0  1  0  1  1  0  0  1  3  2  1  1  0  2  1  3  0  0  1  1  1  3  2  3  1  1  3  3  1  1  2  2  1  2  1  2  2  1  1  1  2  0
 0  0  1  1  1  0  1  0  1  3  2  1  2  1  1  0  1  3  1  1  0  1  0  1  1  0  0  1  1  0  2  2  2  3  0  1  3  3  0  0  3  0  3  0
 0  0  0  2  0  0  0  0  0  2  0  0  0  0  0  0  2  2  2  2  0  0  0  2  0  0  0  2  2  2  2  2  0  0  2  2  2  0  2  0  2  0  0  0
 0  0  0  0  2  0  0  0  0  0  0  0  2  2  2  2  2  0  2  2  0  0  0  0  2  2  0  0  0  0  2  0  2  2  2  0  2  0  0  2  0  0  2  0
 0  0  0  0  0  2  0  0  0  0  2  0  0  0  0  0  0  0  2  2  0  0  2  0  0  2  2  2  2  2  2  0  2  2  2  0  2  0  2  2  2  2  2  0
 0  0  0  0  0  0  2  0  0  0  0  0  0  2  0  0  2  2  2  0  0  2  0  2  2  2  2  2  0  2  0  0  2  2  2  0  0  2  2  2  2  0  0  0
 0  0  0  0  0  0  0  2  0  2  2  2  0  0  2  0  2  0  0  0  2  0  0  2  2  0  2  0  0  2  0  0  0  2  2  0  2  2  2  0  2  0  2  0
 0  0  0  0  0  0  0  0  2  0  0  2  2  2  2  2  0  2  2  0  2  2  2  0  2  0  2  2  0  2  2  0  2  0  2  2  0  2  0  2  0  0  2  2

generates a code of length 44 over Z4 who´s minimum homogenous weight is 34.

Homogenous weight enumerator: w(x)=1x^0+45x^34+50x^35+132x^36+140x^37+174x^38+212x^39+199x^40+284x^41+304x^42+344x^43+340x^44+340x^45+286x^46+284x^47+256x^48+212x^49+165x^50+118x^51+76x^52+48x^53+42x^54+16x^55+15x^56+6x^58+4x^60+2x^62+1x^64

The gray image is a code over GF(2) with n=88, k=12 and d=34.
This code was found by Heurico 1.16 in 1.48 seconds.