## 40 Reputation

0 years, 195 days

## To Obtain the Column Rank of a Matrix Ov...

Maple 15

Consider the finite field G:=GF(p,k) for some prime p and a positive integer k. Let H be an nxm matrix over G.

My question: How to obtain the minimum number of linearly dependent columns of the H over G.

## The computation of the power of large bi...

Consider A is an nxn binary matrix where n>1000. Assume that there is an integer k<100 such that the entries of the kth power of A, denoted A^k, are all positive integer numbers.

My problem is that the computation of A^k takes too times and I want to ask: Is there some techniques in Maple for obtaining the matrix A^k.

One technique that I am used is based on the boolean semiring with 1+1=1. In fact, I evaluate A^k until I reach the full-1 matrix. But the mentioned technique dose not effect the time of computation.

For example consider the 1024*1024 binary matrix which is given as an attachment. It takes 2278 seconds to find out the entries of the 20th power of this matrix are all positive integer numbers.

I am used the following version of Maple 15.
Thanks for any suggestions.

Edition(1): The following command maybe useful. From the next command you can upload the given matrix in your worksheet and test it for my claim.

## Generating a Random Permutation in Maple...

Maple 15

I want to write a procedure P such that the input of P is a positive integer number n and the output of P is a random permutation of the numbers {1,2,..,n}.

I know that there is a command in Maple 2017 such that the command produces the mentioned request (random permutation in the combinatoric package), but I should work with Maple 15 and there is no the random permutation command in Maple 15.

One of the solutions that I am used is based on the random number and check that if the produced numbers are pairwise distinct or not. The problem of this method  is that for n>128, it takes too time to generate a random permutation of the length n.

Thanks for any suggestions.

## How to get an nxn binary matrix with som...

Maple 15

Let A=(a_{i,j}) be an nxn non-singular matrix over GF(2). Assume that we have a positive integer number s and an irreducible polynomial f of degree n over GF(2).

My question: How to get nxn binary matrices such as A provided that the characteristic polynomial  of these matrices over GF(2) is f and also sum(a_{i,j}) is equal to s with 1<=i<=n and 1<=j<=n.

For example, consider n=8 and s=10 and f= x^8+x^7+x^5+x+1. Then I applied the following Maple code to generate the mentioned matrices.

restart
with(LinearAlgebra):

randomize();
roll := rand(1 .. 64);
roll1 := rand(1 .. 8);
roll2 := rand(9 .. 16);
roll3 := rand(17 .. 24);
roll4 := rand(25 .. 32);
roll5 := rand(33 .. 40);
roll6 := rand(41 .. 48);
roll7 := rand(49 .. 56);
roll8 := rand(57 .. 64);

u := 1; while u > 0 do
L := [roll(), roll1(), roll2(), roll3(), roll4(), roll5(), roll6(), roll7(), roll8(), roll()];
if nops({L[]}) = 10 then
A := Matrix(8, 8, 0); s := 0;
for i to 8 do
for j to 8 do
s := (i-1)*8+j;
if evalb(in(s, L)) then A[i, j] := 1 end if
end do; end do;
if Determinant(A) <> 0 then
if evalb(in(sort(mod(Factor(CharacteristicPolynomial(A, x)), 2))[x^8+x^7+x^5+x+1]))
then print(A, sort(mod(Factor(CharacteristicPolynomial(A, x)), 2)));
u := 0 end if; end if;
unassign('A, s, i, j')
end if; end do;
unassign('u, i')

with my computer, it takes less than one minute to generate an 8x8 desired non-singular binary matrix as follows

\left[
\begin {array}{cccccccc}
0&0&1&0&0&0&0&0\\
0&1&1&0&0&0&0&0\\
0&0&0&1&0&0&0&0\\
0&1&0&0&0&0&1&0\\
1&0&0&0&0&0&0&0\\
0&0&0&0&1&0&0&0\\
0&0&0&0&0&0&0&1\\
0&0&0&0&0&1&0&0
\end {array}
\right]

I wish I could find a systematic method to find these kind of matrices.

Thanks for any suggestions

## How to define the smitform of a matrix ...

Maple 15

Suppose that A is an nxn matrix over the finite field Z:=GF(2,q) for some q. I wan to get the smitform of A over Z. First I used the package

with(LinearAlgebra[Generic])

and after that I applied the command

S := SmithForm[Z](A)

but the mentioned command made some errors. In fact, I do not how to define commands igcdex, iquo, irem, sign and abs for SmithForm over finite fields.

Thanks for any suggestions

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