# Questions tagged [coding-theory]

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

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### 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 ...
519 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 ...
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 ...
363 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 ...
4k 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 ...
190 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 ...
308 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 ...
382 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 ...
665 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 ...
252 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 $v\in \{0,1\}^n$, ...
505 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 ...
357 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 ...
596 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 ...
1k 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 error-...
119 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 ...
293 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 ...
2k 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 ...
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### 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 ...
110 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 $k$ ...
225 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 $T'$ ...
266 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. ...
742 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 ...
142 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 ...
806 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 ...
122 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 ...
798 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 ...
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 ...
130 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 $q(q+1)/2 + (q-1)/2$ ...
297 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}) = \#\{c\in\...
2k 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 ...
657 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.
1k 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 ...
408 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 ...
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 $n * m$ bits....