All Questions
Tagged with coding-theory binary
45 questions
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Generate Max Number of Sequences Separated by Hamming Distance of 3
I'm interested in whether there is an algorithm for generating the maximum possible number of DNA sequences that are $7$ nucleotides long that differ by at least $3$...
- 11
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0
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174
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crc table lookup not giving the same result as basic implementation
The basic implementation of CRC uses XOR and left-shift operations to find the remainder. While the index of the leftmost bit of the remainder is greater than the degree of the generator polynomial, ...
- 13
1
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2
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273
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Cyclic Redundancy Check: burst error detection capability
For a given polynomial generator $G(X) = 1 + \cdots + X^r$ of degree $r$, it can detect a burst of errors of length $r$.
So if the polynomial generator does not contain the $+1$ term, I guess it can ...
- 13
1
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1
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275
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How to find equidistant hamming sequences?
Given N as the number of bits, how to find N sequences of (2^N/N) numbers each such that: given an arbitrary number n, there is always one number in each sequence that has hamming distance 1 from n.
...
- 11
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54
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About Cyclic Redundancy Check validity
Summery of Problem:
I've read Proof of Cyclic Redundancy Check validity
I tried to understand with decimal.
And I encounter a problem:
Why
$$M(x)\cdot x^n = Q(x)G(x) + R(x)$$
equivalently
$$R(x) =M(x)\...
- 103
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0
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66
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Coding Theory: Irreducible Polynomial divides x^m+1
I need some help please. This is for a coding theory course dealing with Galois Fields $GF(2^r)$, the problem is stated as follows:
Show that if $h(x) \in \mathbb{F}_2[x]$ is an irreducible ...
- 27
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1
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365
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Dimension and basis of set of linear codes of even weight
I am working on questions about coding theory. The set C is the set of all words in binary code that have an even weight and are of length n.
I have already proven that this is a linear code. Now I ...
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1
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1
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181
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Why the minimum distance of a binary code of length $\geq 3$ is at least 3
Let C a binary code of length $n \geq 3$ with generator polynomial $g(x) \neq 1$, where n is the smallest positive integer for which $g(x)~|~x^n + 1$. Show that C has a minimum distance at least 3.
I'...
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0
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19
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How to judge that a Boolean polynomial cannot be divisible by another?
Let $f(x),g(x)\in\mathbb{F}_2[x]$, and assume that $f(x)\nmid g(x)$, $\deg (f) \ll \deg(g)$
So how to confirm this fact $(f(x)\nmid g(x))$ as soon as possible?
Further more, is there an algorithm ...
- 1
1
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2
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331
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Finding a binary prefix code provided lengths
Firstly, I am relatively new to this particular forum, and I usually use Stack exchange (maths). I do not know if this is the right place to post so please be aware in case, I should ask this question ...
- 325
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3
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323
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Prove that the Hamming distances between three n-tuples cannot be 6,2,7
Let $x,y,z \in \{0,1\}^n$, and let $d_H(x,y)$ be the Hamming distance between codes x and y.
Prove
$d_H(x,y) = 6$,
$d_H(y,z) = 2$,
$d_H(x,z) = 7$
cannot happen.
- 11
1
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0
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164
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Binary coding of Numbers with Minimum Hamming Distance
I am looking for binary encoding of a set of integers that satisfy the following two properties:
The number of 1s in the larger numbers is larger.
The hamming distance between the encoding of two ...
- 48
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1
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565
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Solving binary linear equations - Check matrix
Given the check matrix
$$
B =\begin{bmatrix}1&0&0&1&0\\0&1&0&0&1\\0&0&1&1&1\end{bmatrix}
$$
I need to solve the binary linear equation to 0 to obtain ...
-2
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1
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9k
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How to calculate the generator matrix,parity check matrix and the maximum likelihood decoding
An unknown encoding device for a binary linear block code has 4 bits of input-data pins and 8 bits of output data pins. If we send the messages
$$u_1=(1 1 0 0),\, u_2=(1010),\, u_3=(1001),\, u_4=(...
- 585
0
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1
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253
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Binary string to boolean polynomial
I have to find the corresponding Boolean polynomial to the binary string $1101$ $ 1001$.
To start, I think that I have to use a truth table to find the values for $x_1$, $x_2$, and $x_3$.
We get:
...
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