The generator matrix

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

generates a code of length 15 over F4[X]/(X^2) who´s minimum homogenous weight is 44.

Homogenous weight enumerator: w(x)=1x^0+45x^44+192x^45+15x^48+3x^60

The gray image is a linear code over GF(4) with n=60, k=4 and d=44.
As d=44 is an upper bound for linear (60,4,4)-codes, this code is optimal over F4[X]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in -3.24e-008 seconds.