Invertible $N \times N$ matrix over ${\rm GF}(2)$ having on each row and column $N/2$ ones

Question

As per the title, I'm looking for the name and for a way to construct a ${\rm GF}(2)$ square matrix of size $N$ with the following properties:

All rows/columns should be linearly independent
On each row and each column there should be $N/2$ ones and zeros

Can anybody provide some pointers? It doesn't have to be a general solution (i.e. for all $N$) as long as it works for some $N$ it will be ok.

edit: It has been pointed out that if N is a multiple of 4 the two conditions above are incompatible. Let me relax number 2) by allowing, when N is a multiple of 4, the rows and columns to be (minimally) unbalanced.

edit2: Bonus points for pointing out a way to build matrices as the ones above, with the added constraint that the conditions above should be valid also for their inverse (obviously number 1 is implied).

This is http://math.stackexchange.com/questions/19480/dynamic-programming-a-type-of-balanced-0-1-matrix with an extra constraint, so that might be a useful starting point for algorithm development. — Peter Taylor, Jan 30 '11 at 16:53

score 4 · Accepted Answer · edited Dec 07 '11 at 18:33

4

It's not possible for $N$ divisible by 4, because the vectors with an even number of 1's form a proper subspace of ${\rm GF}(2)^N$. For $N=6$ there are many examples, such as

$$\begin{pmatrix}1&1&1&0&0&0\\1&1&0&1&0&0\\0&0&1&1&0&1\\1&0&0&0&1&1\\0&1&0&0&1&1\\ 0&0&1&1&1&0\end{pmatrix}$$

edited Dec 07 '11 at 18:33

J. M. ain't a mathematician

75,051

answered Jan 30 '11 at 11:40

Derek Holt

90,008

Is there a known algorithm to build such matrices for bigger values of N? – CAFxX Jan 30 '11 at 13:53

Invertible $N \times N$ matrix over ${\rm GF}(2)$ having on each row and column $N/2$ ones

1 Answers1