The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+85x^34+138x^35+198x^36+304x^37+400x^38+438x^39+491x^40+550x^41+512x^42+574x^43+640x^44+662x^45+635x^46+630x^47+524x^48+402x^49+344x^50+240x^51+165x^52+114x^53+65x^54+28x^55+28x^56+16x^57+7x^58+1x^76

The gray image is a code over GF(2) with n=88, k=13 and d=34.
This code was found by Heurico 1.10 in 2.2 seconds.