All Questions
8 questions
22
votes
1
answer
970
views
Grothendieck on polyhedra over finite fields
In Grothendieck's Sketch of a Programme he spends a few pages discussing polyhedra over arbitrary rings and concludes with some intriguing remarks on specializing polyhedra over their "most ...
- 528
13
votes
2
answers
1k
views
An expander (?) graph
For a prime $p$, consider the graph on the vertex set ${\mathbb F}_p$, in
which every vertex $z$ is adjacent to $z\pm 1$ and also to $z^{-1}$ (unless
$z=0$). I was told that this graph is known to be ...
- 23k
5
votes
1
answer
447
views
More expanders?
Having received several exhausting answers to my recent question about
the expansion properties of a certain graph, I now wonder whether anything is
known on the following graphs of a similar nature:
...
- 23k
5
votes
1
answer
163
views
Polynomials vanishing on prescribed layers
Given a prime $p$ and an integer $n\ge p$, what is the smallest possible degree of a polynomial $Q\in\mathbb F_p[x_1,\dotsc, x_n]$ such that $Q$ vanishes on every vector $x\in\{0,1\}^n$ of weight $w(x)...
- 23k
4
votes
1
answer
152
views
Enumerator Polynomials for Linear Anytime Codes
Let $C = \{c \in \mathbb{F}^n_2 : Hc=0\}$ be a binary linear code where $H \in \mathbb{F}^{k \times n}_2$ is a block lower-triangular matrix of full rank called the parity-check matrix of $C$. Clearly ...
- 313
2
votes
1
answer
192
views
A Vandermonde-type system
For a prime $p$ and $a_1,\dotsc,a_n\in\mathbb F_p^\times$, consider the system of equations
$$ \begin{cases}
\begin{align}
a_1 + \dotsb + a_n &= 0 \\
a_1x_1 + \dotsb + a_nx_n &...
- 23k
2
votes
0
answers
100
views
Sum of binary quadratic forms over inputs of equal Hamming weight
$\DeclareMathOperator{\field}{\mathbb{F}}$
Let $n$ be a positive integer, and let $q : \field_2^{n} \rightarrow \field_2$ be a quadratic form, specified in coordinates as
$$q(x)=\sum_{i =1}^n \alpha_i ...
- 980
2
votes
0
answers
157
views
On hypergeometric functions over finite fields
Let $\mathbb{F}_q$ be a finite field of $q$ elements. Let $A,B,C,\cdots$ denote the multiplicative characters over $\mathbb{F}_q$, and let $\overline{A}$ denote the inverse of $A$, i.e., $A(x)\...
- 93