Questions tagged [coding-theory]
The study of data representations that enable error detection, error correction and/or compression.
209 questions
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Hardness for COSET WEIGHT problem in terms of Hamming weight
The COSET WEIGHTS problem is given as follows paper1.
Input: A binary matrix $A\in \mathbb{F}_2^{m\times n}$, a binary vector $y \in \mathbb{F}_2^{m}$, and a non-negative integer $w\in \mathbb{Z}^{+}$....
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Optimal dictionary for coding
Given alphabet $A$ and a sequence $S$, determine the optimal dictionary $D$ of max word length $m$ that can compose the sequence.
Optimality can be given as a two-part cost $(L)$ = dictionary cost $(...
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Prefix-disjoint code
A set $C$ of words is a prefix code if no word in $C$ is a proper prefix of any other. Consider the following stronger property $P$: no two distinct words in $C$ have any proper non-empty prefix in ...
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Optimal coding for uniform distribution
I'm following Elements of information theory by THOMAS M. COVER. The problem is about finding an optimal coding for uniform distribution. The reasoning from the solution manual works like this:
...
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Prefix code and twenty-question games?
This is going to be a simple question. I'm following the book "Elements of information theory" by Thomas M. Cover. He state the equivalence between coding and "designing a series of ...
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How are non-binary codes implemented in practice, i.e. codes over $\mathbb{F}_q^n$ for $q\neq 2$?
I am only familiar with binary implementations of information, though I am aware other implementations (ternary etc.) exist. I was wondering, if a code over $\mathbb{F}_q^n$ is used in practice, where ...
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Why, for Reed-Solomon codes, does the alphabet size depend on block length?
I'm learning about Reed-Solomon codes so apologies if this is a basic question. The alphabet size $q$ in RS codes grows with the block length $n$ - in particular, we have $q \geq n$.
But intuitively, ...
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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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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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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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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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