All Questions
Tagged with finite-fields co.combinatorics
136 questions
5
votes
1
answer
764
views
Conjugacy classes in $GL_{n}(Z / pZ)$
Let $p$ be a prime number and $G=GL_n ( \mathbb{Z} / p \mathbb{Z}
)$. Consider the set $U$ of upper-triangular matrices of $G$
having entries of $1$ on the diagonal. The cardinality of $U$ is $p^{\...
- 463
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
5
votes
0
answers
205
views
Polynomials representing locally constant functions
Let $K$ be a finite field with $p$ elements.
(a) Let $f\in K\lbrack x\rbrack$ be such that (i) $\deg(f)<p$ and (ii) $f(2x) = f(x)$ for $\geq (1-\epsilon) p$ values of $x$ in $K$. What can we say ...
- 20.2k
4
votes
3
answers
240
views
Best strategy for a combinatorial game
Consider the following scenario. We have 20 balls and 100 boxes. We put all 20 balls into the boxes, and each box can contain at most one ball.
Now suppose we are given 5 chances to pick 20 out of ...
- 281
4
votes
1
answer
561
views
Why should it be hard to generalize Dvir's proof of the finite field Kakeya conjecture to the Euclidean case?
Let $q$ be prime and let
$q\delta \sim 1.$ Let $K$ be any set of $C_n\delta$-separated tubes in $B(0,2)$, where $C_n$ is some constant depending on $n$. Let us consider a grid of $q^n$ points scaled ...
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
212
views
Order of a rational function on $\mathbb{F}_p$
Let $a$ be an element of $\mathbb{F}_p$, which is not a quadratic residue.
Define $$f(x) = \frac{x + a}{x+1},$$ which is a rational function on $\mathbb{F}_p$. In fact, if we set $f(-1)=\infty$ and $f(...
4
votes
1
answer
273
views
Dominating sets in subtournaments of the Paley tournament
For a tournament $T$, let $\mathrm{dom}(T)$ be the order of a smallest dominating set in $T$. Let $q$ be a prime power congruent to 3 mod 4 and let $T_q$ be the Paley tournament on $q$ vertices.
Is ...
- 1,701
4
votes
1
answer
271
views
Square-free sets in $\mathbb F_2^n\oplus\mathbb F_2^n$
A square in $\mathbb F_2^n\oplus\mathbb F_2^n$ is a quadruple of the form
$$ (u,v)+\{(0,0),(0,d),(d,0),(d,d)\},\quad u,v,d\in\mathbb F_2^n,\ d\ne 0. $$
A set $A\subset\mathbb F_2^n\oplus\mathbb F_2^...
- 23k
4
votes
1
answer
491
views
What is the permutation group generated by those three given morphisms of the affine space $\mathbb{F}_q^3$?
Let $\mathbb{A}^3 = \mathbb{F}_q^3$. Consider the following three functions $\mathbb{A}^3\to\mathbb{A}^3$:
\begin{eqnarray*}
h: (x, y, z) &\mapsto& (x, y, xy - z) \\
u: (x, y, z) &\mapsto&...
user4324
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 ...
- 313
4
votes
1
answer
382
views
Counting couples of square-free polynomials over finite fields
I have a curve defined by the following equations over the finite field $\mathbb{F}_q$ with $q=p^r$ with $p \geq 3$:
$$C_{h_1,h_2}:\begin{cases} y_1^2=h_1(t) & \\
y_2^2=h_2(t) &...
- 41
4
votes
1
answer
463
views
Covering all, but $k$ points with affine subspaces
For non-negative integer $d\le n$ and $k\le 2^n$, how many affine subspaces of co-dimension $d$ are needed to cover all, but exactly $k$ elements of the vector space ${\mathbb F}_2^n$, and what are ...
- 23k
4
votes
1
answer
266
views
Can we explicitly compute this "shift"-quantity over Boolean functions $u:\mathbb{F}_2^n\rightarrow \mathbb{F}_2$?
This question is a follow-up of this question.
Let $\mathbb{F}_2=\{0,1\}$ be the field with two elements, and suppose that $n$ is odd.
Question: Can we compute the exact minimum $$A:=
\min_{u:\mathbb{...
- 6,741
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
4
votes
1
answer
204
views
Balancing points with lines
$\newcommand{\F}{\mathbb F}$
Suppose that $p$ is a prime, and $k<p/2$ a positive integer.
Consider a system of $k$ distinct directions in the affine plane $\F_p^2$, and the system of $k$ pencils ...
- 23k
4
votes
2
answers
2k
views
probability of having linearly independent sparse vectors over finite fields
Suppose that there are $i< N$ linearly independent $N$-dimensional vectors $\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_i$ over a finite field $\mathbb{F}_q$, whose elements are denoted as $\{0,1,2,...
