Questions tagged [coding-theory]

The mathematical theory of codes, as used in communication, data compression, and cryptography.

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8
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2answers
284 views

Bivariate low-degree polynomial testing of Polishchuk-Spielman

In the seminal paper of Polishchuk and Spielman where they give a construction of nearly linear sized $PCP$ for an $NP$ problem, one of the key ingredients is a low-degree test for bivariate ...
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0answers
12 views

Is it possible to find the huffman code for each symbol given only the encoded string

If i have a encoded string such as this : 01111001100011010111100 And lets say i know the number of symbols, is it possible to find the code for each symbol ?,...
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71 views

How hard is it to approximate distance of linear code

I'm trying to figure out what is the current knowledge about how hard it is, given a generating matrix of a linear code over a field $F_{q}$, approximate it's distance. I of course found that ...
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34 views

Attacks on Niederreiter cryptosystem

Guo et al presented an attack on QC LDPC McEliece cryptosystem. Before that, a similar attack was presented on QC MDPC in Asiacrypt by Guo et al. Do these attacks also work on Niederreiter ...
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Quantum security of cryptosystems

One of the main candidates for PQ cryptography is code based cryptography (other than lattice based). The Niederreiter cryptosystem based on goppa codes is shown to be resistant to hidden subgroup ...
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48 views

Underlying codes in 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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91 views

Where in $PH$ are these problems?

Is 'Given two codes with alphabet in $\mathbb F_2$ with Generator matrices $G_1$ and $G_2$ do they have the same minimum distance?' in $NP$ or is it in $coNP$ (I can see it in $P^{NP}$)? If $G_1$ is ...
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190 views

Geometric Intuition behind Locally testable codes

Conventional coding theory provides a good geometric picture behind linear error correction codes in terms of Hamming distance. What additional geometric requirement one should add to make a code ...
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1answer
129 views

Minimum distance of a code

Is there a way to compute minimum distance of a code given a systematic parity check matrix? I know that min dist is smallest number $d$ such that there exists $d$ linearly dependant columns. I am ...
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0answers
26 views

Minimum distance of a code [duplicate]

Is there a way to compute minimum distance of a code given a systematic parity check matrix? I know that min dist is smallest number $d$ such that there exists $d$ linearly dependant columns. I am ...
2
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0answers
405 views

Lee metric, Gilbert-Varshamov and Hamming bounds for larger relative distance ranges

Gardy and Sole provide a Gilbert-Varshamov lower bound and a Hamming upper bound for the Lee metric when the distance between codewords is smaller than the length of the code (captured by $r = \delta ...
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2answers
2k views

How many different Huffman encoding for a given number of symbols

In Huffman coding, if we have two symbols to be encoded, we will get the result either 01 or 10. If we have three symbols, we ...
8
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1answer
113 views

Can Quarter-Subset Membership be decided space-efficiently?

Consider the following decision problem. Let $q = \sum_{i=0}^{n/4} \binom{n}{i}$ and let $(C_0^n, C_1^n,\dots,C_{q-1}^n)$ be a suitable enumeration of those subsets of $\{0,1,\dots,n-1\}$ that have at ...
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2answers
767 views

Explanation of polar decoding?

Polar decoding, according to Arikan's paper, can be decoded using a successive cancellation (SC) decoder, which is based on calculating the likelihood ratio, as seen in the image below. How can I ...
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2answers
61 views

Bounds for maximum number of code-words in a ternary error correcting code with length n and distance d?

I'm not sure I can find an explicit formula. Wondering if anyone can come up with lower/upper bounds.
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67 views

Explicit Formula of Delsarte's Linear Programming Upper Bound for $A_q(n,3)$

The problem of giving an explicit formula for $A_q(n,d)$ is sometimes referred to as "the main problem in coding theory." The value of $A_q(n,d)$ is given by the maximum number of codewords in a q-ary ...
2
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2answers
202 views

Why can't codes be defined over infinite fields?

