Questions about Information theory, entropy, and information content of various sources

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Applying information theory to processor clocks

Has there been any research on the subject of applying information theory to a processors clock? It occurred to me that a clock is actually transmitting data that is used for synchronization of ...
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1answer
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How can we find the channel capacity from a BSC? (information theory) [closed]

I'm struggling to find how to calculate the channel capacity from a binary symmetric channel, given alpha(mean error) = 0.25 p(x1) = 0.25, p(x2) = 0.75 r = 1.25MBits/sec I found this in the ...
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1answer
35 views

Data transfer at certain frequencies

I am curious as to what the maximum data transmission rates are to a given frequency. Say you have a com channel with a frequency of 10 khz. How many bits/s would you be able to send? Thanks!
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1answer
35 views

One-shot Private Randomness Extractor

Suppose a pair of random variables $(X,Y)\in\mathcal{X}\times \mathcal{Y}$ with joint distribution $P_{XY}$ is given. I am interested in a deterministic mapping $f:\mathcal{Y}\to \{0, 1\}^k,$ for ...
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0answers
38 views

Average redundancy in Huffman or Hu-Tucker codes on random symbol probabilities

Huffman and Hu-Tucker codes are well-known compression schemes, which both come close to the entropy lower bound. It is known that if $L_1$ and $L_2$ are the lengths of a Huffman resp. Hu-Tucker code, ...
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1answer
120 views

Is there a generalization of Huffman Coding to Arithmetic coding

In trying to understand the relationships between Huffman Coding, Arithmetic Coding, and Range Coding, I began to think of the shortcomings of fuffman coding to be related to the problem of fractional ...
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0answers
51 views

Two dimensional parity check

Firstly, I would like to apologize if I misplaced this topic / i think the theory of coding is close to CS / I am little bit confused right now, in the school we were learning about Hamming's code, ...
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1answer
23 views

Prove $\forall c \in \mathbb{N} \, \exists x,y \in \Sigma^* \, [K(xy) > K(x) + K(y) + c]$

I am trying to prove a theorem (title) given in a starred problem in Sipser's book. I have absolutely no idea how I would go about showing it, and after trying a few different approaches came here ...
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1answer
48 views

What does this mean $[X]_1^T$?

I found this in information theory paper, P.3883* the authors states the following Most existing theoretic studies of network coding focus on DAGs due to its simpler structure and dure to the ...
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5answers
2k views

Data compression using prime numbers

I have recently stumbled upon the following interesting article which claims to efficiently compress random data sets by always more than 50%, regardless of the type and format of the data. Basically ...
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1answer
33 views

Question about the Simon's algorithm

This comes from trying to understand the "Simon's algorithm". So we have a set of $2^n$ kets $|x_i \rangle$ one each for $i \in \{0,1\}^n$. Each $x_j \in \{0,1\}^n$. And we have the further ...
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2answers
88 views

Shannon Entropy for Binary Numbers

First of all, I have to mention that I am very new in the field of information theory. I have a question regarding the Shannon Entropy calculation for binary values. As far as I understood, the main ...
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1answer
39 views

What is the relation between differential-privacy mechanism and entropy?

Why do differential-privacy people care whether or not the noise function saturates the lower bound of Shannon entropy? For example : Laplace distribution that is used to model the noise function ...
3
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1answer
26 views

Why is self-information defined the way it is?

The self-information of an event of probability $p_x$ is defined as $I(p_x)=-\log_2(p_x)$.¹ I fully understand this for equiprobable events of the form $p_x = \frac{1}{2^k}$. In that case, we want ...
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2answers
104 views

How much bits need to add to 100bit of data in order to correct up to 10bits?

I'm trying to calculate how much minimum bits need to be added to data of 100bits, in order to correct 10 bits that are messed up by: bits that deleted (Erasure Correcting) bits that corrupted ...
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1answer
23 views

Entropy based progress bar [closed]

Would it be possible to build a progress bar that estimates progress using entropy? Consider a web browser that is downloading a large file (for instance), which displays a progress bar indicating ...
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1answer
30 views

Can one use the PCP theorem to prove correctness of deternimistic algorithms?