- 265
4
votes
0
answers
108
views
Differential duality: Triangular codes vs. VT codes / Single-substitution vs. Single-deletion
Here is the introduction to my problem:
Codes correcting single-deletion. Let $q$ and $n$ be non-negative integers, and let $\mathbf{x}= \left ( x_{1}, x_{2}, \ldots, x_{n} \right )\in\mathbb{Z}_{q}^{...
- 41
4
votes
0
answers
263
views
Cosine Modulo $p$?
Consider the integers modulo a prime $p$. I'm looking for a nice polynomial function that acts as a sort of "cosine" on the integers modulo $p$. Specifically, I'm looking for solutions to ...
- 591
4
votes
0
answers
104
views
Visiting zero-sum triples in a vector space
Is it true that for any set $A\subset\mathbb F_5^n$ satisfying $A\cap(-A)=\varnothing$, there is a subset $A'\subseteq A$ such any triple $(a,b,c)\in A\times A\times A$ with $a+b+c=0$ has exactly one ...
- 23k
4
votes
0
answers
215
views
What is the function like when its Mobius inversion is $\sum_{w|r, (w,t)=1}\mu(w)q^{r/w}$?
Everyone, I am now reading a paper named The Irreducible Factors of $(cx+d)x^{q^m}-(ax+b)$ over $GF(q)$, http://qjmath.oxfordjournals.org/content/14/1/61.extract. And I’m confused with one of its ...
- 41
3
votes
2
answers
235
views
Largest $A\subset \mathbb{F}_2^n$ such that no two $a\neq b$ in $A$ add to an element of $A.$
If such a set $A$ of size $m$ exists, all its admissible pairwise sums must lie in its complement, thus
$$
\binom{m}{2} \leq 2^n-m,
$$
which gives $$m\leq 2^{(n+1)/2}\qquad (1)$$.
Edit: The upper ...
- 10.4k
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
177
views
Double-covering $\mathbb F_p$ with three sets
Let $p$ be a (large) prime.
Does there exist a function $f\colon\mathbb F_p^\times\to\mathbb F_p$ such that the three sets
$$ \{f(z)-z\colon z\in\mathbb F_p^\times\},\ \{f(z)\colon z\in\mathbb ...
- 23k
3
votes
1
answer
210
views
On a problem about $GF(2)^n$
For $A\subseteq {\mathbb F}_2^n$ let
$$
Q(A)=\{\alpha+\beta\mid \alpha,\beta \in A,\ \alpha\neq\beta \}.
$$
I want to prove or disprove that if $|A|=2^k+1$ for some integer $k$, then
$$
|Q(A)|\ge2^{k+...
- 197
3
votes
1
answer
245
views
An upper bound on the number of sets of parallel lines covering points in a finite plane?
Let $\mathbb{F}$ be a finite field of characteristic $2$. Let $L_m$ denote the set of lines in $\mathbb{F}^2$ with slope $m\in\mathbb{F}$, that is, all parallel lines of the form $y=mx+b$. Consider a ...
- 133
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
2
answers
256
views
Picking codewords that are close
I posted this question in https://math.stackexchange.com/questions/1142698/picking-codewords-that-are-close a week back.
Let $[n,k,d]$ be a linear code over $\Bbb F_q$ with minimum distance $d$ and ...
- 13.9k
3
votes
0
answers
293
views
Approximate versions of Segre's Theorem
Consider projective $2$-space over a finite field of odd prime characteristic $p$. We say a set of points, $A$, in this space is an arc if any line meets it in at most two points. We say that an arc ...
- 13k
3
votes
0
answers
147
views
Under what conditions on $A$ and $v$ is the size of the sumset $v \cdot A + A$ over $\mathbb{F}_p$ equal or close to $|A|^2$?
Let $p$ be a prime, let $A$ be a subset of $\mathbb{F}_p$, and let $v \in \mathbb{F}_p \setminus \{0\}$.
Under what conditions is $|v \cdot A + A|$ (that is, $|\{ va + b : a \in A,\ b \in A \}|$) ...
3
votes
0
answers
181
views
On an exercise in The Probabilistic Method : random dilate of a set in a finite field
This is related to Problem $4.6$ in ``The Probabilistic Method'' by Alon and Spencer, where one essentially has to prove the following:
Let $p$ be a prime, and $A$ be any subset of $\mathbb{F}_p$. ...
- 31
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
3
votes
0
answers
73
views
Number of classes $\pmod p$ represented by $b_1s^{n-1} + \dots + b_n$ where $ord_p(s) = n$
Let $n \in \mathbb Z$ with $n \ge 3$ and let $p$ be a prime number such that $n|p-1$. Let $a_1,a_2,\dots,a_{2n-1} \in \mathbb Z/p\mathbb Z$. Suppose that the same class is represented by at most $n-1$ ...
- 169
3
votes
0
answers
230
views
On weight enumerators of codes
Are there $[n,k]_q$ constant rate $\frac kn$ and constant alphabet linear code families with automorphism group of size $\Omega((n-n^\beta)!)$ that have minimum distance $d=O(n^\alpha)$ and number of ...