In Coding Theory, people use $q$-ary alphabets: why do we need a finite set? Why can't we define codes over infinite sets. such as $\mathbb{R}$ or $\mathbb{C}$?
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71 views

Error correction with asymmetric channel

Suppose A is trying to transmit a message to B over a noisy low bandwidth channel, while B has the ability to simultaneously transmit arbitrary amounts of information losslessly to A. Are there ...
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13answers
2k views

Using error-correcting codes in theory

What are applications of error-correcting codes in theory besides error correction itself? I am aware of three applications: Goldreich-Levin theorem about hard core bit, Trevisan's construction of ...
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1answer
100 views

Does a code need at least two symbols to be defined as a code? [closed]

I am wondering whether you could still call a code something that, if transmitting, only transmits one symbol. Or does the formal definition of code require 2 or more symbols? (and would the answer ...
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72 views

Function which detects rotation of bit string

Consider a function $F: \mathbb{F}_2^d \to \mathbb{Z}^n = (f_1,\ldots,f_n)$ with the property that if $y \in \mathbb{F}_2^d$ is a rotation of $x \in \mathbb{F}_2^d$, i.e. $y$ is $x$ permuted by an ...
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1answer
125 views

Why are folded Reed Solomon Codes considered non linear?

This is for my understanding. What am I missing?
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1answer
53 views

Families of LDPC codes with constant error fraction corrected

I am looking for families of error-correcting LDPC codes with a constant error fraction corrected by a decoding algorithm. For example, I know that Sipser and Spielman proved that there is an ...
21
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2answers
3k views

How good is the Huffman code when there are no large probability letters?

The Huffman code for a probability distribution $p$ is the prefix code with the minimum weighted average codeword length $\sum p_i \ell_i$, where $\ell_i$ is the length of the $i$th codword. It is a ...
2
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0answers
134 views

Unit hypercube encodings

How can we chose to place $k$ points in $[0,1]^d$, such that the minimum Euclidian distance between any two points is maximized? Is there a more common term for these combinatorial designs than unit ...
4
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2answers
632 views

Relation between group theory and information theory

Motivation: I am interested about the application of group theory to information theory. To be precise, I am interested in data compression (source coding theory). Question: Is there any paper/survey ...
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1answer
175 views

Structured set of binary words

Definitions: Let $n\in \mathbb N$ be an integer, and consider the field $\mathbb K=GF(2^n)$. For $c\in \mathbb N$, let $S_c$ be a set of $n$ elements from $\mathbb K$ such that: Every element $e$ ...
2
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1answer
88 views

Subspace-evasive set performance in the random case

A subspace evasive set is defined as a large subset of a vector space which has small intersection with any $k$ dimensional affine space. That is, it "evades" all affine subspaces of small enough ...
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1answer
106 views

The noise distribution on $F_2^n$: probability of landing in a subspace versus a coset

Let $V = F_2^n$ be the $n$-dimensional vector space over the field of two elements. The $\epsilon$-noise distribution on $V$, denoted $\mu_\epsilon$, is a probability distribution on $V$ for which ...
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0answers
148 views

The edit distance of BWT of two strings with one difference

Let $BWT$ stand for the Burrows-Wheeler transform on strings. What is the maximal edit distance of $BWT(w)$ and $BWT(u)$, if $w$ and $u$ differ only in one character.
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40 views

Reference request on dynamic flows combined with network coding

I have read some papers about network coding and dynamic flows (flows over time). I think I have made comprehensive searches on google, google scholar and IEEE Xplore. IMHO, the reasons for the ...
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0answers
70 views

Convex hull of codebook (LP-decoding)

So the well-cited article by Feldman et al from 2005 has a method of constructing the convex hull of the feasible set for ML-decoding. Basically, he considers the parity check matrix $H$ as a Tanner ...
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0answers
73 views

Communication complexity of edit resilient synchronization

Supposing we have two strings $A$ and $B$ that are both edit distance $\tau$ from each other at two different sites is there a communication complexity model where synchronizing such strings have been ...
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0answers
60 views

Optimal distribution of integer edge weights

I am not sure whether the following problem has been studied. Any help would be greatly appreciated. I have $L$ sets, $S_1, S_2,...,S_L$, each of $n$ elements, taken from a universe of $N$ elements. ...
3
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2answers
168 views

Are there Similar Distance Binary Error Correcting Codes?

