6 votes

Why can't codes be defined over infinite fields?

Sphere packings give a nice analogue of codes over $\mathbb{R}$. A sphere packing is a set $\mathcal{P} \subset \mathbb{R}^n$ such that $d_{\mathcal{P}} := \inf_{x,y \in \mathcal{P}, x \neq y} \|x - y\...
Noah Stephens-Davidowitz's user avatar
5 votes

Why can't codes be defined over infinite fields?

It is possible to define codes over infinite fields, but it is usually not as useful as codes over finite fields. The original motivation for error-correcting codes comes from the needs to transmits ...
Or Meir's user avatar
  • 5,370
5 votes
Accepted

Treewidth relations between Boolean formulas and Tseitin encodings

There is a relation between the treewidth of a circuit and the primal treewidth of its Tseitin transformation but you will have to take the fan-in of the circuit into account, which is large when ...
holf's user avatar
  • 2,174
5 votes

How to state the adequacy of an encoding of lambda calculus in itself?

As pointed out by others, the obvious definition of an "adequate" coding is that it is equitranslatable with any standard one. The question is therefore to characterize such codings in terms of more ...
Andrew Polonsky's user avatar
4 votes

Treewidth relations between Boolean formulas and Tseitin encodings

Actually, we can transform any Boolean Circuit $C$ that uses only $\wedge$, $\vee$, and $\neg$ gates of treewidth $k$ into a CNF whose primal treewidth is within a constant factor of $k$, where this ...
raki123's user avatar
  • 41
4 votes

How to state the adequacy of an encoding of lambda calculus in itself?

This is not an answer. It is an elaboration of the question, that looks interesting to me and maybe should deserve more attention than it actually received. First of all, let me say that there is an ...
Andrea Asperti's user avatar
4 votes
Accepted

Is the Mendler-encoding in System-F adequate?

$\newcommand{\Alg}{\mathsf{Alg}\ }$ $\newcommand{\NatF}{\mathsf{NatF}\ }$ $\newcommand{\Nat}{\mathsf{Nat}}$ $\newcommand{\map}{\mathrm{map}\ }$ $\newcommand{\Z}{\mathrm{Z}}$ $\newcommand{\S}{\mathrm{S}...
Dan Doel's user avatar
  • 931
4 votes
Accepted

Is subtractive dithering the optimal algorithm for sending a real number using one bit?

Note: See the edit at the bottom for an argument showing that there is an unbiased algorithm which has variance strictly lower than $1/12$ for all $x \in [0,1]$. We can at least prove that if $x$ is ...
zeb's user avatar
  • 376
3 votes
Accepted

Why Asymptotic Equipartition Property theorem proofs assume the source is memoryless?

Before we try to get into ergodic or whatever else, let's try to understand what phenomenon a mathematician or scientist is trying to (or could be trying to) model with AEP. Well Asymptotic for ...
Pedro Juan Soto's user avatar
2 votes

How many different Huffman encoding for a given number of symbols

The answer by Peter Shor is correct. But for an optimal case when the symbols can only be placed at unique leaf nodes the number of possible Huffman codes drops to $C_{n-1}2^{n-1}$.
Qwert_y's user avatar
  • 21
1 vote
Accepted

Entropy of a byte in a compression algorithm?

I define a random variable X which samples a bit uniformly at random from C∗. Should I expect Prob[X = 0] to be close to 1/2? Roughly, yes. The compression algorithm is lossless (bijective), so the ...
usul's user avatar
  • 7,615
1 vote
Accepted

Can we say that Church encoding is a form of Gödelization?

It's possible to do a Gödel numbering that assigns every term in System F to a unique natural number? The answer is yes, pick your favorite way of coding terms in F, simply because they are ...
Kaveh's user avatar
  • 21.6k

Only top scored, non community-wiki answers of a minimum length are eligible