- 13.9k
2
votes
1
answer
119
views
Asymptotic size of largest subset in $\mathbb F_p^2$ defining only lines of different slopes
Suppose that all lines defined by pairs of distinct elements in a subset of $\mathbb F_p^2$ have different slopes. How large can such a subset be asymptotically (for primes $p\rightarrow \infty$)?
...
- 17.9k
2
votes
1
answer
262
views
Randomly fixing elements and transcendence degree
Given $f_1,\ldots,f_n \in \mathbb{F}_q[x_1,\ldots,x_m]$ such that $\deg(f_i) \leq d < q$. Suppose we have for some $1 \leq j \leq m$
$$ \operatorname{trdeg}_{\mathbb{F}(x_1,\ldots,x_j)}\{f_1,\ldots,...
- 341
2
votes
1
answer
217
views
Low-Hamming weight vectors in low-dimensional subspaces of $\mathbb{F}_p^n$
What is the maximum number vectors of Hamming weight at most $w$ in a $d$-dimensional subspace of $\mathbb{F}_p^n$, where $w,d,p$ are constant and $p$ is odd. (The Hamming weight is the number of ...
- 93
2
votes
1
answer
308
views
The number of unitary circulant matrices over a finite field $\mathbb{F}_{q^2}$
I asked this question in MSE few days ago but there was no response.
Suppose $\mathbb{F}=\mathbb{F}_{q^2}$, where $q$ is a prime power. The conjugate of elements in $\mathbb{F}$ is defined by $\...
- 379
2
votes
1
answer
612
views
Inner product over finite field
sorry for informals but is my first post.
In Coding theory (exactly in Coding theory a first course - San ling and Chaoping) found this definition:
$\langle , \rangle :\mathbb{F}_q^n\times\mathbb{F}_q^...
2
votes
1
answer
362
views
Characterization of nilpotent adjacency matrices [closed]
Let $\theta$ be the adjacency matrix of a simple graph (symmetric and zeros on the diagonal). What is the characterization of those $\theta$ which satisfy $$\theta^2 \equiv 0 \pmod{2}$$ i.e. which $\...
- 23
2
votes
1
answer
192
views
A Vandermonde-type system
For a prime $p$ and $a_1,\dotsc,a_n\in\mathbb F_p^\times$, consider the system of equations
$$ \begin{cases}
\begin{align}
a_1 + \dotsb + a_n &= 0 \\
a_1x_1 + \dotsb + a_nx_n &...
- 23k
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 ...
- 21
2
votes
1
answer
128
views
Bases of the special form
Let $R = \mathrm{GF}(q)$, $S = \mathrm{GF}(q^n), \ n\geq 2$ be extension of $R$, $h$ be a primitive element of $S$. I want to count or estimate the number $N$ of bases of the following form.
Let $$\...
2
votes
1
answer
185
views
How many ways are there to pick $n$ points on the finite affine plane $(\Bbb F_q)^2$ such that no three are collinear?
How many ways are there to pick $n$ points on the finite affine plane $(\Bbb F_q)^2$ such that no three are collinear?
For example, how many ways can we pick $5$ points on $\Bbb F_{32}\times\Bbb F_{32}...
- 960
2
votes
0
answers
187
views
Matrix with elementary symmetric polynomials as entries
Let $n\geq 1$, and for each $j=1,\ldots, n+1$ let $\mathbf{X}_{j}=(X_{j1},\ldots, X_{jn})$ be $n$ variables. Let $M$ be the $(n+1)\times (n+1)$ matrix whose $(i,j)$-th entry is $$M_{ij}=(-1)^i e_{i-1}(...
2
votes
0
answers
203
views
Can the relation between count of commuting pairs and conjugacy classes for finite groups be generalized to semigroups?
It is well-known that number of pairs of commuting elements in finite group G is equal to number of conjugacy classes multiplied by cardinality of G.
More generally here (MO275769) Qiaochu Yuan ...
- 24.9k
2
votes
0
answers
100
views
Sum of binary quadratic forms over inputs of equal Hamming weight
$\DeclareMathOperator{\field}{\mathbb{F}}$
Let $n$ be a positive integer, and let $q : \field_2^{n} \rightarrow \field_2$ be a quadratic form, specified in coordinates as
$$q(x)=\sum_{i =1}^n \alpha_i ...
- 980
2
votes
0
answers
157
views
On hypergeometric functions over finite fields
Let $\mathbb{F}_q$ be a finite field of $q$ elements. Let $A,B,C,\cdots$ denote the multiplicative characters over $\mathbb{F}_q$, and let $\overline{A}$ denote the inverse of $A$, i.e., $A(x)\...
- 93
2
votes
0
answers
111
views
Standard interpretation of permanents (of orthogonal included) over finite fields
Given a $0/1$ matrix in $\mathbb Z^{n\times n}$ the standard interpretation of permanent of the matrix is the number of perfect matchings in the underlying $2n$ vertex balanced bipartite graph with ...
- 13.9k