error-correcting codes, error-detecting codes and related algebraic and/or combinatorial constructions

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2
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2answers
15 views

Channel code for multiple bit errors

I've been exploring information theory out of personal interest and have a cursory understanding of Hamming Codes. From what I can tell, they're designed to exclusively detect the location of a single ...
2
votes
0answers
33 views

Sizes of Hamming balls on the discrete torus

Consider the discrete torus $\mathbb Z^2_k $, with $k$ even, i.e. the graph with vertex set $\{0,1,\dots, k-1\} \times \{0,1,\dots, k-1\}$ and edges between any pair of vertices which differ in ...
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1answer
19 views

Why the only binary MDS codes are trivial ones?

Why the only binary MDS codes are trivial ones? I have been thinking how to draw a contradiction by assuming the MDS code is not trivial. Thank you very much!
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1answer
61 views

Coding theory (parity check matrix) [closed]

***my question is : Let n ≥ 4 and let H be a parity-check matrix for a binary linear code C of length n. Suppose that the columns of H are all distinct and that the weight of every column of H is odd. ...
0
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0answers
21 views

Description length in model coding

In class, our professor posted the following: We will discretize $\theta$ (some model) into $1/\sqrt{n}$ distinct values. Intuitive argument: with N data points, our estimation error for $\hat ...
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0answers
19 views

How many different binary prefix-free encoding schemes with given lengths?

For length =3: Here is my thought process, since every word is length 3, they are all prefix-free from each other. Then, there are 8 choices for the first word, 7 for the second, etc. Therefore, ...
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0answers
19 views

List all vectors of the binary [9,2] cyclic code

How do I know how many vectors there will be? I know the length(dimension?) is 7. I also know the generator matrix is: ...
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0answers
14 views

Give the idempotent generators of the four binary QR codes C1 , C2 , C3 , C4 , of length 7.

I'm having trouble on some homework. This is the last problem and I can't figure it out. Can anyone help or point me in the right direction? Thanks! For each code Ci , 1 ≤ i ≤ 4, from part (a), give ...
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0answers
6 views

Generator polynomials of idempotent binary QR code

I'm doing some homework and I ran into a question that I just do not know how to do. I have to give the generator polynomials of the four binary quadratic residual codes of length 7. Quite honestly I ...
1
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1answer
29 views

Generator matrix of a binary cyclic code

I need to find the Generator and Parity check matrix of a binary cyclic [9,2] code. If I calculated right, the Generator polynomial is x^7 + x^6 + x^4 + x^3 + x + 1 and the check polynomial is x^2 - x ...
2
votes
0answers
20 views

Error correcting codes for asymmetric channels

Most work in error correction coding theory (Hamming, Cyclic, BCH, Reed-Solomon, Turbo Codes, LDPC...) deals with linear codes. Now, a linear code binary code is a good fit (only?) for a symmetric ...
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3answers
111 views

Irreducible factors of x^16 - 1 over GF(3)

Just want to double check my work. I'm trying to list the irreducible factors of $x^{16} − 1 $ over $GF (3)$ of degree $1$ and $2$ . Here's what I have: $$x + 1, x + 2, x^2 + x + 2, x^2 + 2x + 2$$ ...
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0answers
36 views

Finding a generator polynomial of all binary cyclic codes

I need to find the generator polynomials of all binary cyclic codes of length 7 that contain the vector (1, 1, 1, 0, 0, 1, 0). From what I know of generator polynomials of a cyclic code divides x^n - ...
2
votes
1answer
46 views

Show that the map $\varphi $ is an isomorphism.

We defined the Generalized Reed-Solomon codes the following way: $\alpha=(\alpha_0,\alpha_1,\ldots,\alpha_{n-1})\in \mathbb{F}_q$, distinct elements of the finite field $\mathbb{F}_q$, $n\leq q$, ...
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0answers
42 views

Idempotent generators of the four binary QR codes of length 7

I have a coding theory assignment and I thought it would be a good idea to double check before I hand it in. I'm asked to find the idempotent generators of the four binary QR codes C1, C2, C3, C4, of ...
2
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0answers
31 views

Give the generator polynomial of a binary cyclic [9, 2] code.

