All Questions
37 questions
14
votes
2
answers
655
views
Number triangle
This question arose just out of curiosity. Note the triangle of 0-1's below, whose construction is as follows. Choose any number, say 53 as done here. The first line of the triangle is the binary ...
- 1,216
13
votes
1
answer
468
views
Near-linear mappings from $\mathbb F_p$ to $\mathbb R$
$\newcommand{\F}{{\mathbb F}}$
$\newcommand{\R}{{\mathbb R}}$
$\renewcommand{\phi}{\varphi}$
Let $p\ge 5$ be a prime.
If the functions $\phi_1,\phi_2,\phi_3\colon\F_p\to\R$ satisfy $\phi_1(x)+\...
- 23k
11
votes
2
answers
387
views
Bounds for the difference in the number of ones in $M$ and $M^{-1}$
If $M$ is a full rank $n$ by $n$ binary matrix over $\mathbb{F}_2$, how much larger or smaller can the number of $1$s in $M^{-1}$ be, compared to the number of $1$s in $M$?
Clearly the identity matrix ...
- 3,377
10
votes
1
answer
807
views
How many Lie and associative algebras over a finite field are there?
This question is related to the following general question:
Given a variety of (non-associative) algebras $\mathcal V$, a finite field $\mathbb{F}_q$, with $q$ elements, and a positive integer $n$, ...
- 398
9
votes
1
answer
543
views
On the number of polynomials that divide $x^{q-1}-1$ in some subspaces of $\mathbb F_q[x]$
Let $\mathbb F_q$ be a finite field (where $q$ is in general a power of a prime), and let $e, k$ be positive integers with $k \leq e < q-1$. Let $f_0(x), \ldots,
f_k(x) \in \mathbb F_q[x]$ be ...
- 307
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 $k < n$ ...
- 2,618
8
votes
1
answer
531
views
How large can the dimension of a 'Span of powers of a finite field basis' be?
Let $p$ be a prime. For finite field $\mathbb{F}_{p^k}$ and $d\in\mathbb{Z}^+$, I am considering the following quantity, where we interpret the field $\mathbb{F}_{p^k}$ also as a $\mathbb{F}_p$-vector ...
- 89
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 ...
- 1,271
7
votes
2
answers
589
views
On a matrix problem in the field $\mathbb F_2$
Given $M$ a symmetric matrix in $\mathbb F_2^{n\times n}$ having $\mathsf{det}_\mathbb R(M)\neq0$ (non-singular in reals) and satisfying $PMP'=(M+J+I)$ or $P(M+J+I)P'=M$ where $P$ is a permutation ...
- 13.9k
7
votes
1
answer
253
views
Enumerating subspaces of $\mathbb{F}_q^n$ in terms of words and inversions
When $q$ is a prime power, then on the one hand the $q$-binomial coefficient $\binom{n}{k}_q$ equals the number of $k$-dimensional subspaces of $\mathbb{F}_q^n$, and on the other hand it is the ...
- 63.1k
7
votes
2
answers
440
views
How to count the number of tensors over a finite field of tensor rank $r$?
For simplicity, work over $\mathbb F_2$ and only consider order-$3$ equilateral tensors. For $r\in\mathbb Z_{>0}$, how many tensors $\mathfrak{T}\in\mathsf{Ten}_{n}^{\otimes 3}(\mathbb F_2)$ are ...
- 161
7
votes
0
answers
300
views
Arrangement of subspaces over finite fields
I'm trying to find out what is already known about the following setup.
Let $V$ be an $n$-dimensional vector space over a finite field $F_q$ (I'm mostly interested in the case where $q$ is prime), and ...
- 1,062
6
votes
1
answer
457
views
Vector with many non-zero coordinates
Given finite field $\mathbb{F}_q$, positive integers $n$ and $k<n$. Given $k$-dimensional subspace $X$ of $\mathbb{F}_q^n$, for which $m=m(q,k,n)$ may we find for sure a vector in $X$ with at least ...
- 109k
6
votes
1
answer
297
views
Covering the finite plane with lines
This is, essentially, a geometrically rendered version of the question I asked a week ago, with the emphases slightly shifted; it seems more natural and appealing (to me, at least) in this form.
Let ...
- 23k
5
votes
1
answer
394
views
Disjoint union of affine subspaces contains a larger affine subspace
I'd like to say that a large structured subset of the $n$-dimensional Boolean cube $\{0,1\}^n$ contains a non-trivial affine subspace. To be more specific, I want to prove/disprove that for some ...
- 211
4
votes
1
answer
237
views
Intersection of the general linear group $\text{GL}_n(\mathbb{F}_q)$ and an affine subspace over a finite field
Let $\mathbb{F}_q$ be a finite field of $q$ elements, let $\text{M}_n(\mathbb{F}_q)$ be the vector space of $n \times n$ matrices over $\mathbb{F}_q$, let $\text{GL}_n(\mathbb{F}_q)$ be the group of $...
- 43
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\...
- 13.9k
3
votes
2
answers
303
views
Asymptotics of A000613
The general linear group $GL_n(\mathbb{F}_2)$ acts on the powerset $2^{{\mathbb{F}_2}^n \setminus \{0\}}$ by multiplication: $A \cdot S := \{Ax \in {\mathbb{F}_2}^n : \, x \in S\}$, for an invertible ...
- 331
3
votes
3
answers
611
views
On MDS code property
Is there a code that is Maximum Distance Separable and not isomorphic to Reed Solomon Codes? When is a MDS code isomorphic to Reed Solomon Code?
