Questions tagged [hamming-distance]
32 questions
6
votes
1 answer
Find binary number with max hamming distance wrt given set of binary numbers
Suppose we have a set $A$ of binary numbers with the same length $n$.
For example (with $n=8$):
$A = \{ 10010011, 01011011, 00010010, 11110001\}$
Now, I want to find the binary number $z$ (also with $n$ bits) such that the minimum hamming distance…
Math-E-Mad-X
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4
votes
2 answers
Finding a vector of maximum Hamming distance from a subspace of $(\mathbb{Z}/2\mathbb{Z})^n$
Let $W$ be a linear subspace of the vector space $V = (\mathbb{Z}/2\mathbb{Z})^n$. Let $k = \dim(W)$. For $v \in V$, define the distance from $v$ to $W$ to be $d(v,W):=\min_{w\in W} d(v,w)$ where $d(v,w)$ is the Hamming distance.
Given such a…
Ben
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3
votes
1 answer
Finding a word that minimizes the sum of squared Hamming distances in a data set of words
Suppose we have a set of words of common length randomly sampled from a dictionary of words; assume you have access to that dictionary. Let $d(x,y)$ be the Hamming distance between words $x$ and $y$. The Fréchet mean word minimizes the sum of…
cgmil
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3
votes
2 answers
How to recognise the number of errors that can be detected and corrected of a large set of codewords (k) each having a specified number of bits (n)?
I am struggling to find how can I know the number of errors that can be corrected and detected using (n=10) bit code with a (k=550) codewords.
As far as I know, to calculate the number of errors to be corrected and/or detected someone must know the…
Powerful blaster
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3
votes
1 answer
Algorithm for cyclic $n$-string Hamming distance with constant sized language $\Sigma$
Suppose we are given a language $\Sigma$ where, suppose, $|\Sigma| = O(1)$. Consider two fixed strings $A, B \in \Sigma^n$. Define the Hamming metric between these strings as
$$d_{H}(A,B) = \sum_{i=1}^n \boldsymbol{1}\lbrace A(i) \neq…
spektr
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2
votes
1 answer
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 bound, at least?
A somewhat-trivial lower bound is…
TLW
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2
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1 answer
Polynomial generator required to detect single bit error in Cyclic Redundancy Check codes
I was reading about CRC coding from two books:
Data Communication and Networking by Forouzan Page 294
Computer Network by Tanenbaum Page 188
They use following notations:
$d(x)$: dataword to be sent (as a polynomial)
$c(x)$: codeword sent (as a…
Mahesha999
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2
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1 answer
How Can the Bounded Search Tree Algorithm for Closest String run in $\mathcal{O}(kd)$ per node?
I am trying to understand an algorithm for solving Closest String using bounded search trees, as found in Parameterized Algorithms (Cygan et al., 2015).
Assume we have a set of $k$ strings $x_1, ..., x_k$ all of length $L$ and an integer $d$. The…
LemongrabThree
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2
votes
1 answer
What is the difference between Hamming Distance and Manhattan Distance for non-binary data?
What is the difference between Hamming Distance and Manhattan Distance for non-binary data (specifically I am comparing points in $\mathbb{R}^2$)? I understand Manhattan sums the absolute difference in the and x and y directions but doesnt hammming…
John D
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2
votes
1 answer
Calculate number of error-correcting code check bits
To design a code with $m$ data bits and $r$ check bits which allow all single-bit errors to be corrected, the formula
$$(n + 1) 2^m \leq 2^n$$
with $n = m + r$ and $(m + r + 1) \leq 2^r$ is used.
Why is $+ 1$ used here, doesn't $n$ contain already…
Road
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2
votes
1 answer
Largest set of 10-digit numbers where none have Hamming Distance = 1 with any other
I'm working on a system that will require manual data entry of 10-digit numbers (Σ = 0123456789). To help prevent data errors, I want to avoid generating any two strings that have a hamming distance of 1.
For example, if I generate the string…
Nic
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2
votes
1 answer
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 original codeword is still closer than any other…
RajS
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2
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1 answer
How does this answer for automata and Hamming distance not lead to inconsistencies?
I had already been given the answer by the TA in class, but I don't understand it. I'm not asking for the answer on a homework problem or anything.
The problem:
The Hamming distance ("distance") of a word w to v of the same size is the number of…
pikecha
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1
vote
1 answer
Finding a function $\hat{\mu}:[0,1]\to\{0,1\}$ for a data set $X_i:[0,1]\to\{0,1\}$ that minimizes the sum of squared distances to that function
Not long ago I requested an algorithm that can find the word minimizing the sum of squared Hamming distance of all words in a data set. The answer to this question is that the minimization problem is NP-hard, though there are some approximation…
cgmil
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1
vote
0 answers
Name for class of error-correction codes
I'm considering a binary error-correction scheme, but I'm missing the correct terms to dig further into it.
The idea is to decide the code-rate during encoding, but for the decoder to decide how that's used, depending on its resources (and estimate…
cloudfeet
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