All Questions
Tagged with algebraic-combinatorics finite-fields
8
questions
1
vote
0
answers
47
views
Hyperplane arrangement : The Shi arrangement
I have been lately reading Hyperplane arrangement lectures by Richard Stanley on https://www.cis.upenn.edu/~cis610/sp06stanley.pdf . In lecture 5, Theorem 5.16 we define the characteristic polynomial ...
- 11
14
votes
3
answers
324
views
Family of sets with $|F_i| \equiv 2\pmod 3$ and $|F_i \cap F_j| \equiv 0 \pmod 3$
Let $p$ be a prime. By considering the incidence vectors of subsets $F_1,\ldots,F_m$ of $\{1,2,\ldots,n\}$, such that $|F_i| = a \not\equiv 0 \pmod p$ and $|F_i \cap F_j| \equiv 0 \pmod p$ for all $1\...
- 2,853
11
votes
0
answers
297
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Clubs whose intersections are multiples of six (Oddtown variant)
This is a question about generalizing the famous "Clubs in Oddtown" problem. The original setup is that a town has $n$ people, and $m$ clubs each consisting of a subset of the population. ...
- 78.4k
4
votes
0
answers
118
views
Constant sum of characters
Let $q$ be a prime power and $\omega=\exp(2\pi i/q)$. For a fixed $y\in\mathbb{Z}_q^n$, the map
$$\mathbb{Z}_q^n\ni x\mapsto \omega^{x\cdot y}=\omega^{x_1y_1+\dots+x_ny_n}$$
is a character of $\...
- 450
1
vote
1
answer
85
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Cyclotomic scheme is a Association scheme
I try to show that the following defines an association scheme:
Let $\mathbb{F}_q$ be a field, $\omega$ a primitive element of $\mathbb{F}_q^\times$ and $s$ divides $q-1$. Define $r=\frac{q-1}{s}$, $...
user239922
2
votes
1
answer
359
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Counting monic irreducible polynomials of a particular degree without use of finite fields?
I know that every irreducible polynomial over $\mathbb F_p[x]$ of degree equal to the degree $m$ of a finite field $\mathbb F_{p^{m}}$ has a root in the field. Using this result we can count the ...
- 1,508
1
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0
answers
316
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Necklace polynomial recurrence relation
Let the number of monic, irreducible polynomials of degree $n$ over $F_q$ be $f(n)$, then $f(1)=q$, and to calculate $f(n)$, I would count the number of monic, reducible polynomials of degree $n$ this ...
- 3,806
8
votes
2
answers
312
views
Given a list of $2^n$ nonzero vectors in $GF(2^n)$, do some $2^{n-1}$ of them sum to 0?
Let $G=(\mathbb{Z/2Z})^n$ written additively, $n>1$. (you can think of it as $\mathbb{F}_{2^n}$ but I didn't find that useful... yet)
Let $v_i$ be nonzero elements of $G$ for $i \in \{1 \dots 2^n \...
- 786