I'm trying to find a low distortion embedding of the trivial metric space into hamming space. It seems this should be doable by using a large set of low dimensional vectors, with approximately equal ...
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0answers
50 views

Reference for randomized GMD decoding

The GMD decoder is an algorithm for decoding concatenated codes up to half their minimal distance. The standard presentation of this algorithm usually proceeds in two steps: First, one shows a ...
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0answers
62 views

The curve used in Parvaresh-Vardy decoding

Consider the Parvaresh-Vardy list decoder. As I understand it, the idea is to decide on a curve over an extension field of the form $(f,f^h mod E, f^{h^2} mod E,\dots)$ and then evaluate each of ...
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0answers
44 views

Number of different cycles in cyclic codes with length n

I am studying Information theory, coding theory in particular at the moment, and I am having trouble determining how many different cycles are defined by a certain generator polinomial? Given a ...
4
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0answers
68 views

What is the largest noise rate $\eta(n)$ for which learning parities with noise is easy?

Learning Parity with Noise (LPN) is usually stated with constant noise rate $\eta < 1/2$ on the labels, and it is believed to be hard to learn because of the high statistical dimension of the ...
2
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0answers
113 views

Universal constant for bivariate testing

In the seminal paper of Polishchuk and Spielman where they give a construction of nearly linear sized $PCP$ for an $NP$ problem, one of the key ingredients is a low-degree test for bivariate ...
5
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0answers
124 views

Applications of small Kakeya sets over finite fields

It was proved by Dvir that a Kakeya set in $\mathbb{F}_q^n$ has size at least $q^n/n!$, a bound which was later improved to $q^n/2^n$. For $n = 2$ and $q$ odd the exact bound is $q(q+1)/2 + (q-1)/2$ ...
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1answer
65 views

Complicated Huffman coding [closed]

I am trying to figure out how to code these symbols. I am pretty sure I have it, but it gets a little tricky. Let A,B, and C have probabilities .71, .16, and .13 respectively. I am trying to code the ...
4
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2answers
406 views

On the need for a self-correcting function in the PCP theorem

Original proof of the PCP theorem, uses self-correction property of linear functions. Assume we have $f: \{0,1\}^n \rightarrow \{0,1\}$, a function or table of values, that is $(1-\delta)$-close to ...
3
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1answer
347 views

Distance of arbitrary vectors to Hadamard code

Let $n$ be a positive integer and $N = 2^n$. The Hadamrd code of "block length" $N$ can be generated using the inner product $f_u(x) = \langle u,x \rangle$ mod $2$ for all $u \in \{0,1\}^n$. It is ...
4
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1answer
289 views

Approximation algorithms for min vector subset-sum over GF(2)

In this question vzn asked about the following problem, which I'll call Vector-Subset-Sum. Given a set of vectors $v_i$ over GF(2) and a target vector $y$, is there a subset of the $v_i$ summing to ...
2
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1answer
106 views

Reference request: Classical analog of quantum threshold theorem

For quantum circuits, once the gate error is below a threshold, the error probability of an entire computation can be driven exponentially small with polylog costs in time and space: http://arxiv.org/...
2
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1answer
269 views

Question about “typical set” in Shannon's source coding theorem

I was following the textbook by David Mackay: Information theory inference and learning algorithms. I have question on asymptotic equiparition' principle: For an ensemble of $N$ $i.i.d$ random ...
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1answer
302 views

Source Coding Theorem: what happen when we go below N*H(x) bits?

I was following the textbook by David Mackay: Information theory inference and learning algorithms. I have question on Shannon's source coding theorem (p81): $N$ i.i.d. random variables each with ...
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2answers
136 views

Determining the distribution of results of a simple algorithm

Setting Consider repeating the following process on the numbers $N=\{1, 2, 3, \ldots, n\}$: Pick an integer $k \in N$, uniformly at random. Pick a subset of $k$ elements from $N$, uniformly at ...
27
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1answer
516 views

Good codes decodable by linear-sized circuits?

I'm looking for error-correcting codes of the following type: binary codes with constant rate, decodable from some constant fraction of errors, by a decoder implementable as a Boolean circuit of ...