All Questions
10
questions
0
votes
2
answers
103
views
Inverse of a matrix in $\mathbb{F}_5^{4\times4}$
Let
$f, \, g, \, h \in \mathbb{F}_5[X]$ where
$$f=X^9+X^8+ \cdots +X^2+X+1,\\ g=X^4+X-2 = X^4+X+3,
\\ h = 3X^2+4X+3.$$
$h$ is the greatest common divisor of $f$ and $g$.
It holds that
...
- 635
1
vote
0
answers
43
views
Counting the powers of a companion matrix that possess nonzero leading principal minors
Let $p$ be prime, let $A$ be the $n\times n$ companion matrix associated to a primitive polynomial $f(x)$ of degree $n$ with coefficients in ${\mathbb Z}_p$, and let $T=\{A^j\mid 1\leq j\leq p^n-1\}\...
- 11
1
vote
1
answer
62
views
Left inverse of a matrix $3 \times 2$ in $\mathbb{F}_7[x]$
Do you know a method to calculate inverse matrix in $\mathbb{F}_7[x]$?
I want to calculate left inverse the following matrix of $3 \times 2$ in $\mathbb{F}_7[x]$
\begin{bmatrix}
x^2+1 & x-1 \\
...
- 504
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
3
votes
1
answer
2k
views
Generator matrix of a binary Goppa code
My goal is to construct a generator matrix for a classical binary $[8,2,5]$ Goppa code $\Gamma(L,G)$, with $L=\mathbb{F}_8$ and Goppa polynomial $G(x)=x^2+x+1$.
So far I have successfully been able to ...
- 532
3
votes
1
answer
790
views
What is the relationship between the generator matrix and polynomial of a FEC block code?
When talking about FEC (forward error correction) block codes, some literature uses matrix terminology and some talks about polynomials. I know that the same block code could be expressed with either ...
- 143
5
votes
1
answer
314
views
Algebraic or Analytic Proof of a Polynomial Identity
Let $m$, $n$, and $r$ be integers with $0\leq r \leq \min\{m,n\}$. Define
$$f_{m,n,r}(q):=\left(\prod_{j=1}^r\,\left(q^m-q^{j-1}\right)\right)\,\left(\sum_{\substack{{j_1,\ldots,j_r\in\mathbb{Z}_{\...
- 49.8k
1
vote
1
answer
6k
views
Construct generator matrix given generator polynomial?
How would I take a generator polynomial and construct a generator matrix out of it for a cyclic code?
For example, I have a cyclic code in:
$R_{15}=GF(2)[x] / \langle x^{15} + 1\rangle$
This is ...
- 307
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
1
vote
1
answer
284
views
Matrix polynomial factorization
This is about exercise 1207 from the book "Problems and Solutions in Mathematics", 2nd edition, by Ta-Tsien.
Let $p$ be a prime and let $V$ be an $n$-dimensional vector space over the finite field $...
- 1,894