I am thinking of the equality "PCP(O(log(n)),0) = P" Say I have a deterministic polynomial time algorithm $A$ whose correctness I can't prove immediately. But say I create a probabilistic version of ...
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3answers
71 views

Which fingerprinting/hashing algorithms support compounding?

The definition of fingerprinting algorithms in Wikipedia describe a property called compounding as you can see here as: Some fingerprinting algorithms allow the fingerprint of a composite file to ...
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3answers
348 views

Compression functions are only practical because “The bit strings which occur in practice are far from random”?

I would have made a comment, as this pertains to Andrej Bauer's answer in this thread; however, I believe it is worth a question. Andrej explains that given the set of all bit strings of length 3 or ...
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2answers
102 views

Why Shannon's Entropy is said to be a measure of information?

I got a bit of how Shannon explained to find the number of bits required to represent a message and Shannon's Entropy.But it's natural to know that to code alphabet letters you need ...
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3answers
239 views

Compression of Random Data is Impossible?

A few days ago this appeared on HN http://www.patrickcraig.co.uk/other/compression.htm. This refers to a challenge from 2001 - where someone was offering a prize of \$5000 for any kind of reduction to ...
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4answers
6k views

Is Morse Code binary, ternary or quinary?

I am reading the book: "Code: The Hidden Language of Computer Hardware and Software" and in Chapter 2 author says: Morse code is said to be a binary (literally meaning two by two) code because ...
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1answer
43 views

Entropy notation: What does this mean?

If you look at page 13 of the lecture slides here there is this line $H(Y) = H((1-\pi)(1-\alpha), \alpha, \pi(1-\alpha))$ I don't really understand what the term on right hand side is. At first I ...
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2answers
144 views

Is 95% of code really non-semantic fluff? [closed]

According to A Study of Wheat and Chaff in Source Code, 95% of code is "chaff", or "non-core functionality", whatever that means. Is this really a sensible study? Does the IT World article correctly ...
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1answer
31 views

Lower bounds for space with some probability of error

There is an information theoretic lower bound of $\log_2 {U \choose x}$ for the number of bits to represent a subset of $x$ elements chosen from a universe of size $U$. We can in principle use this ...
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1answer
48 views

Doubt in derivation of a proof in Information Theory

In my class we were trying to derive Shannon's Source Theorem, first by proving the equivalent form in a weaker version. The question is: Consider a biased coin with probability of heads $p \geq ...
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1answer
27 views

Using all the entropy in an biased bit

Suppose we have $n$ bits of random-looking data, and we want to encode it in such a way that instead of 1/2 the bits being 1's, we have (say) 3/4 the bits being 1's. The entropy of each bit in the new ...
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0answers
34 views

Name of a type of code similar to block codes

I've encountered a system where I need to construct a sort of quasi block code: We want to communicate a symbol $s$ from a finite-sized alphabet $\mathcal{S}$ using $N$ segments of information. ...
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2answers
23 views

Why are long block lengths commonly assumed/used in channel coding proofs?

As the title states, why are long block lengths commonly assumed or used in channel coding proofs?
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8answers
12k views

Is Morse code without spaces uniquely decipherable?

Are all Morse code strings uniquely decipherable? Without the spaces, ......-...-..---.-----.-..-..-.. could be Hello World ...
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27 views

Information content of computational problems

The notion of low information content is used to describe sparse sets and tally sets in complexity theory. Such sets can not be $NP$-complete unless $P=NP$. I am not aware of a formal ...
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2answers
66 views

An example of a code that is neither p-code or s-code but is uniquely decodable?

As the title says, code can't be a prefix-code and can't be a suffix-code, but it must be uniquely decodable. One possible code is this: {1, 101, 1001, ... }. Number of zeroes corresponds to the index ...
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Can independence numbers of box products of cycles increase after stabilizing?

Is there an evidence or a proof that the independence of strong products of graphs can increase after stabilizing? I am interested in odd cycles only. Let $C_n$ be an odd cycle and $\alpha(G)$ ...
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1answer
24 views

Inferring Shannon Capacity of pentagon

Can we infer from the fact that the number of independent sets in product of $5$-cycle two times is $5$ and in product of $5$-cycle four times is $25$, that the capacity of pentagon is $\sqrt{5}$? ...
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1answer
54 views

Is the Source Coding Theorem straightforward for uniformly distributed random variables?

