# Questions tagged [coding-theory]

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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### Verify if array is orthogonal

This is a repost from the computer science stackexchange. The question has been offered a bounty, but received no answers. Therefore, I would like to ask this question here. Orthogonal arrays often ...
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### Bipartite version of Hamming bound (two families of codewords with large Hamming distance)?

Update: In light of Fedor Petrov's answer, I added an additional requirement that all strings in $A$ and $B$ have Hamming weight exactly $n/2$, which hopefully makes the question more interesting. ...
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### Mutual benefits of coding theory and the reconstruction conjecture

Let $G_n$ denotes all graphs with $n$ nodes. For any graph $G$ in $G_n$, the adjacency matrix $A(G)$ can be viewed as a codeword of length $n^2$. Also, the codes arises from $G_n$ is a linear binary ...
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### How many Hamming spheres of radius 1 does it take to cover the cube?

I am looking for the sharpest known upper bound on $K(n, 1)$ as $n \rightarrow \infty$. This is the minimal cardinality of a (not-necessarily linear) covering code of $\{0, 1\}^n$ of radius 1. In ...
1 vote
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### Computational complexity of rate $\frac{1}{2}$ codes

We know from Berlekamp, McEliece and Van Tilborg [On the inherent intractability of certain coding problems, IEEE Trans. Information Theory, 24 (1978)] that computing the minimum distance of a (binary)...
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### Promise version of minimum distance

It has been known for some time that computing minimum distance of a linear code (minimum weight codeword) is NP-hard. This immediately also says that given a code $C$, calculating minimum hamming ...
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### Dense and decodable lattices in high dimensions

We are currently looking for both dense and decodable lattices. Precisely, we want a lattice which CVP can be solved in polynomial time like $O(n^2)$ or $O(n^3)$ where $n$ is the dimension like 128 or ...
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### Are there any homomorphic analog error correction code?

Are there any analog error correction codes that are additively and multiplicatively homomorphic?
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### Maximum number of vectors with upper bound on pairwise inner products

I have a collection $\{v_1,...,v_k\}$ of vectors in $\{\pm 1\}^n$ with the property that for all $i\neq j$ we have $\langle v_i, v_j \rangle \le c\log_2(n)$. I am looking for an upper bound on $k$ in ...
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### Upper bounds for high-dimensional spherical codes given the covering radius

I assume that this sort of question has already been considered at great length. Nevertheless, I could not find an answer to this question in the related literature. Given a constant $a\in (0,2]$, ...
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### A variant of Huffman code

Given an alphabet of $n$ symbolswith probabilities $p_i$ for symbol $i$, we need to encode the symbols (in a prefix-free way) to binary codewords $c(i)$ with length $\ell(c(i))$ to minimize the ...
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### error correcting huffman code [closed]

I am looking for a code that can correct errors with variable and limited length like huffman code. I am not an expert in coding theory. Is there any code or related literature on this?
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### Linear independence of Vandermonde matrix in systematic Reed-Solomon code

My question is about using a Vandermonde matrix vs a Cauchy matrix in erasure coding. In the Reed-Solomon (RS) code, encoding is done by multiplying a $N\times K$ ($N>K$) matrix with the code words ...
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### When are Hamming codes cyclic?

I've asked this question on math.stackexchange before, but it has not been solved. The following statement appears to be true: The $q$-ary Hamming code of codimension $r$ over $\mathbb{F}_q$ is ...
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### Cyclic codes: sparse codewords not orthogonal to the all-ones vector

Is it true that for any sufficiently large prime $p$, there exists a prime $q\ne p$ and a cyclic code of length $p$ over $\mathbb{F}_q$ that contains a codeword of Hamming weight at most ord$_p(q)$ ...
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### Ternary error correction codes

Let`s define ternary ECC as a code that its codewords can be defined by $\{ xyz f(y,z) f(x,z) f(x,y) | x,y,z \in \{0,1\}^m \}$ for some function $f$. $f$ returns bitstring of constant length. ...
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### Linear-time logspace encodable error correcting code with constant

Is there a binary code with (quasi)constant rate, constant relative distance, and an encoder that takes (quasi)linear time and logspace simultaneously? Note that there are no constraints on ...
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### 3-uniform hypergraphs and their circuit space

So, I'll break this post into two questions. Both concern 3-uniform hypergraphs. A 3-uniform hypergraph $H=(V,E)$ consists of a set of vertices $V$ and a set of edges $E$, where each edge $e\in E$ is ...
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### Lower bounding decoding error in a noisy adversarial channel

Problem description Suppose we have a finite alphabet $\mathcal{X}$, where each letter $X \in \mathcal{X}$ indexes into some fixed set of distributions, $\{P_{1},\ldots,P_{|\mathcal{X}|}\}$. For ...
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### How do I check if two linear binary codes are equivalent?

Suppose I have a list of generator matrices $G_i$, $i=1,\ldots N$, of the same size (each defines an $n$-bit linear binary code encoding $k$ logical bits). I consider two codes to be equivalent if ...
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