I'm new to Cyclic Codes and I'm not sure the process to find a generator polynomial of a cyclic code. I know what a cyclic code is, but not sure how to find a generator polynomial. Since the question ...
2
votes
3answers
41 views

question on morse code

The morse code is made up of marks called dots and dashes."Q", for example is (--,--).Is it possible to make up such a code so that every letter of the alphabet is represented by at most three marks? ...
0
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0answers
15 views

Encode A11B modulo 37

Encode the word A11B modulo 37 using the encoding 0=0, 1=1, . . . , 9=9, A=10, B=11, . . . , Z=35, blank space=36. I took the weighted sum: 5(10) + 4(1) + 3(1) + 2(11) + 1(c) ≡ 0 mod 37 Solving, i ...
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0answers
56 views

Why does the eigenvector have to be positive here?

The following Pictures are taken from "Symbolic Dynamics and Coding" by Lind and Marcus.! Can you tell me why it is so important that the eigenvector $v$ is positive? Where exactly is this ...
8
votes
0answers
279 views

How can I construct a solution for this system of many inequalities?

Let there be types $\omega\in\{0,1\}^n$ drawn according to some probability distribution. Suppose that these types are relayed through some imperfect message service. Specifically, any type $\omega$'s ...
0
votes
0answers
28 views

self orthogonal binary code

I know the definition of a self-orthogonal (or weakly self-dual) code; however, I am experiencing a bit of a trouble in building such codes - well, except trivial ones, but that is not interesting. ...
2
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0answers
27 views

Leech Lattice and Golay Code

Consider the following Miracle Octat Generator or MOG. Choose the sign $\pm 3$ and fill in the blanks $\pm 1$ to create a point $x$ in the Leech lattice $\Lambda_{24}$ with $||x||^2=8$. $ ...
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1answer
21 views

Construct a 2-error correcting Reed-Solomon code over GF(11).

I'am trying to construct a 2-error correcting Reed-Solomon code over GF(11). Cna anyone help me to start?
0
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1answer
15 views

Self-dual code from parity-check matrix

I am trying too make a self-dual code from this parity-check matrix: _ _ | 1 1 1 1 1 1 1 1 | | 1 1 1 1 0 0 0 0 | H= | 1 1 0 0 1 1 0 0 | |1 0 1 0 1 0 1 0| ...
2
votes
1answer
54 views

Existence of a BCH-code?

I would like to ask about existence of a $[31,11,12]$ binary BCH-code. How to prove its existence, if it does i have to find the generator polynomial? Can i use some specific bound that could show or ...
2
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1answer
36 views

Code Correction-Detection

From theory: If, for a code C, d(C)≥ s+ 1 then C can detect up to s errors. If d(C) ≥ 2t+ 1 then the code C can correct up to t error. Assume that we have C={00001, 00010, 00100, 01000, 10000} ...
2
votes
1answer
23 views

Parity Bit Detecting Odd Bit Errors

I'm going over a past paper which has a true or false question with the following statement. A single parity bit computed over 128 data bits can detect an error when bit-flips occur in exactly 93 ...
2
votes
1answer
21 views

Undetected Errors in 2 Dimensional Parity

Given a two dimensional parity with a data block of two rows and two columns what is the probability that a four bit error goes undetected? The naive method would be to look at all ways in which an ...
2
votes
2answers
56 views

Coding theory - polynomial

Let $C \in (\mathbb{Z}/5)^3$ be the code consisting of all elements $(x_1,x_2,x_3)$ satisfying $x_1 +3x_2 +2x_3 = 0$ Show that this is a 1-error detecting code. What is the minimal distance of $C$? ...
0
votes
1answer
34 views

(32,7) Reed Muller code [closed]

I was wondering if someone can give me a hint on the problem I have. I came across (32,7) RM code that I am trying to decode. I was given the generator matrix that obviously contains R(1,5) and an ...
1
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1answer
43 views

Code that can be generated from 3 of 5 trusted people?

Suppose a computer contains sensitive data protected by a 3-digit passcode. (I understand this does not provide much security in the real world, but for the sake of the problem, assume only 3 digits.) ...
0
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1answer
54 views

Explicit formula for Nth string of Gray Code.

From Wolfram MathWorld, we have: "A Gray code is an encoding of numbers so that adjacent numbers have a single digit differing by 1. The term Gray code is often used to refer to a "reflected" code, ...
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2answers
61 views

Show that four codewords is the maximal size for a code in V^8 = {(a1,…a8) | ai is in {0,1}} that corrects 2 errors

Here's what I have: The size of the set $V^8$ is $2^8=256$. If the code can correct 2 errors, the minimum distance of the code must be 5. Then the size of an open ball of radius 5 around the code ...
0
votes
1answer
27 views

Highest pairwise Hamming distance between k bitvectors of length n

What is the highest achievable pair-wise Hamming distance $d$ between all possible pairs from $k$ bitvectors each having a length of $n$ bits? The content of each bitvector can be arbitrary, only the ...
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1answer
32 views

Find the number of primitive elements

How can i find the number of primitive elements over the field of order q? GF(27) for example. Is there a formula that I can follow? I'm really confused on how to find them. Any help would be much ...
2
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2answers
20 views

Palindromic Hypotenuses?

