

A344674


a(n) is the maximum value such that there is an n X n binary orthogonal matrix with every row having at least a(n) ones.


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OFFSET

1,4


COMMENTS

The inverse of an orthogonal matrix is its transpose. This implies the dot product of a row with itself must be 1. This further implies the number of ones in each row must be odd. Given that orthogonal matrices form a group, it must be the case the transpose is also an orthogonal matrix. This requires every column of a binary orthogonal matrix also have an odd number of ones. As a result, there will always be an orthogonal matrix of size n X n having rows with n1 number of ones if n is an even number, namely an allones matrix except for zeros down the main diagonal. An n X n orthogonal matrix cannot exist with n1 ones in each row if n is odd, since n1 is even.
a(n) = n1 if n is even.
a(n) < n1 if n is odd.


LINKS



EXAMPLE

There exist 10 X 10 binary orthogonal matrices such that every row has at least 9 ones, but no 10 X 10 binary orthogonal matrix exists with 10 ones in each row, so a(10) = 9.
There exist 9 X 9 binary orthogonal matrices such that every row has at least 5 ones, but no 9 X 9 binary orthogonal matrix exists with 6 or more ones in each row, so a(9) = 5.


CROSSREFS



KEYWORD

nonn,hard,more


AUTHOR



EXTENSIONS



STATUS

approved



