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13 votes
0 answers
188 views

Why is $ULU=NU$ (a refinement of $|N|=q^{n^2-n}$)?

Let $G=GL_n(\mathbb{F}_q)$, $U$, $L$, $N$ the subsets of upper-triangular unipotent, lower-triangular unipotent, all unipotent matrices respectively. Then $ULU=NU$ means that for any $g\in G$ the ...
Anton Mellit's user avatar
  • 3,772
9 votes
0 answers
270 views

The number of non-singular $n\times n$ matrices over $\mathbb{F}_2$ with exactly $k$ non-zero entries

Suppose $M_{n}^{k}$ is the number of non-singular $n\times n$ matrices over $\mathbb{F}_2$, that have exactly $k$ non-zero entries. Is there some sort of formula to calculate $M_n^k$? If $k < n$ ...
Chain Markov's user avatar
  • 2,618
7 votes
2 answers
843 views

Dimension of incomplete matrix over finite fields.

Hi, Suppose one has an incompletely specified $2^n \times 2^n$ matrix over some fixed finite field $\mathbb{F}_{p^k}$. In fact, one knows that the diagonal entries are zero and all other entries are ...
Rob Myers's user avatar
  • 1,291
4 votes
1 answer
152 views

Enumerator Polynomials for Linear Anytime Codes

Let $C = \{c \in \mathbb{F}^n_2 : Hc=0\}$ be a binary linear code where $H \in \mathbb{F}^{k \times n}_2$ is a block lower-triangular matrix of full rank called the parity-check matrix of $C$. Clearly ...
Nikola Kovachki's user avatar
4 votes
1 answer
317 views

On the real and finite field rank of a $0/1$ matrix - I

Let $M\in\{-1,0,+1\}^{n\times n}$ be a matrix of rank $r$. Consider the matrix $f(M)\in\{0,+1\}^{mn\times mn}$ where $0$ in $M$ is replaced by $m\times m$ all $0$ matrix, $+1$ in $M$ is replaced by $m\...
Turbo's user avatar
  • 13.9k
2 votes
1 answer
308 views

The number of unitary circulant matrices over a finite field $\mathbb{F}_{q^2}$

I asked this question in MSE few days ago but there was no response. Suppose $\mathbb{F}=\mathbb{F}_{q^2}$, where $q$ is a prime power. The conjugate of elements in $\mathbb{F}$ is defined by $\...
Groups's user avatar
  • 379
2 votes
1 answer
368 views

The minimum rank of a matrix over GF(2) when part of non-zero off-diagonal elements are set to be zero

Given an $n\times n$ matrix $A$, whose elements are over $GF\left(2\right)$ and all diagonal elements are $1$. There are $m\ (m\leq n^2-n)$ non-zero off-diagonal elements in $A$. If we are allowed to ...
Tang's user avatar
  • 21
1 vote
0 answers
162 views

Number of full-rank binary matrices with given column Hamming weights [closed]

What is the number of $m \times n$ matrices over $\mbox{GF}(2)$ that share the following constraints : They have full rank ($\mbox{rank} = m$, given that $m<n$). Their columns have the given ...
Sapiens's user avatar
  • 111
0 votes
0 answers
54 views

Rank decomposition of matrices over $\mathbb F_2$

Given an integer matrix $M\in\mathbb Z^{n\times n}$ of real rank $k$ what is the minimum and maximum number of rank $1$ matrices $B_1$ to $B_t$ we require so that $M\equiv\sum_{i=1}^tB_i\bmod 2$? If $...
Turbo's user avatar
  • 13.9k
0 votes
0 answers
299 views

Question on rank of matrices over $\mathbb F_2$

$A$ is a square matrix in $\mathbb F_2^{n\times n}$ of rank $k\leq n-1$. $B$ is a square matrix in $\mathbb F_2^{n\times n}$ of rank $n$. $T$ is a square matrix in $\mathbb F_2^{n\times n}$ of rank $1$...
Turbo's user avatar
  • 13.9k