Questions tagged [coding-theory]

The study of data representations that enable error detection, error correction and/or compression.

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Understanding a CRC32 Implementation

I'm currently trying to understand an implementation of CRC32 about which I have a question. On this page at section 6, there is the following code: ...
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Why is channel capacity of AWGN infinite?

My professor taught us that channel capacity of AWGN channel is infinite without any input power constraints. The noise is $Z \sim \mathcal{N}(0,\sigma^2) $. There is no constraint on input signal. I ...
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Total weight of Huffman Code

We are given the following letters with the respective frequencies: \begin{equation*}\begin{matrix}a/2 & b/4 & c/7 & d/6 & e/4 & f/5 & g/8 & h/10 & i/3 & j/11\end{...
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Channel coding and Error probability. Where are these probabilities from?

From where are the following probabilities? We consider BSCε with ε = 0,1 and block code C = {c1, c2} with the code words c1 = 010 and c2 = 101. On the received word y we use the decoder D = {D1,D2} ...
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Worked out example of Slepian-Wolf Theorem

Note: First posted this on Theoretical Computer Science Stack Exchange, but deleted it from there since it seems to be off-topic. The Slepian-Wolf theorem states that sequences of outputs from two ...
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Typical sets size

I am currently studying Shannon's entropy and I have just come across an exercise related to typical sets. More specifically, given a certain type $t$ for the set, the exercise asks to demonstrate ...
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Bob has to find Alices hidden gold by questioning yes/no questions

Suppose that Alice has $n$ places to hide the gold $v_1, ..., v_n$ and that Bob knows the probability of each place. Bob has to ask Alice a series of yes/no questions to find the gold. I have done ...
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Coding Theory Optimal Code with given max length

Why do we need at least 2 Code Words of the max length in optimal Codes? Any why do they just differ in their Prefix? Could someone give me more insight into this? Have to proof that for a given ...
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Number of independent parity checks satisfied by a code

I have been studying quantum error correction codes and as background, I am currently studying the theory of classical linear codes. Despite many efforts, I am unable to understand the following ...
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Complexity of maximization of entropy of Hamming distance of bitstrings

We have a set of possible "key"s $S$ represented by bitstrings of length $k$. In other words, $S$ contains an arbitrary subset of all bitstrings of length $k$. For example, when $k=3$, it can be $S = \...
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Smallest set of balls under hamming distance that covers all $n$-bit strings

Suppose we defined a set $S = \{x\mid0 \leq x \leq 2^n-1\}$. Notice that all element in $S$ can be represented with a $n$-bit binary string. Now consider subset $S_i$ such that, $$S_{y_i} = \{y \in S\...
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Name of binary encoding scheme for integer numbers

I once found on Wikipedia a nice technique for encoding $k \in (2^{n-1}, 2^n)$ uniformly distributed integer numbers with less then $\log_2n$ average bits/symbol, thanks to a simple to compute ...
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Data Compression :Compress a Compressed File

Suppose we have file A that has been compressed by the the method B and the output-file is C, now if I am not wrong We can not compress C more by method B, but there might another method=algorithm D ...
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Minimum basis for the nullspace of sparse matrices

Let $A\in\mathbb{F}_2^{m\times n}$ and denote its nullspace as $V=\{x\in\mathbb{F}_2^m:xA=0\}$. The weight of a basis $B=\{b_1,\dots,b_l\}$ for $V$ is the total weight of vectors in the basis, denoted ...
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Encoding System that Assign Same Number of Bits for Each Character

I am trying to get a binary string that has been converted from text of a text file, I am able to get that but the problem is, I need each character to be represented by same number of bits, but that ...
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Understanding connection between length of codeword and hamming distance in Hamming code

I came across following in Huffman coding: Minimum Hamming distance to correct up to s errors is $2s + 1$ because that way the legal codewords are so far apart that even with $s$ changes the ...
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What is the difference between rateless and online encoding?

