Questions tagged [coding-theory]

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

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126 views

Computational complexity of minimum distance of rate $\frac{1}{2}$ codes

We know from Berlekamp, McEliece and Van Tilborg [On the inherent intractability of certain coding problems, IEEE Trans. Information Theory, 24 (1978)] that computing minimum distance of a (binary) ...
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1answer
343 views

Is subtractive dithering the optimal algorithm for sending a real number using one bit?

Consider the problem of sending a real number $x\in[0,1]$ using a single bit $X\in\{0,1\}$ in an unbiased manner. We assume that the sender and receiver have access to shared randomness $h\sim U[-1/2,...
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Is there a standard interpretation of this quantity?

Define an encoding scheme by a pair of functions: $$\mathsf{encode} : \mathcal{M}\to\mathcal{X},\quad \mathsf{decode} : \mathcal{X}\to\mathcal{M}$$ Typical examples of encoding schemes are things like ...
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1answer
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What are the general direction and target question in the field of quantum error correction?

After quantum error correction was introduced in mid '90s, in subsequent years many of the classical analogues regarding the structure of code (such as singleton bound, GV bound etc) were obtained in ...
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1answer
90 views

Survey on Quantum error correction

Are there any standard recent survey articles on quantum error correction (and may be including fault Tolerant computing)? The most standard ones that many people refer to are this and this. Both of ...
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48 views

Damerau–Levenshtein distance with transposition of non-adjacent characters?

Wondering if it's possible to calculate Damerau–Levenshtein distance with transposition of non-adjacent characters (DL distance allows transposition of immediately adjacent characters only). I want ...
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1answer
139 views

Explicit Bits-back Coding (a.k.a. Free Energy Coding) applied to Gaussian mixtures

I've been trying to understand Bits-back coding (Frey, B. J., and G. E. Hinton. 1997.) a bit more (pun intended), which can be used to encode data with latent variable models. This tutorial by Pieter ...
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114 views

Quantum error correction and graph codes

I was reading combinatorial approach towards quantum correction. A lot of work in this is on finding diagonal distance of a graph. Let me add definition of diagonal distance so that this remains self-...
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236 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 ...
2
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1answer
100 views

Why Asymptotic Equipartition Property theorem proofs assume the source is memoryless?

I do not understand the assumption $X_1, X_2, \cdots$ are i.i.d. ~p(x) in the AEP proofs I have seen. I have read some different sources for understanding the Asymptotic Equipartition Property. Using ...
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100 views

Notation of sequences in rate distortion theory

I have been reading whatever sources I could get my hands on today, regarding this problem. Most notes online about rate distortion theory come from the book Elements of Information Theory by Thomas ...
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55 views

Relation between automorphism group of a linear code and its dual code

Are there any strong connections between automorphism groups of codes that are dual codes of each other? I am looking for statements like one charcterizes other or one gives bounds on other etc. In ...
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77 views

Explicit binary codes with block length n, distance n / log n, rate 1 - o(1)?

What are the best (i.e., highest-rate) explicit binary codes with block length $n$ and minimum distance $d$, in the regime $d = n^{1 - o(1)}$? The "redundancy" of a code is the difference between the ...
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1answer
52 views

weights in low density codes

Generally, low density parity codes are decoded using sum product decoder (also known as decoding under belief propagation). Such codes are usually decoded nicely if there are no short length cycles ...
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1answer
167 views

Complexity of finding automorphism group of code

What is the computational complexity (may be both classical or quantum) for finding automorphism group of a general linear code? Is there better bound on complexity if structure of code is known for ...
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342 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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78 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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85 views

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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51 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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95 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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30 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
414 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 ...
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1answer
120 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
813 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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134 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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80 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 ...
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2answers
244 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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75 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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3k 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
127 views

Why are folded Reed Solomon Codes considered non linear?

This is for my understanding. What am I missing?
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1answer
62 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 ...
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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 ...
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139 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 ...
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865 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
177 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 $e$ ...
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1answer
95 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
107 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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149 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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76 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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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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64 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. ...
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2answers
183 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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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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63 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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50 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 ...