Questions tagged [finite-fields]

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How is a "low-degree polynomial" precisely defined in Algebrization?

I'm going through papers which present algebrization as a barrier and I'm trying to understand how "low-degree" polynomials are precisely defined, i.e. are they low with respect to the ...
5
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0answers
221 views

$\#$P hardness of computing weighted sum of degree $2$ polynomials

Consider polynomials $f: \{0, 1\}^{n} \rightarrow \{0, 1\}$ over $\mathbb{F}_2$ (addition and multiplication are taken modulo $2$.) Consider integers $x \in \{0,1\}^{n}$, written in binary. Let $\...
5
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0answers
111 views

Algebraic dependence of roots of irreducibles over a finite field

I asked this question in Math SE too, but I have since modified it to make it more suited here. Also, in hindsight, the question itself was more algorithmic and was a better fit here. https://math....
5
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140 views

Applications of small Kakeya sets over finite fields

It was proved by Dvir that a Kakeya set in $\mathbb{F}_q^n$ has size at least $q^n/n!$, a bound which was later improved to $q^n/2^n$. For $n = 2$ and $q$ odd the exact bound is $q(q+1)/2 + (q-1)/2$ ...
2
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64 views

Which invertible linear transformations can be computed reversibly without ancilla/garbage bits just as easily as they can be computed irreversibly?

Suppose that $L:F_{2}^{n}\rightarrow F_{2}^{n}$ is an invertible linear transformation. Then define $w(L)$ to be the gate count of the smallest reversible circuit on $n$ bits without ancilla/garbage ...
2
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0answers
62 views

A random ensemble of sparse boundary operators

The following question arises from the study of quantum error correction, and high-dimensional expanders: Is there an algorithm that for given numbers $n>0,d≤n,r≤n$ samples uniformly a linear ...
2
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0answers
88 views

Is berklemap-massey works on extention field

According to the articles that i read (also wikipedia [1]), they refer that the above algorithm works on finite field $GF(q)$. But I cannot find an evidence that it still works on extension field, i.e:...
2
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0answers
85 views

Module Independence Graphs

Given $n, k \in \mathbb{N}$, we create a set $S \subseteq \mathbb{Z}_m^k$ by treating it as an $n\times k$ grid and choosing each $S_{ij}$ by sampling from $\mathbb{Z}_m$, such that $P(S_{ij} = z) = ...
1
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71 views

BCH codes and polynomials with many values in a subfield

For points $P=\{x_1, \ldots, x_n\} \subset {\mathbb F}_{2^m}$ define $$\mathcal{C}(P, t) =\{(f(x_1), \ldots, f(x_n)) \mid \mbox{$f\in {\mathbb F}_{2^m}[X]$ has degree $t$}\}$$ and $\mathcal{C}'(P, t) =...
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64 views

The curve used in Parvaresh-Vardy decoding

Consider the Parvaresh-Vardy list decoder. As I understand it, the idea is to decide on a curve over an extension field of the form $(f,f^h mod E, f^{h^2} mod E,\dots)$ and then evaluate each of ...
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27 views

Searchable finite field

Let $F$ be a large finite field, where the elements are strings of length $n$. We require, addition, multiplication, and division to be efficient (polynomial in $n$). We say that $F$ is searchable if ...
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91 views

On permanent mod $3^t$ computation

By $'$ I mean transpose. I gather the info here from rjlipton.wordpress.com/2014/12/21/modulating-the-permanent. We know that if $U\in\Bbb F_{3^t}^{n\times n}$ satisfies $UU'=I_n$ in $\Bbb F_{3^t}$ ...