Questions tagged [coding-theory]
The study of data representations that enable error detection, error correction and/or compression.
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Why is the block size chosen to be q-1 for Reed-Solomon codes?
Consider a Reed-Solomon code over a finite field of $\mathbb{F}_q$. Why is the typical block size chosen to be $q-1$ [1][2][3]? The reasoning I saw around this is ...
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Implementing algorithms from online courses in code to solidify understanding
I came across the online course Design And Analysis Of Algorithms and the assignments have several questions that ask you to design algorithms, but you are not asked to implement them in code. I think ...
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Data structure for prefix covering
I have a list $[1, 2, \ldots, T]$. I want to create a collection of subsets, such that:
each element belongs to a small number of subsets
each prefix is a union of small number of subsets (these ...
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Multi Edge Type LDPC codes - How to construct H?
I need to create a parity-check matrix, H, for a MET-LDPC code.
I know that H will still be two-dimensional and have only 0s and 1's, just like "normal" LDPC codes.
I am aware of the ...
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Calculation of compression ratio using arithmetic encoding?
Arithmetic encoding is one of the most famous entropy encoding techniques, and I am using it to encode an image.
For this, I am using the built-in function of Matlab that also gives other values such ...
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2
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How do I find the most even combination for two arrays
I have two arrays that both contain $n$ elements (positive, non zero, not negative)
$\{x_1\dots x_n\}$
$\{y_1\dots y_n\}$
I want to pair them up optimally, one from each array, so that the pairs come ...
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Am I right that Reed-Solomon codes can be used to implement arbitrary-parity RAID schemes?
I guess the question does not apply just to CS as I'm trying to understand how it applies to RAIDs, but I guess it's maybe the most suitable place to ask anyway.
There's a lot of info that RS codes ...
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Consider a ternary channel with the following channel matrix:
\begin{bmatrix}
1-\alpha & 3\alpha/4 & \alpha/4\\
\alpha/4 & 1-\alpha & 3\alpha/4\\
3\alpha/4 & \alpha/4 & 1-\alpha\\
\end{bmatrix}
I was told that the probability of error of ...
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Algebra of error models and error correcting codes?
In coding theory we typically consider the situation where we have a
channel that connects a sender and receiver. The messages flowing from
the sender to the receiver are corrupted by an error source ...
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DJNZ command in Universal Register Machine
How do I represent DJNZ command of counting machine via commands of Universal Register Machine, those commands are CLR JNE INC and TR, via this commands i have to represent DJNZ command, any help ...
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matching vector families that form a group
Is there any research/information on matching vector family sets (the U list or the V list or both) that form a group (under addition)?
You can find the definition of MV families here:
https://homes....
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Number of binary words that form a group of Hamming weight at most d
Consider binary words in {0,1}^n whose Hamming weight is at most some constant d. We want to select some of these words such that they form a group under addition. How many words can we choose at most?...
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What is the name of the following binary encoding?
S is a the set of binary strings in Shortlex order: [0,1,00,01,10,11,...]
I want to encode / decode natural numbers with the following scheme:
Encoding N: For ...
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How to show that $A(3k, 2k)=4$?
Denote by $A(n,d)$ the maximal size of a binary code of length $n$ with distance $d$.
How to show that $A(3k, 2k)=4$? From Plotkin bound: $$2k > \dfrac{3k}{2} \Rightarrow A(3k, 2k) \leq 4$$ But I ...
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Upper bound of sum of codewords lengths
I need to show that for any binary optimal code for $n$-letter source the following inequality
holds:
$$\sum_{i=1}^n l_i \leq 0.5(n + 2)(n −1).$$
By $l_i$ denoted the length of the sequence ...
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temporal compression of binary data
I have a sequence of source files that are very similar (akin to frames in a video), and each file can be compressed by a codec independently, but there is no temporal compression.
I want to further ...
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What does zsh: suspended ./a.out mean? [closed]
I was writing a program to count lines from an input text stream.
This is the program. It compiles perfectly but when I execute ./a.out to get the output my ...
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How to decode shortened Reed-Solomon code?
I am working with a shortened version of $[n,k,d]$ Reed-Solomon code. I am encoding a message of size $k−l$ which gives a shortened code of size $n−l$ (this is equivalent to encoding the same message ...
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Number of signatures of each type in a fixed column set of the Hadamard matrix
Consider a $2^n \times 2^n$ Walsh-Hadamard matrix (via Sylvester's construction). Fix a set $S \subset [2^n]$ of $k\leq 2^{n-1}$ columns.
