All Questions
Tagged with co.combinatorics finite-fields
136 questions
2
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111
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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
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0
answers
277
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Counting special metrics on finite fields
Define a Galois coding norm of degree n as a map $|\space| : \Bbb F_{2^n}\rightarrow {\Bbb Z}$ with the following properties :
(I) $(\Bbb F_{2^n},|\space|)$ is a self-orthogonal code ; i.e. $(x,y)\...
- 173
2
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0
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186
views
Sum of reciprocals in finite fields
Let $p$ be an odd prime number which large enough. I am interested in the study of the sums of reciprocals in the field $\mathbb{F}_p$.
In particular, I have the following question:
which primes $p$ ...
- 347
2
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0
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162
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Determining the multiple solutions for $\mathrm{GF}(2)$ discrete logarithms of polynomials with partially known coefficients
I have an LFSR, essentially $x^k \bmod p(x)$ for some characteristic primitive polynomial of degree $N$ with coefficients in $\mathrm{GF}(2)$, as outlined in Clark and Weng's article: it has a period $...
- 663
2
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0
answers
113
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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 $$v=...
- 13.9k
2
votes
0
answers
337
views
Enumerating certain types of permutation polynomials
Given a prime power $q$, I would like to enumerate (preferably up to isomorphism*) all the permutation polynomials $f(x)$ on $K = GF(q^3)$ satisfying the following conditions:
$f(ax) = af(x)$ for all ...
- 1,197
1
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1
answer
541
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Number of zeros of quadratic equation over finite fields
Let $\mathbb{F}_q$ denote the finite field with $q$ elements and Ch$\mathbb{F}_q\neq 2$. What is the number of solutions of the quadratic equation
$X_1^2+\cdots + X_r^2=0$ in $\mathbb{F}_q^m$ for $1\...
- 179
1
vote
1
answer
82
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question about sets of polynomials with a special agreement guarantee
Let $\mathbb{F}$ be a finite field and $S\subset\mathbb{F}_{\leq d}[x,y]$, a set of bivariate polynomials over $\mathbb{F}$ of degree at most $d\ll|\mathbb{F}|$. Assume the linear span of $S$ is all ...
- 125
1
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1
answer
333
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Szemerédi–Trotter type theorem in finite field
This question is about the content of this paper by J. Bourgain, N. Katz, T. Tao.
In the final step (page 18) of the proof of Szemerédi-Trotter type theorem, we have already known
$$|A''+A''|\lesssim ...
- 13
1
vote
1
answer
338
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Polynomial form/Fourier transform of rational function over finite affine space
I am certainly going to make a mess of any serious algebraic terminology, so bear with me as I present my problem arising from a probability problem.
Consider the space of sequences of $n$ zero-one ...
- 526
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
133
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Constant term of a power modulo a polynomial
I'm interested in the constant term of $$(x+k)^m \in F_p[x]$$
modulo a polynomial $q(x)$ over the field $F_p$. The polynomial $q(x)$ is relatively simple in practice, take $q(x) = x^6 -2x^3+3$, for an ...
- 591
1
vote
1
answer
153
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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
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1
answer
145
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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
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0
answers
71
views
Bias of $a^k / q$ modulo $q$?
Let $q$ be a prime. Let $0< a < q$ be an integer so that it is primitive modulo $q$. Let $k$ be a random integer up to $q-1$. Consider
$$a^k = b_k + q * c_k$$
as $k$ varies modulo $q^2$. So $b_k$...
- 591
1
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0
answers
91
views
Functions that take quadratic residues to non quadratic residues
Let $p$ be prime and $Q$ be the set of integers $x$ mod $p$ so that $x^2-1$ is a quadratic residue. Let $Q^c$ be the complement of $Q$. If we don't consider $x = 1$ then these two sets have the same ...
