# Questions tagged [coding-theory]

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

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### 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 ...
585 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 ...
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### 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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### 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 ...
107 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/...
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### 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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### 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 ...
135 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 ...
73 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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### 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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### 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 ...
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 ...
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 ...
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### 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. ...
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### 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 $\#P-Complete$, ...
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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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 ...
108 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 ...
133 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 ...
332 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 ...
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### 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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### 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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### 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 ...
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### 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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### 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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### 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 ...
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 ...
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### 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 ...