slide #4 lecture 6 'Recursive Bound on B(N,k)'

[skwong]

I have difficulty in understanding this. I tried to think about it by using
numerical example, say with the Perception, k = 4; and using N = 4 (0 = -1,
1 = +1).

Say, for alpha, I ignore x4 (don't care terms), then I can list all 8
combination of x1 to x3; when x4 is 0, then only 3 case works; and the
opposite is same. Therefore, I can come up with following:

x1 x2 x3 | x4

0 0 0 1
1 0 0 0
0 1 0 0
1 1 0 1
0 0 1 0
1 0 1 1
0 1 1 1
1 1 1 0

0 0 0 0
0 1 1 0
#1 0 1 0 (that should be left out)
1 1 0 0

1 1 1 1
1 0 0 1
#0 1 0 1 (that should be left out)
0 0 1 1

Then, I found that I mis-understood. Because:

1) The total is 14 (which matches with what was found in previous lecture),
but does not match with the B(N, k) calculation result of 15.

2) According to the numerical table in slide 8/18, the beta is 7, alpha
should be 1. I try to list them out like the above, but cannot know
which 7 combination to pick for beta rows, because I think I need to
kick out the 1010 and 0101 pattern. But then the 2 sections of beta
in the table will not be same for the x1 to x3.

Therefore, I actually do not understand it. Can you please enlighten me ?

Thanks in advance.

SK