The generator matrix

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

generates a code of length 93 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 278.

Homogenous weight enumerator: w(x)=1x^0+108x^278+96x^279+36x^282+3x^292+9x^296+3x^308

The gray image is a linear code over GF(4) with n=372, k=4 and d=278.
This code was found by Heurico 1.16 in 0.14 seconds.