What is the largest seven-digit palindrome which can be expressed as the sum of two perfect squares? I tried Java but couldn't get the right answer, and unfortunately OEIS ends at around 5 digits in. ...
0
votes
1answer
46 views

How does an alphabet size relate to the Kraft-McMillan Inequality?

I'm trying to figure out how the alphabet size m relates to the McMillan Inequality. I'm using Norman L Bigg's equation which is $\sum_{i=1}^M \frac{ni}{b^i}$$=\frac{n_1}{b^1}, \frac{n_2}{b^2}, . . . ...
1
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1answer
29 views

Calculation of polynomial in the finite field

I'm trying to understand the McEliece cryptosystem and I'm looking to this paper http://www.mif.vu.lt/~skersys/vsd/crypto_on_codes/goppamceliece.pdf On page 26 they are calculating syndrome and ...
0
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0answers
23 views

Explaining of lost probalbity over random loss channel

I am reading a paper about packet loss probability over random loss channel. In this paper, the author give a equation about loss probability as $(1)$. However, I cannot understand the meaning of it. ...
0
votes
1answer
31 views

Counting binary words distance one from codewords

If you have a [15,11] hamming code, how would you count the number of binary words that are distance 1 from codewords? I know this code will have 2048 codewords, and there are 32768 binary words of ...
2
votes
0answers
13 views

ternary code from Hadamard matrix

I am looking for a direct proof of this statement: "A $12\times 12$ Hadamard matrix is the generator matrix of a ternary selfdual (linear) $[12,6,6]$ code $C$." That the length is 12 is clear. As ...
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0answers
25 views

Constant-weight code for error correction

I need some Constant-weight code for error correction. Understanding how these codes generated is really hard for me. The papers of this topic are focus on lower bound and upper bound. What I need is ...
4
votes
0answers
23 views

How to number the natural numbers lexicographically with minimal overhead (and provide a lower bound for the overhead)?

Working in binary, note that the number 100 is lexicographically smaller than the number 11 even though $100 > 11$. How can we devise a function $f$ such that $f(a)$ is lexicographically smaller ...
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0answers
36 views

Binary Polynomial Factoring

I just need confirmation that I've done my math right. If $a(x) = x^4 + x^3 + x + 1$ and $b(x) = x^2 + x + 1$ are binary polynomials, find binary polynomials s(x) and r(x) such that $x^4 + x^3 + x + ...
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1answer
40 views

Primitive elements of GF(8)

I'm trying to find the primitive elements of GF(8), the minimal polynomials of all elements of GF(8) and their roots, and calculate the powers of α^i for x^3 + x + 1. If i did my math correct, I ...
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0answers
15 views

Generator matrix of a Reed-Muller code [duplicate]

I need to find a generator matrix (2,4) of the Reed-Muller code (2,4), the dimension of R(2,4) and the minimum distance of R(2,4). I know that R(r,m) of order r, then length: n^m, dimension k = 1 + ...
0
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1answer
25 views

Find out the primitive polynomial GF(3)

1.) $x^2 + 2x$ 2.) $x^2 + 1$ 3.) $x^2 + 2$ 4.) $x^2 + 2x$ 5.) $x^2 + 2x + 1$ 6.) $x^2 + 2x + 2$ 7.) $x^2 $ 8.) $x^2 + x + 2$ 9.) $x^2 + x + 1$ Can any one help me in listing out primitive polynomials ...
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1answer
24 views

When Errors Go Undetected in CRC

I understand that CRC will not be able to detect errors if: The remainder of $E(x) / G(x) = 0$ $E(x) = G(x).Z(x)$ for some polynomial $Z(x)$ I understand the first point, which means that if the ...
0
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1answer
50 views

How many primitive elements does GF(256) have?

I know the answer for this is 36 but I don't exactly know how to reach to this. Can you any one help me in understanding this.
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1answer
65 views

Polynomial Arithmetic Modulo 2 (CRC Error Correcting Codes)

I'm trying to understand how to calculate CRC (Cyclic Redundancy Codes) of a message using polynomial division modulo 2. The textbook Computer Networks: A Systems Approach gives the following rules ...