Tagged Questions

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

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1answer
35 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: ...
1
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1answer
64 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 ...
0
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1answer
80 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 ...
1
vote
1answer
47 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 ...
0
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2answers
129 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 ...
1
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0answers
26 views

Decoding of Gabidulin codes

Consider the space of matrices in $\mathbb{F}_q^{n \times m}$ where $\mathbb{F}_q$ is the finite field with $q$ elements. We can define a metric on this space, given by $d(A,B) := rank(A-B)$, called ...
3
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1answer
49 views

Are there good locally decodable _erasure_ codes

Are there known locally decodable erasure codes with linear codeword length and $\:n^{o(1)}\:$ query complexity? According to pages 1 and 4 of this link (which annoying does not give its own ...
2
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0answers
40 views

Complexity of counting codeword length distribution

Suppose I have a code $C$ over $GF(2)$. I would like to count exactly the number of codewords of $C$ of weight $k$. Here $k$ should be thought of as small compared to the dimensions of the code. ...
2
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4answers
316 views

Research in Coding Theory

I have just started learning about coding theory. Hence, I would like to ask for your suggestions and guidance for a very beginner like me. Which books are good for beginning coding theory? (I start ...
6
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1answer
145 views

Coding theory and complete problems

Coding theory is an useful topic in theoretical computer science. There are known examples of problems coming from coding theory which turn out to be NP complete. My questions are the following: ...
2
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0answers
57 views

Weight enumerator and levels of polynomial hierarchy

Let $A_i$ be the number of codewords in a binary linear code $\mathcal{C}$ of weight $i$. It is known that: $A_k$ is in $P$, where $k = \mathcal{O}(\log_2 n)$. $A_{n}$ is in $\#P-Complete$, ...
3
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2answers
148 views

The length of non perfect binary 1 error correcting codes

I am interested in the best known number of code words in binary 1 error correcting codes of length $n$. I am aware of the Hamming code when $n=2^r-1$, but i would like to get lower bounds for other ...
1
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0answers
20 views

Number of non-isomorphic codes

We have number of non-isomorphic graphs given in http://planetmath.org/enumeratinggraphs Over an alphabet $q$ how many non-isomorphic $[n,k,d]_q$ codes are possible particularly in the special case ...
12
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0answers
165 views

Name and references for balanced variant of the long code?

The long code (arising in PCP theory etc) is an encoding of a set of $k$ values, using binary strings of length $2^k$ (double exponential in the number of bits needed to specify a value), with one ...
6
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0answers
90 views

Codes that are both locally testable and locally decodable

Are there any known constructions of binary locally testable codes with very low (e.g., independent of the length of the codeword) query complexity and "good" rate (e.g., mapping strings of length $k$ ...
1
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0answers
86 views

On Labeling the cube

Regularly in a Hamming hypercube, the vertices are labelled so that edge difference equals path difference. That is greater the edge difference, greater the hamming distance of the labelled vertices ...
5
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0answers
181 views

Weight enumerator of a binary linear code

The weight enumerator polynomial of a $(n,k)$ binary linear code $\mathcal{C}$ is defined as $$WE(\mathcal{C}) = \sum_{i=0}^{n}WE_{i}(\mathcal{C}) x^{i}$$ where $$WE_{i}(\mathcal{C}) = ...
3
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4answers
204 views

Optimal encoding of $k$-subsets of $n$

A colleague asks the following question: Let $S_k$ denote the subsets of $n$ of size $k$. Is there an optimal and efficient encoding of these subsets ? Namely, is there an function $f$ from ...
4
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2answers
92 views

Reference Request: Error correction code for a random projection

I'm interested in a variant of the Binary Channel, where the bits aren't flipped with some probability, but rather they are completely erased. Output words therefore vary in length. I'm quite sure ...
5
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0answers
237 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 ...
3
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0answers
79 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 ...
2
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0answers
64 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 ...
3
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2answers
541 views

Comparison Huffman Encoding and Arithmetic Coding dependent on Entropy

Where can I get an understanding of how Arithmetic Coding and Huffman Encoding compare as entropy increases. I know Arithmetic Coding is better for low entropy distributions, but how can I get a sense ...
3
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1answer
144 views

Upperbound on cardinality of product of two string sets at pairwise Hamming distance $> 1$

I am considering products $U\times V$ of subsets $U, V\subset \{0, 1\}^p$ with a pairwise Hamming distance greater than 1 : $\forall uv\in U\times V, D(u,v) \geq 2$. Given $p$, I am looking for a ...
3
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2answers
316 views

Asymptotically good codes

In short my question is what are all known explicit constructions of asymptotically good codes over finite alphabet? In more details: A sequence of codes codes $C_i: F^{k_i}\rightarrow F^{n_i}$ with ...
3
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1answer
203 views

Surveys on Algorithmic Problems in Coding Theory

Coding Theory has many algorithmic problems that enriched theoretical computer science. I want to learn specifically the connection between coding theory and the hardness of computational problems. ...
2
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0answers
80 views

Bayesian compression

Suppose you have a sequence generated by an i.i.d. process (such as repeatedly rolling a die and recording the values in order) parameterized by some K-dimensional vector $\vec{\gamma}$ (the ...
2
votes
1answer
118 views

Ambivalent Coding

Consider the following situation, I want to send one of two bitstrings, A or B, to a receiver. Clearly, I can do this by sending the shortest, but is there a better way? It seems that the requirement ...
8
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2answers
203 views

How do I construct an optimal affix code?