Shannon's source coding theorem states the following: $n$ i.i.d. random variables $X_1,\dots,X_n$ each with entropy H(x) can be compressed into more than n⋅H(x) bits with negligible risk of ...
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1answer
42 views

Understanding the flaw in a proof attempt of the Communication Complexity of Equality

I'm new to communication theory and I've been wondering where the following simple argument fails: Equality Problem We have two players, player 1 Alice who gets an $n$-bit vector $X$ and player 2 Bob ...
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2answers
158 views

Lovasz theta of even cycle

How does one show Lovasz theta of even $n$-cycle ($n$ is even) is of form $\frac{n}{2}$? Why is the Lovasz theta of such cycles not of form $\frac{n \cos(\frac{\pi}{n})}{1+\cos(\frac{\pi}{n})}$. Could ...
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28 views

G1, G2, H1, H2 are submatrices of the generator and parity check matrices of a code as described below. Can G1 * H2' and G2 * H1' both be all zeroes?

$ G $ and $ H $ are the generator and parity check matrices respectively of a linear block code. Let $ G_1 = G(:, 1:n-s) $ (Matlab style representation of sub matrices). That is, $ G_1 $ is equal to ...
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1answer
79 views

Is a very long plain text password harder to crack than a short complicated password? [closed]

Is it true that a password consisting of the alphabet, even of common known names is much harder to find for a computer program than a short password, even though it uses numbers and other characters? ...
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1answer
62 views

Mutual information intuition

I was creating an example for a casual talk on mutual information. I considered a system of two coins, which with probability 1/2 are copies of each other, and with probability 1/2 are independent. ...
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1answer
26 views

Entropy sources: Weaver (1949) typo?

In Recent Contributions to The Mathematical Theory of Communication (Weaver 1949), aka The Mathematics of Communication (Weaver 1949) (various copies exist online), and also published as Part I of The ...
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1answer
45 views

Why the alphabet of the digital information is composed of 2 elements? [duplicate]

please don't offer an answer about "it's an electronic thing" or something like that, just keep reading. I don't understand why we use a dictionary, a lexicon, with just 2 elements to express the ...
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1answer
88 views

Relationship between message entropy and complexity of the best algorithm

Is it possible to estimate number of steps in best possible algorithm for classification of messages, using entropy of messages? E.g. linear search problem. We have an ordered set of incomparable ...
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1answer
140 views

Error-correction code for transmission only with bit-flipping from 0 to 1

I am using a transmission system that uses a Bloom filter (this part is out of my control). I want to send a small amount of data (32 bits) using this system. For each bit [0,31], I add its index to ...
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1answer
183 views

Conceptual question about entropy and information

Shannon's entropy measures the information content by means of probability. Is it the information content or the information that increases or decreases with entropy? Increase in entropy means that ...
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56 views

How to compare conditional entropy and mutual information?

I am solving a problem of information theory. The problem reads, Consider a stationary memoryless channel specified by the channel matrix $T = \begin{pmatrix}1-q&q\\r&1-r\end{pmatrix}$. ...
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52 views

Number of phrases of LZ compression

It is known that for the number $c(n)$ of phrases / tupel of the LZ compression for binary words of length $n$ the following relation holds: $$c(n)\leq\frac{n}{(1-\epsilon_n)\log_2 n}$$ With ...
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3answers
311 views

Can a transcendental number like $e$ or $\pi$ be compressed as not algorithmically random?

The related and interesting fields of Information Theory, Turing Computability, Kolmogorov Complexity and Algorithmic Information Theory, give definitions of algorithmically random numbers. An ...
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158 views

Bayesian Nets & Markov Blanket

As i passed PHD entrance exam, some days ago, i want to find solutions for challenging problem. In Bayes network on X={X1,...Xn} each random variable has P parents and Q child's. for Xi we want to ...
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Mutual Information in a Binary Erasure Channel

Imagine a Binary Erasure Channel as depicted on Wikipedia. One equation describing the mutual information is: $$ \begin{align*}I(x;y) &= H(x) - H(x|y) \\ &= H(x) - p(y=0) \cdot 0 - p(y=?) ...