The generator matrix

 1  0  1  1  1  1  1  1  1  X  1  1  1  1 a*X  1  1  1  1 a^2*X  1  1
 0  1 a^2*X+1  a a^2*X+a^2  X a*X+1 X+a a*X+a^2  1 a*X X+1 a*X+a X+a^2  1 a^2*X  1 a^2*X+a a^2  1  0  X

generates a code of length 22 over F4[X]/(X^2) who´s minimum homogenous weight is 64.

Homogenous weight enumerator: w(x)=1x^0+27x^64+192x^66+24x^68+12x^72

The gray image is a linear code over GF(4) with n=88, k=4 and d=64.
As d=64 is an upper bound for linear (88,4,4)-codes, this code is optimal over F4[X]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in -3.62e-008 seconds.