The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+126x^34+413x^36+801x^38+965x^40+1116x^42+1299x^44+1219x^46+998x^48+690x^50+399x^52+139x^54+19x^56+4x^58+1x^60+1x^62+1x^80

The gray image is a code over GF(2) with n=88, k=13 and d=34.
This code was found by Heurico 1.16 in 6.35 seconds.