Definitions of Rateless encoding and Online encoding are as follows. Error-correcting codes that employ no fixed block length are called rateless or fountain codes. Online encoding refers to the ...
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What's the decoding time complexity of LT codes?

LT codes are practical fountain codes that are near-optimal erasure correcting codes. Simply stated, for encoding a $n$-block message, each packet first chooses a degree $d\in\{1,\ldots,n\}$ ...
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Is arithmetic coding restricted to powers of $2$ in denominator equivalent to Huffman coding?

With restriction to $\frac{k}{2^n}$ as line segment ends, does arithmetic coding degrade to Huffman coding? As far as I can tell, each symbol will be encoded with an integer amount of bits, which is ...
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Effects of parity bit on odd code length regarding its size after alternation

I am trying to understand code distance, but I am not sure regarding the following scenario: Assume that you have an information word M with m bits, that You code into a coding word using the ...
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finding amount of errors that can be fixed based on code length [closed]

i tried to look online and search this site and others but haven't found any good explanation to the following simple question: how many errors can a code with length k(k>2)fix?
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distance of a code name in scheme

i wonder: is it true that if we take a information word, call it M(with m bits) for example, and code it by first coding M using a code, that we don't know anything about, except of it a length of k, ...
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Fundamental motivation behind the use of bits and binary representation

This is a naive question, but what makes binary representation special from a theoretical standpoint and from the standpoint of information theory? If for technical reasons building ternary computers ...
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Decoding problem and conditional probabilities

I'm reading the book by MacKay "Information theory, inference and learning algorithms" and I'm confused by how he introduces the decoding problem for LDPC codes (page 557). given a channel ...
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Where can I find someone that will sit with me and explain the analytics reports from my iPhone and computer? [closed]

This may seem like a dumb question to most people, BUT PLEASE READ THIS!! I AM BEYOND DESPERATE to get rid of a hacker that is digitally stalking me and terrorizing me. I picked your forum because ...
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Example of a code that can be decoded using bounded distance decoder

In the book Information theory, Inference and Learning Algorithm, in chapter 13, MacKay defines bounded distance decoding A bounded distance decoder is a decoder that returns the closest codeword ...
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What is a symbol code?

I am a physicist learning a bit of information theory. I have encountered a term ("symbol codes") on Wikipedia, and cannot find what it means: Source coding theorem for symbol codes Let $\...
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MDS codes with a given proportion of weights

I was trying to understand the hamming weight distribution of codewords in MDS codes. I read (https://wiki.cse.buffalo.edu/cse545/content/mds-codes) the following: Let C be a $[n, k, d]$ MDS code . ...
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44 views

Which algorithm to detect errors in 32bit data with 8bit parity

I want to transmit a 32bit message in eight groups of 5bit each. This leaves me with 8bits to use for error checking. Overall, a group is likely transmitted without error, but when there is an error ...
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35 views

Underlying codes for Niederreiter cryptosystems

Niederreiter cryptosystem is usually described by a parity check matrix $H$ over $\mathbb{F}_{2^n}$. The minimum distance $d$ is given by $d := min\lbrace k \text{ such that there are $k$ linearly ...
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59 views

using 2 short CRC(s) VS one longer CRC

I'm doing some research on the CRCs, but i can't find informations about the use of two (or more) short CRCs compared to the use of a longer CRC: Suppose that i have a dataword A of some length and 3 ...
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How to construct a random error-correcting code (its generator matrix) according to the code parameters?

