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2
votes
0answers
32 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
votes
4answers
272 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
votes
1answer
138 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
votes
0answers
56 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
votes
2answers
131 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 ...
0
votes
0answers
15 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 ...
11
votes
0answers
148 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
votes
0answers
88 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
vote
0answers
84 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
votes
0answers
137 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
votes
4answers
183 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
votes
2answers
89 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
votes
0answers
206 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
votes
0answers
76 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
votes
0answers
57 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 ...
2
votes
2answers
184 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
votes
1answer
138 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
votes
2answers
292 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
votes
1answer
191 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
votes
0answers
79 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 ...
0
votes
0answers
92 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
votes
2answers
175 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
150 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
142 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
107 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
votes
2answers
560 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 ...
6
votes
3answers
876 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
202 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
261 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
votes
0answers
313 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
votes
0answers
184 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 $T'$ ...
-4
votes
2answers
367 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
votes
1answer
245 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
votes
2answers
275 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
277 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
votes
2answers
510 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
vote
0answers
98 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 ...
11
votes
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
548 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
votes
0answers
256 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
votes
2answers
260 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
votes
3answers
419 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 ...
26
votes
0answers
352 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 ...
4
votes
1answer
110 views

Lower bounds on 2-query locally decodable codes

Does any one knows if there is a non-quantum proof of the fact that non-linear 2-query LDC must have exponential size?
16
votes
2answers
1k 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 ...
6
votes
0answers
225 views

Huffman “terminator” bitstring

Motivation Imagine a huffman compressed file that gets downloaded partially, like in P2P software, so we allocate disk space for the whole file first and then start downloading random file chunks. ...
8
votes
2answers
331 views

Finding out a set by intersection comparison

The following problem recently emerged from my research and I would like to ask if anyone knows if this problem was considered before or has heard of anything that might be related. The general ...
1
vote
0answers
325 views

Soft decoding of linear block (20,10) codes - what methods are used ?

What algorithms are advised for soft-decoding of linear block codes (20,10) ? What are advised references ? Sincerely Yours Alex PS By soft-decoding - I mean that input - is set of 20 real ...
6
votes
1answer
643 views

Dual BCH codes of design distance $d$

The SODA 2008 Ailon-Liberty paper on fast Johnson-Lindenstrauss transforms uses a "dual BCH code of design distance 5" as part of the construction. They cite the MacWilliams-Sloane book on ...
35
votes
11answers
1k 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 ...