An affix code is a code that is simultaneously a prefix and suffix code. That is, no codeword is neither the prefix nor the suffix of any other codeword. Affix codes can be instantaneously decoded in ...
6
votes
1answer
153 views

Best upper bound on rate for q-ary codes

Among the many upper bounds for families of codes in $\mathbb F _2 ^n$, the best known bound is the one by McEliece, Rodemich, Rumsey and Welch (derived through a linear programming relaxation ...
1
vote
1answer
147 views

fast range summable hash functions

I'm finding is there any range summable hash function. ADD: The hash function I refer to is the one that is typically used in tug-of-war sketch(AMS sketch). Please refer to The space complexity of ...
4
votes
2answers
113 views

Gift bits when encoding a sequence of messages, how is that?

Recently a friend of mine asked a question I couldn't give immediate answer to. Say we have $ n $ messages of length $ m $ bits each. Now we can pack them in a single message of length $ n * m $ ...
5
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2answers
675 views

Transposition of any characters in Damerau–Levenshtein edit distance computation

Is it possible to modify the computation of Damerau–Levenshtein distance to take into account not only the transposition of adjacent characters, but the transposition of any characters? Maybe some ...
7
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3answers
1k views

Maximum subset of words with Hamming distance ≥ D

For all words of fixed length L over a given alphabet, I am interested in a practical algorithm that can give me a subset of maximal cardinality such that the Hamming distance between any two words in ...
10
votes
1answer
204 views

Boolean error correcting code over $\mathbb{F}_q$

Is there any known construction of a linear error correcting code $\mathsf{ECC}:\mathbb{F}_q^n \to \mathbb{F}_q^m$ (with reasonable parameters), such that when given a Boolean vector $v\in \{0,1\}^n$, ...
11
votes
1answer
279 views

Constructing vectors in general position

Let a real $k\times n$ ($k\le n$) matrix ${\bf A}$ with the property that any collection of $k$ columns is full rank. Q: Is there an efficient way to deterministically find a vector ${\bf a}$ such ...
15
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0answers
320 views

Looking for an operator on polynomials

I have a small, self-contained, math question, whose motivation is from theoretical computer science (specifically, list decoding of algebraic codes, derivative/multiplicity codes, etc). I wonder ...
6
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0answers
194 views

Tree rotation, a problem similar to Huffman coding

I am not sure whether the following problem has been studied. We have a undirected tree $T$. We would like to construct another tree $T'$. $T'$ is a binary tree. Each inner nodes of $T'$ ...
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2answers
379 views

Covering Codes with Game Theory Application

Here is a question I came up with and i have been pondering for a while. It relates to covering codes, a subset of coding theory. I could not come up with an adequate solution, so here I am, asking ...
3
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1answer
247 views

reduction of maximum independet set to minimum distance of code

Is there a reference for direct reduction of computing maximum independent set of a suitably constructed graph to computing minimum distance of a linear code when the code is specified by its parity ...
8
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2answers
297 views

Applications of Spectral Graph Theory in Information and Coding Theory

I wanted to find out what are some application of SGT in the area of information and coding theory and maybe communications. The most related that comes to mind is the work on Expander Codes Michael ...
2
votes
1answer
283 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 ...
4
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2answers
518 views

“long code test” and “dictatorship test”

Why is "long code test" also called "dictatorship test"? I got really confused when I read about it in Arora's survey.
1
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0answers
103 views

Information theory and Tsfasman-Manin's problem

Yuri Manin recently posted an interesting paper on computability of boundary regions of distance-rate trade-offs for error correction codes. http://arxiv.org/PS_cache/arxiv/pdf/1107/1107.4246v1.pdf I ...
12
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5answers
2k views

Why does Huffman coding eliminate entropy that Lempel-Ziv doesn't?

The popular DEFLATE algorithm uses Huffman coding on top of Lempel-Ziv. In general, if we have a random source of data (= 1 bit entropy/bit), no encoding, including Huffman, is likely to compress it ...
9
votes
4answers
562 views

Surveys on Network Coding

I want to start learning about Network Coding: http://en.wikipedia.org/wiki/Network_coding Do you know any good survey (e.g. from IEEE Surveys and Tutorials) on the above subjects. I found some ...
0
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0answers
289 views

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

Smaller version of the cross-posting from http://mathoverflow.net/questions/70524/lee-metric-constructive-and-asymptotic-bounds The following link provides a Gilbert-Varshamov lower bound and a ...
11
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2answers
264 views

Solvability of matrix filling

Matrix $A$ has dimension $n \times n(n-1)$. We want to fill $A$ using integers between $1$ and $n$, inclusive. Requirements: Each column of $A$ is a permutation of $1, \dots, n$. Any submatrix ...
2
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3answers
437 views

Is there a lower bound of number of redundant bits necessary to encode a word with certain Hamming distance?

Is there a lower bound (in coding theory or elsewhere) of number of redundant bits necessary to encode a word with certain Hamming distance? There is some known data for parity checks, CRC, Hamming ...
27
votes
1answer
443 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 ...