Is there an easy test? If so, could someone provide ...
- 13.9k
3
votes
1
answer
427
views
Ranks of higher incidence matrices of designs
In 1978 Doyen, Hubaut and Vandensavel proved that if $S$ is a Steiner triple system $S(2,3,v)$ then the $GF(2)$ rank of its incidence matrix $N$ is
$$
Rk_{2}(N)=v-(d_{p}+1),
$$
where $d_{p}$ is the ...
- 6,962
3
votes
0
answers
116
views
Trace of Symmetric matrices in fixed rank
I am solving some problem related to symmetric matrices over a finite field $\mathbb{F}_q$ and I am stuck at the following problem:
For every $a\in\mathbb{F}_q $, let $S_a(t,m)$ be the set of all $m\...
- 179
2
votes
1
answer
612
views
Inner product over finite field
sorry for informals but is my first post.
In Coding theory (exactly in Coding theory a first course - San ling and Chaoping) found this definition:
$\langle , \rangle :\mathbb{F}_q^n\times\mathbb{F}_q^...
2
votes
0
answers
111
views
Inseparable field extensions of degree p and linear independence
Let $F$ be a field of characteristic $p$; let $\alpha \in F$ such that $\alpha \neq \beta^p$ for any $\beta \in F$, and let $K := F(x)$ where $x=\sqrt[p]{\alpha}$.
Is it true that the elements $1,(x-...
- 21
2
votes
0
answers
113
views
Complexity of tensor decomposition vector over $\Bbb F_q$ or $\Bbb Z$
Suppose we have a matrix $$T\in\Bbb K^{n^k\times m}$$ and a target vector $v\in\Bbb F_q^m$ where $m<n^k$ and $1<k$ holds.
We need to find $k$ vectors $u_1,\dots,u_k\in\Bbb K^n$ such that $$v=...
- 13.9k
1
vote
1
answer
391
views
System of equations - Proof that a solution exists
Let $ a = (a_1,a_2, \ldots,a_{10})\in \{ 0,1\}^{10}$ be a binary vector of length $10$.
Question: Without using a computer-aided method, how to prove that there exists binary vectors $x_{i,j} \in \{ ...
user83947
1
vote
1
answer
153
views
Schur complement and depermuting an algorithm for $\mathsf{determinant}\bmod2$
Let $$M=\begin{bmatrix}A&B\\C&D\end{bmatrix}$$ be a matrix in $\mathbb F_2^{n\times n}$ where $A\in\mathbb F_2$ and $D\in\mathbb F_2^{(n-1)\times(n-1)}$ are square.
Through the determinant ...
- 13.9k
1
vote
1
answer
145
views
How many matrices $C \in \mathrm{M}_3(\mathbb{F}_q)$ such that $\det(C-A)=\det(C-B) = 1$?
I am studying the special unit-graph $G$ on $M_3(\mathbb{F}_q).$ Now, I want to estimates the upper bound for the second largest eigenvalue of adjacency matric of $G.$ One of questions that I need is ...
user148117
1
vote
0
answers
162
views
Number of full-rank binary matrices with given column Hamming weights [closed]
What is the number of $m \times n$ matrices over $\mbox{GF}(2)$ that share the following constraints :
They have full rank ($\mbox{rank} = m$, given that $m<n$).
Their columns have the given ...
- 111
1
vote
0
answers
150
views
Counting non-zero Gramians of Grassmanians over finite field
In case of $\mathbb{F}_{2}$, we can obtain the number of all reduced row echelon forms (so called Grassmannians) for some m$\times$n full rank matrices by the following gaussian polynomial,
$$
\binom{...
- 11
1
vote
0
answers
88
views
On the real and finite field rank of a $0/1$ matrix - II
Let $M\in\{-\ell,\dots,-1,0,+1,\dots,+\ell\}^{n\times n}$ be a matrix of rank $r$ where $\ell\geq1$ such that there is a permutation matrix in $\{0,1\}^{m\times m}$ of order $2\ell$.
Fix a permutation ...
- 13.9k
1
vote
0
answers
75
views
Symmetric matrices of hyperbolic and elliptic type with certain kind of trace zero
I have been working on a problem related to determinantal varieties in symmetric matrices. I am stuck at the following point and would like to get some reference/help for the following question.
Let $\...
- 179
1
vote
0
answers
413
views
Combinatorial Interpretation of an Extension of Gaussian Polynomials
It is well-known that the Gaussian polynomial (or Gaussian coefficient, q-binomial coefficient) $\binom{n}{k}_q$ counts the number of $k$-dimensional subspaces of an $n$-dimensional vector space over $...
- 443
0
votes
1
answer
146
views
Does the Plotkin bound mean that one can not achieve the Singleton bound asymptotically?
I am a little confused with the relationship between various bounds for error correcting codes.
Does the Plotkin bound mean that one can not achieve the Singleton bound asymptotically? That is, is ...
- 11
0
votes
1
answer
390
views
An upper bound for skew symmetric rank 2 matrices
Earlier, I had asked a similar question but that was not the correct problem where I got stuck. After a few quick answer, I realized that and I apologize for that.
Let $B_m$ be the space of all skew-...
- 179
0
votes
0
answers
99
views
Invertible matrices with bounded nonnegative coefficients
I am teaching a class in linear algebra and I asked myself the following question: what is the chance to get an invertible matrix if I write a random one? My impulsive answer is "very likely"...
- 2,152
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 $...
- 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 $1$...
- 13.9k