The generator matrix

 1  0  1  1  1  1  1  1  0  1  1  1  1  1  0  1  1  1  1  1  1  1
 0  1  1  a a+1  0 (a+1)X+1  a  1 (a+1)X+a+1 (a+1)X+1  0  a (a+1)X+a+1  1  a (a+1)X+a+1 X+a (a+1)X+1 (a+1)X+1 aX+a  0
 0  0 (a+1)X  0  0  X  0  X  X  X (a+1)X  X aX  0  0  X aX aX (a+1)X  0 aX  0
 0  0  0  X  0 (a+1)X (a+1)X (a+1)X  X aX (a+1)X aX aX aX aX  0 (a+1)X (a+1)X  0  0 aX  0
 0  0  0  0 (a+1)X  X (a+1)X aX (a+1)X (a+1)X (a+1)X aX  0  0 (a+1)X (a+1)X  0 (a+1)X aX (a+1)X  X  X

generates a code of length 22 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 52.

Homogenous weight enumerator: w(x)=1x^0+42x^52+24x^54+60x^55+294x^56+48x^57+288x^58+240x^59+516x^60+432x^61+1296x^62+840x^63+1098x^64+1296x^65+2592x^66+1200x^67+1275x^68+1296x^69+1944x^70+732x^71+684x^72+114x^76+51x^80+21x^84

The gray image is a linear code over GF(4) with n=88, k=7 and d=52.
This code was found by Heurico 1.16 in 0.368 seconds.