All Questions
Tagged with finite-fields co.combinatorics
136 questions
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 $$\...
11
votes
1
answer
258
views
Counting nonzero hyperdeterminants over $\mathbb{F}_q$
The hyperdeterminant $D(A)$ is a multidimensional generalization of the
determinant. It is a polynomial in the entries of a $(k_1+1)\times
(k_2+1)\times\cdots \times (k_n+1)$ array $A$. The ...
- 50.8k
11
votes
2
answers
788
views
Blocking sets in three dimensional finite affine spaces
What is the smallest possible size of a set of points in $\mathbb{F}_q^3$ which intersects (blocks) every line?
Clearly the union of three affine hyperplanes that intersect in a singleton, say $x = 0,...
- 1,197
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
5
votes
1
answer
459
views
$(n-2)$-blocking sets in $AG(n,2)$
Let's define $k$-blocking set in affine space $AG(n,q)$ a set that meets every coset (translate of subspace) of dimension $k$.
I have seen a lot work related to minimal $(n-1)$-blockings set.
...
- 337
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
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
6
votes
1
answer
458
views
Applications of small Kakeya sets over finite fields
It was proved by Dvir that a Kakeya set in $\mathbb{F}_q^n$ has size at least $q^n/n!$, a bound which was later improved to $q^n/2^n$.
For $n = 2$ and $q$ odd the exact bound is $q(q+1)/2 + (q-1)/2$ ...
- 1,197
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
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 $p \geq 3$:
$$C_{h_1,h_2}:\begin{cases} y_1^2=h_1(t) & \\
y_2^2=h_2(t) &...
- 41
3
votes
2
answers
256
views
Picking codewords that are close
I posted this question in https://math.stackexchange.com/questions/1142698/picking-codewords-that-are-close a week back.
Let $[n,k,d]$ be a linear code over $\Bbb F_q$ with minimum distance $d$ and ...
- 13.9k
22
votes
1
answer
969
views
Grothendieck on polyhedra over finite fields
In Grothendieck's Sketch of a Programme he spends a few pages discussing polyhedra over arbitrary rings and concludes with some intriguing remarks on specializing polyhedra over their "most ...
- 528
1
vote
0
answers
207
views
Polynomial existence over finite field
Denote $\mathcal{F_n}$ as collection of multiaffine polynomials $f\in\Bbb F_2[x_1,\dots,x_n]$.
Denote total degree of $f\in\mathcal{F_n}$ as $deg(f)$ (note $deg(f)\leq n$).
Denote $e_i=(0,\dots,0,\...
- 13.9k
6
votes
1
answer
372
views
A parity counting problem for subsets over finite fields
Let ${\mathbb F}_p$ be the prime field of $p$ elements and $b$ be an element in ${\mathbb F}_p$.
For a subset $T\subseteq {\mathbb F}_p$, define
$$Bias(T)=|N_e( {\mathbb F}_p,b)-N_o( {\mathbb F}_p,b)|...
- 121
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
8
votes
0
answers
304
views
A strong sum-product "for translates" in finite fields
In the course of some recent research, I've sketched out a proof of the following result. My basis question is: is the result interesting?
Proposition There exists an absolute constant $c$ such ...
- 11.2k
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
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
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
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
8
votes
1
answer
375
views
A Balog-Szemeredi-Gowers-type question
A nice Lemma due to Konyagin asserts that for any subset $B \subset \mathbb{F}_p$ holds
$$
|B.B - B.B + B.B - B.B + B.B - B.B| \geq \frac{1}{2}\min\{p, |B|^2 \},
$$
where the standard notation for the ...
- 960
17
votes
1
answer
381
views
Do mutually dual finite vector spaces have the same orbit cardinalities under a linear group action?
Let $G$ be a finite group acting linearly on a finite dimensional vector space $V$ over a finite field. By Burnside's lemma,
$$
|V/G| = \frac 1{|G|} \sum_{g\in G} q^{\dim(ker(g - I))}.
$$
Since $g-I$ ...
- 5,654
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
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
5
votes
1
answer
447
views
More expanders?
Having received several exhausting answers to my recent question about
the expansion properties of a certain graph, I now wonder whether anything is
known on the following graphs of a similar nature:
...
- 23k
13
votes
2
answers
1k
views
An expander (?) graph
For a prime $p$, consider the graph on the vertex set ${\mathbb F}_p$, in
which every vertex $z$ is adjacent to $z\pm 1$ and also to $z^{-1}$ (unless
$z=0$). I was told that this graph is known to be ...
- 23k
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
17
votes
4
answers
1k
views
A mixing property for finite fields of characteristic $2$
In connection with this MO post, here is a question somewhat implicitly contained in a joint paper of S. Kopparty, S. Saraf, M. Sudan, and myself.
Let ${\mathbb F}$ be a finite field, and suppose ...
- 23k
9
votes
2
answers
543
views
A mixing property of linear map over finite fields
Let $F$ be a finite field of odd size $q$, and $\phi_0 : F \mapsto F$ be any map from $F$ to itself. For each $a \in F$, set $\phi_a : x \in F \mapsto \phi_0 (x) + ax $.
When $\phi_0 : x \mapsto x^2 ...
- 235
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
20
votes
2
answers
2k
views
Sums of powers mod p
For prime $p > 7$ with $p-1=rs$, $r>1$, $s>1$, let $A=\{x^r|x \in \mathbb{Z}_p\}$ and $B = \{x^s|x \in \mathbb{Z}_p\}$. If $g$ is a primitive root mod $p$ then $A = \{0\} \cup \{g^{ir}|0 \leq ...
- 619
67
votes
6
answers
7k
views
How to recognise that the polynomial method might work
A couple of days ago I was at a nice seminar given by Christian Reiher, during which he told us about a short proof of the following special case of a theorem of Olson.
Theorem. Let $(a_1,b_1),\dots,(...
- 29k
16
votes
3
answers
2k
views
Periodic orbits and polynomials
There are two simple and classic enumerations that still I'm puzzled about. Let's start with a simple counting problem from a well-known dynamical system.
fact 1 Consider the "tent map" f:[0,1]→[...
- 60.5k
7
votes
2
answers
476
views
A quadratic form
Let $q$ be a power of 2. Let $P$ be the set of polynomials in
$F_q [x]$ of degree d or less.
Let $\mathbb{Z}$ be the ring of integers. For any $f \in P$, let $\psi(f)$ be the
number of distinct ...
- 503
23
votes
4
answers
3k
views
Exhibit an explicit bijection between irreducible polynomials over finite fields and Lyndon words.
Let $q$ be a power of a prime. It's well-known that the function $B(n, q) = \frac{1}{n} \sum_{d | n} \mu \left( \frac{n}{d} \right) q^d$ counts both the number of irreducible polynomials of degree $n$...
- 118k