- 591
1
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0
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162
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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
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0
answers
84
views
Constructing k-wise independent variables over a general set
We have seen in class a polynomials based construction that builds in $O(n^k)$ time, $n$ random variables, $k$-wise independent, over a field with $n$ elements. More specifically, you generate all the ...
- 11
1
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0
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130
views
Maximal subsets of affine or projective space with no three collinear points
Let ${\mathbb A}^n_q$ and ${\mathbb P}^n_q$ be affine and projective spaces of dimension $n$ over a field of order $q$.
Say that a subset of either ${\mathbb A}^n_q$ or ${\mathbb P}^n_q$ is generic if ...
- 23k
1
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0
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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
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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
96
views
Polynomial composition utilizing polynomials in two different finite fields
At every $n\in\mathbb N$ (all polynomials are of degree $O(1)$) is there $g_{3,1}^{(n)},\dots,g_{3,k}^{(n)}\in\mathbb F_3[x_1,\dots,x_n]$ at $k=\mathsf{poly}(n)$ and $g_2^{(n)}\in\mathbb F_2[x_1,\dots,...
- 13.9k
1
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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
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0
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154
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On the set $\{\sum_{k=1}^n \lambda_ka_k:\ a_1,\ldots,a_k\ \text{are distinct elements of}\ A\}$
For a field $F$ let $p(F)=p$ if the characteristic of $F$ is a prime $p$, and $p(F)=+\infty$ if $F$ is of characteristic zero.
In 2007 I considered the linear extension of the Erdos-Heilbronn ...
- 15.6k
1
vote
0
answers
207
views
Polynomial existence over finite field
Denote $\mathcal{F_n}$ as collection of multiaffine polynomials $f\in\Bbb F_2[x_1,\dots,x_n]$.
Denote total degree of $f\in\mathcal{F_n}$ as $deg(f)$ (note $deg(f)\leq n$).
Denote $e_i=(0,\dots,0,\...
- 13.9k
1
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0
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413
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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
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2
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295
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Permutations of squares and finite fields
Let $S_n$ be the symmetric group of all permutations of $\{1,\ldots,n\}$, and let
$$S(n)=\bigg\{\sum_{k=1}^nk^2\pi(k)^2:\ \pi\in S_n\}.$$
Motivated by Question 316142 of mine, here I ask the following ...
- 15.6k
0
votes
1
answer
189
views
Constructing a family of $3$-wise independence functions from $\mathbb{Z}_p^n \rightarrow \mathbb{Z}_p$
A family of function hash functions $\mathcal{H}:\{h:N\rightarrow M\}$ is call $k$-wise independent if whenenver $h$ is drawn uniformly from $\mathcal{H}$, let $x_1,\ldots,x_k$ be distinct elements of ...
- 121
0
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1
answer
146
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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
1
answer
190
views
$k$-covering $\mathbb F_p$ with $k+1$ sets
Let $p$ be a (large) prime.
How large can a set $C\subset\mathbb F_p$ be given that there is a function $f\colon\mathbb F_p^\times\to\mathbb F_p$ such that for every element $g\in \mathbb F_p$, ...
- 23k
0
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0
answers
82
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Solution modulo $9$ of certain linear equation implies triviality modulo $3$
Question: Let $k \geq 2$ and $r \geq 4$ be two natural numbers. We are given eight integers $\nu_{ij} \geq 0$ for every $1 \leq i \leq k$ and $1 \leq j \leq r$ such that the following two conditions ...
- 119
0
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0
answers
53
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A question on bounding the size of the polynomial
Suppose we are given the following n polynomials in $\bar{\mathbb{F}}_2[x_1,...,x_n]$:
$f_1 = x_1 + x_n^2$
$f_2 = x_2 + x_1^2$
$\cdot$
$\cdot$
$f_{n-3} = x_{n-3} + x_{n-4}^2$
$f_{n-2} = x_{n-2} + x_{n-...
- 341
0
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0
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100
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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
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0
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54
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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
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0
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299
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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