I need to construct a random code which corrects T errors, has R check bits and has N maximum bits in the transmitting word. I have researched the topic and found a few theorems about the bounds (The ...
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Prove that the upper bound in the Noiseless-coding theorem is strict

Given a probability distribution $p$ across an alphabet, we define redundancy as: Expected Length of codewords - entropy of p = $\ E(S) - h(p)$ Prove that for each $\epsilon$ with $0 \le \epsilon \...
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Showing that a source is not Markov

If I have some source sending codewords and I take a large sample of codewords in order to construct an empirical distribution of the codewords sent. Using this empirical distribution, what methods ...
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Optimal prefix-free binary code with several codewords already assigned

The classic Huffman algorithm, as Wikipedia states, finds an optimal prefix-free binary code with minimum expected codewords length, given a set of symbols and their weights. Now, suppose codewords ...
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Coding for data compression with large target's symbol set (where the target symbol set is larger than the source symbol set)

For data compression, every codding that I've seen is binary. It means we convert a language with $N$ symbol size to a language with $M=2$ symbol size. For example, in Huffman coding, the goal is to ...
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finding the overhead and distance of an unknown code based on message making algorithm

for an information word M with m bits that is coded as following: M is coded into a word A using an unknown code that allows detection of not more than one error. the code word is the word obtained ...
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40 views

Sphere packing inequality for error-correcting codes

i am wondering if the following inequality is correct: if a code allows repairing of no more than k errors (inclusive, included) and m is the number of information bits and r the check bits, then $$∑^...
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Difference between a regular and a stationary source?

As far as I understand a stationary source is a regular source but it's not necessarily true the other way around. And a stationary source is a source for which its distribution is unaffected by a "...
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arithmetic coding for generating random number with desired distribution

Hi i want to convert random number with uniform distribution to desired distribution using arithmetic coding. It has been done in the following research paper called arithmetic distribution coding ...
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Does a binary code with length 6, size 32 and distance 2 exist?

The problem is to prove or disprove the existence of $C$, s.t., $|c| = 6,\forall c\in C$; $|C| = 32$; $d(c_i,c_j)\geq2,1\leq i<j\leq32$. ($d$ stands for hamming distance) I tried to construct a ...
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Prove the MinAveCodeLen of a product information source is less than the sum of that of the multiplicand and multiplier source?

The product of 2 independent sources $(S_A,P_A)$ and $(S_B,P_B)$ is defined as $$ (S,P)\text{ s.t. }S = \{s_As_B|s_A\in S_A,s_B\in B\}\text{ and }\ P(s_As_B) = P_A(s_A)\cdot P_B(s_B)\,\forall s_A\in ...
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How to show that in noiseless coding theorem, the bound $\mathrm{MinACL}<H(P)+1$ is tight?

The theorem states that $$ H(P)\leq\mathrm{MinACL}(P)<H(P)+1 $$ where, $\mathrm{MinACL}$ means the minimum average code word length of a given information source, i.e. the average code word ...
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Trouble understanding theory for error detection and correction in repetition code

Question 1 Consider a repetition code to detect $m$ errors. What is the smallest repetition parameter $k$ (i.e., the number of repetitions per bit) it should be used so that the code can always ...
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Media Codec Error Resilience

I'm an entire outsider to computer science eventhough I've been programming for so many years. As we know, modern audio-visual media codecs are essentially entropy codings of subjective preceptual ...
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LT codes with constant degree on information symbols

I have just read a Luby’s paper on the very basic idea of the LT codes, which may be of interest for me. For my application, encoding k information symbols is done in a process that generates a big ...
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Counting the number of multiples of number A that perfectly divides the number B

What is the best way of counting the number of multiples of number A that perfectly divides the number B. This is the sub problem for one of the questions I am solving on codechef. This is the best I ...
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Are Huffman codes self-synchronizing?

A code is (statistically) self-synchronizing if, given that the transmitted string is long enough, the receiver is guaranteed to eventually synchronize with the sender, even if bit flips or slips have ...
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How many prefix code we can find for a given distribution of probability?

A source emits 5 signals s1, s2, s3, s4 and s5 whose probabilities are as follows: 1/3, 1/3, 1/9, 1/9, 1/9, 1/9. How many prefix codes we can construct on A={a, b, c}? And how many are there with the ...
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what does the redundancy of a code means?

I was reading a paper about a transposition and single deletion error correcting code and they claim that the redundancy of the code was only $\log(6n-3)$ bits. But what does that means? I was ...