5
votes
Accepted
number of rational points of hyper elliptic curve $y^5=-x^2+x$ over $\Bbb{F}_{121}$
The job is easily done with computer support, here sage. Let us do this first.
...
- 28.2k
3
votes
Accepted
$\mathfrak{B}=((1,2,0)^t ,(2,1,2)^t ,(3,1,1)^t)$ is a basis of $\mathbb{R}^3$. For what prime numbers p is $\mathfrak{B}$ a basis of $\mathbb{F}^3_p$?
If you have $n$ column vectors, each with $n$ coordinates, you can form a matrix from them and then the determinant is $0$ if and only if the columns are linearly dependent over the field you're ...
- 19.4k
2
votes
Let $V = \mathbb F_p^9$ and $W \subset V$ a dimension $5$ subspace. Find the number of subspaces $U \subset V$ with $\dim(U) = 6, \dim(W \cap U) = 3$
We can reduce this problem to a simpler one using the correspondence theorem.
The Simpler Problem
Given:
a finite-dimensional vector space $V$ over a finite field $F$,
a subspace $W \leq V$, and
a ...
- 12.1k
2
votes
Accepted
Largest number of shards for linear erasure codes over a finite field
In the framing of $D×P$ matrices over $\mathbb{F}_q$, the following points can be exhaustively tested and shown to be true for all $q\le 27$:
$D×1$ and $1×P$ are possible, for any $D$ or $P$
$D×P$ is ...
- 46
1
vote
Question on smooth affine curves over finite fields and Hasse-Weil
Unfortunately, no. The curve determined by $f(x,y)=x(x-1)$ is smooth in the plane, singular at infinity, and not absolutely irreducible. For something that's a little less cheat-y, consider $f(x,y)=x^...
- 56.6k
1
vote
Accepted
Why doesn't linearity of squaring over Galois Field imply linearity of cubing?
How would you mathematically define $ \bar A \cdot \bar B$? For $GF(2^{128})$, inner product produces a single bit scalar, outer product produces a 128 by 128 bit matrix.
Although a matrix can be used ...
- 486
1
vote
Generating de Bruijn sequence over Galois fields using primitive polynomial
Looking at the de Bruijn sequences with alphabet $GF(4)$ and over $GF(16)$ only, as that seems to be your main interest.
I rather think of $GF(4)$ as
$$GF(4)=\{0,1,\beta,\beta+1\}.$$
If only to match ...
- 127k
1
vote
Polynomial irreducibility over $\mathbb{F} _{7^n}$
$X^3 + 2$ does have a root in $\mathbb F_{7^3}$. There are a few ways to see this. For example:
$\mathbb F_{7^3}$ is the splitting field for the polynomial $X^{7^3} - X$ over $\mathbb F_7$. (Indeed, ...
- 26.8k
1
vote
Accepted
Structural description for the order of $\mathrm{GL}_n(\mathbb F_q)$
Let $F_q$ be a finite field of . $q$.
Let $T_n(F_q)$ the set of upper triangular matrices (the semi-dircet product of digoanl matrices and unipotent radical).
The quotient $GL(n,F_q)/ T_n(F_q)$ it the ...
- 6,827
1
vote
Finding ${\rm Aut}_{\mathbb{F}_{27}}(\mathbb{F}_{19683})$
Remember the characterization $$\mathbb{F}_{3^n}=\{ x\in \overline{\mathbb{F}_3}\mid x^{3^n}=x\},$$ namely this is the fixed element set of $\mathrm{Fr}_3^n$, the $n$-th power of the Frobenius map, ...
- 1,688
1
vote
Largest number of shards for linear erasure codes over a finite field
Rephrasing the question: we're trying to find $D×P$ matrices over $\mathbb{F}_q$ such that every square submatrix is invertible. Note that this is using the broadest possible definition of square ...
- 46
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