All Questions
14
questions
2
votes
2
answers
318
views
Minimum weight of ternary Golay code in cyclic form
Motivation
One of the various approaches to the perfect Golay codes is via cyclic codes. From the cyclotomic cosets, one computes the corresponding cyclotomic coset (2 possibilities each) and can use ...
- 23.1k
1
vote
1
answer
296
views
Number of solutions of a polynomial over finite fields
Consider in $\mathbb{F}_q[x_1,\dots,x_n]$, where $r$ is a positive integer dividing $n$, the polynomial
$$
f(x_1,\dots,x_n)=x_1x_2\dots x_r+x_{r+1}x_{r+2}\dots x_{2r}+\dots+x_{n-r+1}x_{n-r+2}x_{n}.
$$
...
- 71
2
votes
2
answers
480
views
Number of n-tuples in $\{0, 1, 2\}$ with sum less than or equal to $d$.
I would like to know if there is an expression for the number of n-tuples of $\mathbb{Z}$, where each component is an integer between $0$ and $2$, and the sum of the components is less than or equal ...
- 97
1
vote
0
answers
225
views
Is it Possible to Apply the Finite Field Method to Hyperplane Arrangements in $\mathbb{R}^n$?
I have a question regarding the finite field method to compute the characteristic polynomial of a hyperplane arrangement. I am using the book "Enumerative Combinatorics, Volume 1" second edition by ...
- 167
1
vote
0
answers
27
views
What is the expected difference in the gap of two polynomials in $Z_2[x_1, ..., x_n]$ when adding a randomly chosen variable $x_i, i \in [n]$?
I define the gap of a polynomial $f \in Z_2[x_1, ..., x_n]$ as
$gap(f) = |f^{-1}(0)| - |f^{-1}(1)|$.
I'm curious about the quantity $\mathbb{E}_i[|gap(f) - gap(f + x_i)|]$. I'm interested in this ...
0
votes
0
answers
63
views
Rank of a matrix in Ellenberg/Gijswijt proof
In a paper by Ellenberg and Gijswijt on the cap set problem, the proof of proposition 2 relies on the claim that certain matrices have rank 1. This is not obvious to me. Why?
https://arxiv.org/pdf/...
- 1,372
1
vote
1
answer
68
views
On minimum Hamming weights in polynomial arithmetic progressions over $\Bbb F_q[x]$
Given $a(x),b(x)\in\Bbb F_q[x]$ consider the arithmetic progression $$a(x)+\gamma(x) b(x)$$ where $\gamma(x)\in\Bbb F_q[x]$. Is there a $\gamma(x)$ such that Hamming weight of $a(x)+\gamma(x) b(x)$ is ...
- 6,245
1
vote
0
answers
316
views
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
0
votes
1
answer
302
views
Find the number of monic square-free polynomials of degree j over finite field [duplicate]
Find the number of monic square-free polynomials of degree j >=1 over the finite field GF(q) ?
I have no idea how to approach this. I was thinking if there was a way to write a monic polynomial ...
- 860
3
votes
1
answer
55
views
Polynomials in two variables over $\mathbb{F}_p$ fixed by $\operatorname{SL}_2(\mathbb{F}_p)$
Let $A=(a_{ij})\in\operatorname{SL}_2(\mathbb{F}_p)$. Consider the ring map $A:\mathbb{F}_p[x,y]\to\mathbb{F}_p[x,y]$ defined by
$$A(x)=a_{11}x+a_{21}y$$
$$A(y)=a_{12}x+a_{22}y$$
and extended ...
- 31.7k
10
votes
1
answer
253
views
On the maximum number of polynomials in a certain subspace
I've already asked this question on mathoverflow, but no one answered. So I put this problem also here. Sorry for that.
Let $\mathbb F_q$ be a finite field and let $e, k$ be positive integers with $...
- 1,549
0
votes
1
answer
77
views
Maps preserving roots of a polynomial function over finite fields
Let $P(x_{1},\ldots,x_{n}):\mathbb{F}_{2}^{n}\rightarrow \mathbb{F}_{2}$ be a polynomial function with degree $d$ and with variables $x_{1},\ldots,x_{n} \in \mathbb{F}_{2}$.
Let $S(P)=\{ (x_{1},\...
- 1
19
votes
2
answers
765
views
How many pairs of nilpotent, commuting matrices are there in $M_n(\mathbb{F}_q)$?
As a follow-up to this question, I've been doing some work counting pairs of commuting, nilpotent, $n\times n$ matrices over $\mathbb{F}_q$. So far, I believe that for $n=2$, there are $q^3+q^2-q$ ...
- 31.7k
8
votes
2
answers
9k
views
Determine the number of irreducible monic polynomials of degree 3 in $\mathbb F_p[x]$
Determine the number of irreducible monic polynomials of degree 3 in $\mathbb F_p[x]$ where $p$ is a prime.
I'd like to start off by acknowledging that I know there are many posts relating to similar ...
- 9,361