Consider the rows in the $2^n \times k$ submatrix $H'$ that's ...
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Bound on the number of signed sums of a non-zero vector that can all equal zero
Let $u$ be a real vector of $m$ entries, and $A$ be a $\pm 1$ matrix of dimension $N\times m$, and real rank $\operatorname{rank}(A) = r$.
What are some conditions on $A$ (e.g. in terms of its rank $r$...
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Huffman Coding as optimal
In our lecture, the Huffman coding was described as optimal. Optimal with regard to the minimum information content. When asked, the professor explained to me that the length of a fixed code word ...
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Information theory - Expurgation step to go from average error to worst case error in the large error regime
Consider a discrete memoryless channel $N$. We use a code to send messages over this channel.
Shannon showed that if we have a code $C$ with a finite number of codewords $|C|$ such that the average ...
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Asymptotically Optimal Universal Code In Other Bases
Universal codes are fairly well studied, and many asymptotically optimal universal codes exist for binary data (see https://en.wikipedia.org/wiki/Universal_code_(data_compression) especially https://...
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Example on Referential transparency (wikipedia)
I have a rather foolish question on an example
explaining the idea behind Referential transparency
Here is given an example i not understand:
Consider a function that returns the input from some ...
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101
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Maximal prefix codes and maximal length
Let $X$ a maximal prefix code on an alphabet $A$, $m(X)$ its maximal length, $F = X \cap A^{m(X)}$ and $F’ \subseteq A^{m(X)}$. Let $X’ = X \setminus F \cup F’$ a maximal prefix code. Why is it true ...
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Error correction for windowed reads from cyclic tapes
I have an array of N symbols written on a cyclic tape. I read a sequence of M symbols starting from a random place on the tape. What error correcting scheme and even a coding scheme should I use for ...
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On the definition of Error-Correcting Codes
Let us start with the following well-known definition:
Definition 1. Let $C\subseteq A^n$ be a code over $A$ and let $t\in \Bbb Z^+$ be a positive integer. We say that the code $C$ is
$\boldsymbol t$...
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Unique decipherability of infinite regular language
Can we design an algorithm to test if a infinite regular language is a code?
We have the S-P algorithm to determinate if a finite language is a code. But how about the infinite part of regular ...
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143
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Epsilon balanced Code
A linear code is termed as an $\epsilon -$balanced code if all the codewords are having fractional hamming weight $\in (1/2-\epsilon,1/2+\epsilon)$. I want to show that for every $\epsilon\in (0,1/2)$,...
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How to build 4 codewords with a code distance of 5?
I wonder how can I construct 4 (distinct) codewords given the fact that code distance is 5. As far as I know that the code distance is the number of distinct bits between any 2 codewords. How to ...
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Converse proof for random coding capacity of AVC
I want to see the converse proof for the random coding (shared randomness) capacity of AVC. All I can find online is Csiszar Narayan's AVC paper which looks at deterministic coding. Further, the proof ...
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Error correction code without error detection
Error detection and correction codes require many bits of redundancy for correcting even a modest number of erroneous bits. However, we often have out-of-band methods to determine when and where the ...
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Intuitive explanation on why stochastic encoding performs better in channel coding
I am a little confused about stochastic encoding in channel coding. For example, in the identification problem (R. Ahlswede and G. Dueck, “Identification via channels”), the authors claim that we can ...
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"pure" explanation of Reed-Solomon?
I encountered two applications of RS codes - one in group testing, and another time someone said that a solution to an interview question was using it. But when I search for explanations, it's all ...
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Algorithm suggestion to order data with specific condition
Suppose, we want to rearrange all possible $n$-bit binary strings (i.e., we have $2^{n}-1$ possible strings) in a 1-D array $X$; given that stings with smaller hamming distance should be placed as ...
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Does RLNC (Random Linear Network Coding) still need interaction from the other side to overcome packet loss reliably?
I'm looking into implementing RLNC as a project, and while I understand the concept of encoding the original data with random linear coefficients, resulting in a number of packets, sending those ...
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How do I decode a received polynomial code with an error?
As a message I get (5,0,1,3), which is coding a sequence of numbers of length 2 in $\mathbb{F}_7$ as polynom with the 4 support points a1 = 0, a2 = 1, a3 = 2, a4 = 6. In the transimission occured an ...
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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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