Questions tagged [coding-theory]

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

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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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What is an equivalent command to grep without piping? [closed]

Write equivalent command(s) to the following command that don’t use pipes The output should be similar, include grep and not have the piping symbol This is the command: ls -l | grep test
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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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Opening a .mp4 file with notepad++ [migrated]

I am just a curious person and one day i was watching a video on my laptop and when I right clicked on it, a window popped up and out of several options one was edit with notepad++, out of curiosity I ...
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23 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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50 views

Complexity of nearest codeword in cyclic codes

Is it $NP$-complete given $c(x),g(x)\in\mathbb{F}_2[x]$ where $g$ generates a cyclic code of length $n$ (so $g\mid x^n-1$), and $\deg c<n$ to find the nearest codeword to $c$? This is related to ...
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Is there a fundamental difference in LIFO and FIFO entropy coding?

The top two competing entropy coders at the moment are Arithmetic Coding (AC) and Asymmetrical Number Systems (ANS). A very interesting difference between the two is that AC is FIFO and ANS is LIFO. ...
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32 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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36 views

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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87 views

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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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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55 views

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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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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30 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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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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28 views

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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How to decode multiple-digit gamma codes and get the gap sequence?

How to decode gamma code ($\gamma$ code): 1110001110101011111101101111011 and get the gap sequence? Detailed information about Gamma codes ($\gamma$ codes) ...
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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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49 views

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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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 ...
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Are there codes that detect the position of an error?

I am looking for code that detect an error and it's position (or an aproximation of it), this is more than an error detector code but a little less than a correcting code. Do you know something like ...
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411 views

How can I estimate Bhattacharyya parameter for BSC channel, used for Polar codes

In polar codes, the frozen bits in each message are determined through the worst channel, where the relevant parameter is the Bhattacharyya . How can I estimate Bhattacharyya parameter for BSC ...
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Minimum number of strings to cover entire space within Hamming distance

Given $(n, k)$: What is the minimum number $x$ of (binary) strings such that all $n$-bit (binary) strings are within $k$ Hamming distance of some string? Is there an asymptotic expansion or lower ...
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Nearest codeword in a translation-invariant code over $\mathbb{Z}^d$

Let $c_1,...,c_n,c':\mathbb{Z^d}\rightarrow \{0,1\}$ all have finite support. Let $C$ be the linear, shift-invariant code generated by $c_1,..,c_n$. It is possible to calculate the nearest codeword $...
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Is there a database of linear codes with “small” covering radius

Does anyone keep a database of (or including) linear codes with small covering radius? I'm specifically interested in the smallest-dimension codes known of covering radius R ($2\le R \le 4$ say) for ...
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Simplification of Sum of Hamming Weight Sets

Let $k,n,K,m \in \mathbb{N}$ such that $k<n$. Let $l\in \mathbb{R}$, such that $0 \leq l \leq 1$. I am analyzing an algorithm and I need $O(N)$. Could you help me with the reduction of the ...
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Parameters of a linear code

Consider the code $C=\{c=(c_1...c_n): c \in \Bbb F_q^n, c_1=c_n\} \subset \Bbb F_q^n$. I was able to prove that the code is a linear code because it is closed under addition and scalar multiplication....
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Conditions to apply Source Coding Theorem

I was wondering what are the conditions to apply source coding theorem (SCT). 1. Is it applied only to uniform-length coding, what about variable-length coding, does it also satisfy SCT? I was ...
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What's the error-correcting capacity (randomly errors, or burst errors) of polar code?

Polar codes, invented by Arikan, with low encoding and decoding complexity. But the minimum distance of polar codes is not great. The normalized minimum distance goes to 0 with the block size. Yet, ...
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Showing that a binary linear code $C$ is self-dual

Let $C*$ be the length 8 binary code obtained by adding a parity check symbol to each word in $C$. (so a word $c_1, c_2, c_3, c_4, c_5, c_6, c_7$ is extended to the word $c_1, c_2, c_3, c_4, c_5, c_6,...
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487 views

What is the error-detection-probability of CRC

I have a table which stores all serial numbers of devices in my system: ...
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Hamming code extension with additional parity bit

Let $Ham$ be the $[7,4,3]_2$ Hamming code. It is known that $\{w(c):c\in Ham\}\subseteq\{0,3,4,7\}$, where $w(c)$ is the Hamming weight of the word $c$. A code $C$ of length $n$ is called cyclic if ...
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Algorithm to find number of covering sets

UPD: The following problem comes from error correction codes, to be precise -- either maximum-likelihood or belief-propagation decoding when the spectrum of special failing subsets is known (to some ...
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224 views

Binary code and Hamming distance

I'm learning about CRC and Hamming distance and I have three questions. Lets say we have binary code described by ($+$ refers to sum modulo $2$): \begin{alignat*}{1} a_1 &+ a_2 &+ a_3 &+ ...
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94 views

How to compare the efficiency of two encoding schemes or hypothesis languages?

My question is pretty basic, I'm looking for a named method if you know one, but also proper terminology, further reading, and anything this reminds you of if you don't. (I'm new to this, don't have ...
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2k views

Difference between fixed-to-variable length codes and variable-to-fixed length codes?

I am a bit confused by the difference between the two. Can someone clarify the difference between the two?
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How an insertion can be decoded in Varshamov and Tenengolts (VT) codes

I have a good idea of how deletion can be decoded in VT codes using Levenshtein algorithm. However, I don't have any idea how can an insertion be decoded? Can anybody give me a small example using a ...
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Soft binary ECC? How to maximize the minimum distance over a channel?

Given that The channel is tiny, it has message size of 72 bits Each bit is probabilistic, meaning that I have a value between 0 and 100 of how likely that bit is on/off1 Only 24 bits of acutal data ...
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Vandermonde matrix and its binary representation

Say one is given a Vandermonde matrix (https://en.wikipedia.org/wiki/Vandermonde_matrix) of dimension $2^q \times k$ such that the elements of the first column of it are $\{0,1,2,..,-1+2^q\}$. (This ...
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Error detection code supporting arithmetic

I am looking for error detection codes, which support addition in the encoded domain and are separate (a tuple of ($N$, $R(N)$), where $N$ denotes the functional value and $R(N)$ its redundancy). So ...
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77 views

Code families with efficient decoding algorithms

Which families of the error correcting codes have an efficient decoding algorithm? I know that decoding a general linear code is NP hard (the general decoding problem). I also know that Goppa codes ...
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206 views

What is the complexity of Hamming nearest neighbor to a subspace …?

Suppose that $F_2$ denotes the field with $2$ elements. We are given $m$ vectors $\{x_1, \ldots, x_m\}$ in $F_2^d$ which are a basis for a subspace $W$. Suppose we have a vector $v \in F_q^m$, and ...