The generator matrix

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

generates a code of length 89 over F4[X]/(X^2) who´s minimum homogenous weight is 260.

Homogenous weight enumerator: w(x)=1x^0+69x^260+258x^264+627x^268+30x^272+15x^276+6x^280+9x^284+6x^288+3x^352

The gray image is a linear code over GF(4) with n=356, k=5 and d=260.
This code was found by Heurico 1.16 in 0.125 seconds.