# Tag Info

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
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

### 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
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

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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