The generator matrix
1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 X 1 1 X 1 aX 1 1 1
0 1 1 a (a+1)X+a+1 0 (a+1)X+1 a X (a+1)X+a+1 1 X (a+1)X (a+1)X (a+1)X+1 aX+1 aX+1 X+a aX+a+1 aX+a 1 X+a aX+a+1 1 a+1 1 1 aX+a a+1
0 0 (a+1)X X aX X 0 (a+1)X aX (a+1)X (a+1)X X aX 0 (a+1)X 0 aX (a+1)X aX X 0 0 X X (a+1)X (a+1)X aX 0 X
generates a code of length 29 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 84.
Homogenous weight enumerator: w(x)=1x^0+129x^84+576x^85+60x^88+60x^92+192x^93+3x^96+3x^116
The gray image is a linear code over GF(4) with n=116, k=5 and d=84.
As d=84 is an upper bound for linear (116,5,4)-codes, this code is optimal over F4[X,sigma]/(X^2) for dimension 5.
This code was found by Heurico 1.16 in 0.031 seconds.