# Tagged Questions

The theory of error-correcting codes stems from Shannon's 1948 _A mathematical theory of communication_, and from Hamming's 1950 "Error detecting and error correcting codes".

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### A covering problem for the Hamming cube

Consider the set of all $k$-subsets of $\{1,\dots,n\}$, naturally identified with a subset $A$ of $\{0,1\}^n$ where each element has exactly $k$ ones. Is there a sharp bound known for $\epsilon$-...
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### Can you cover the Boolean cube {0,1}^n with O(1) Hamming-balls each of radius n/2-c*sqrt(n)?

(where c>0 and the balls need not be disjoint?) This is an embarrassingly simple question, yet somehow I couldn't find an answer (not even, "this is a well-known open problem") after spending some ...
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### Will “error locating codes” have higher rates than ECCs?

I'm wondering to detect all the errors (i.e. their positions) in a codeword $(c_0, c_1, \cdots, c_{n-1})\in Q$ where $Q$ is an alphabet set with size $q$, i.e., to know whether $c_i$ is faulty or not, ...
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### How small parallelograms are we guaranteed to get, when we select the two sides from different plane lattices?

Title question description: Select two lattices $\Lambda_1$ and $\Lambda_2$ (here a lattice=additive free abelian group without accumulation points) of maximal rank two in the real plane. We normalize ...
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### Partial backups

Suppose you have some storage medium of a given size M, and can make some kind of backup on another medium of size B with M > B. You can choose the scheme to determine the contents of the backup. ...
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### Reed-Muller-Codes

Let $F$ be the field with two elements, $V_m=F^{2^m}$.Let $R(r, m)\subset V_m$ be the binary Reed-Muller Code. Define $R_m:=R(1, m)$. Then the dimension of $R_m$ is $1+m$ and its minimal distance is ...
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### Computing channel capacities for non-symmetric channels

I'm studying information theory right now and I'm reading about channel capacities. I know that there are known expressions for computing the capacities for some well known simple channels such as ...
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### Hot-topics in error correcting coding related to interesting math. ?

What are topics in error-correcting coding which are related to interesting math. ? I am primarely interested in nowdays hot topics, but old days topics are also welcome. Let me try to mention what ...
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### Are algebraic geometry error correcting codes (Goppa codes) “good” ?

Question (informal version): Are algebraic geometry error correcting codes (V.D. Goppa codes) "good" ? Some details. There is certain construction of error-correcting codes by means of algebraic ...
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### Asymtotic Complexity Analysis using logarithms and binomial coefficients

On page 11 of "Smaller decoding exponents: ball-collision decoding" by Berstein et.al. they have the formula \lim_{n \rightarrow \infty} \frac{1}{n}\log_{2}\left(\dbinom{k_{1}}{p_{1}}\...
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### What is “automorphism group of an error-correcting code” ?

Here in Wikipedia is written: "The automorphism group of the binary Golay code is the Mathieu group M23." What is "automorphism group of code" ? PS Are there other nice examples of relation ...
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### Ax=0, estimate min(Hamming(x)) ? Equivalently: Bipartite graph. How to find (estimate) minimal number of vertices1 which are connected with EVEN number of vertices2 ? Equivalently: estimate minimal weight of error correcting code ?

Consider system of linear equations Ax=0 over $F_2$ (field with two elements {0,1}). Where number of variables is bigger than equations - so we have many solutions $x$. Question How to estimate ...
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### If graph is tree what can be said about its adjacency matrix ?

Question If graph is tree what can be said about its adjacency matrix ? And vice versa ? Especially I am interested in case when graph is bipartite graph. Such graphs are related to error-...
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### Can you hide a letter without losing information?

Consider the following game between Alice and Bob. $\Sigma$ is a finite nonempty alphabet, $\Delta \notin \Sigma$ denotes a special symbol, and $k > 0$ is a positive integer constant representing ...
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