Coding Theory : Inequivalent design.

[dreamer.ande]

Construct two inequivalent 2-(16,6,2) designs. Show that they are inequivalent by proving that one cannot obtained from the other by any permutations of the rows or columns of the incidence array or by showing that the codes produced by the span of the rows have different parameters.

I have construct 2-(16,6,2) design. How to show they are inequivalent. Can someone helps?
A = (0 0 1 1) C = (0 1 1 0)
(0 0 1 1) (1 0 0 1)
(1 1 0 0) (1 0 0 1)
(1 1 0 0) (0 1 1 0)

B = (0 1 0 1) D = (0 0 0 0)
(1 0 1 0) (0 0 0 0)
(0 1 0 1) (0 0 0 0)
(1 0 1 0) (0 0 0 0)


First Design
M = ( A B C D )
( B A D C )
( C D A B )
( D C B A )

E = neg(A)
Second Design
N = (E B C D)
( B A D C )
( C D A B )
( D C B A )

How to show M and N are inequavient.

 

[lippyka]

To show that two designs are inequivalent, you need to prove that one cannot be obtained from the other by any permutations of the rows or columns of the incidence array or by showing that the codes produced by the span of the rows have different parameters. In this case, we can see that the two designs have different columns, so they cannot be obtained from each other by any permutations of the columns. To show that they produce codes with different parameters, we can compute the ranks of the matrices M and N, which are 8 and 7 respectively. This shows that the codes produced by the span of the rows have different parameters, thus proving that the two designs are inequivalent.