The generator matrix

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

generates a code of length 94 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 278.

Homogenous weight enumerator: w(x)=1x^0+216x^278+168x^279+18x^280+144x^282+144x^283+27x^284+216x^286+72x^287+6x^292+9x^296+3x^308

The gray image is a linear code over GF(4) with n=376, k=5 and d=278.
This code was found by Heurico 1.16 in 0.328 seconds.