# 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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### On weight enumerators of codes

Are there $[n,k]_q$ constant rate $\frac kn$ and constant alphabet linear code families with automorphism group of size $\Omega((n-n^\beta)!)$ that have minimum distance $d=O(n^\alpha)$ and number of ...
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### Finding minimum weight codeword of MDS RS code

For a $[n,k,n-k+1]_q$ Reed Solomon code is there a polynomial time algorithm to find at least one minimum weight $(n-k+1)$ codeword? I searched in literature and I could not find one and hence I am ...
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### Partitions of $\mathbb{F}_2^n$ related to perfect $1$-error correcting binary codes

Edit. After a computer search found an example for $n=8$, I've rephrased my original question as a conjecture. This question is motivated by the existence of perfect $1$-error correcting binary codes,...
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### Relations between the parameters of the Best Linear Code

Let the triple $(n,k,d)$ be length and dimension and minimum distance of a code, respectively. For a fixed given numbers $n$, $k$ and $d$, what relations there are between BKLC$($GF(2)$,n,k)$ ,BLLC$($...
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### Extended Hypercube Graph

Definition 1. The $n$-hypercube graph has vertices which are the elements of the set $\lbrace 0,1\rbrace^n$ of $n$-bit binary strings, and an edge is drawn between each pair of vertices representing a ...
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### is any closed form relation that can state the error probability of code versus its variable and check node degree distributions?

In Low Density parity check code design, when bit (or frame) error probability of code is the objective of the design, we need a closed form relation between error probably (or even an approximate or ...
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### Maximum number of vectors in a hypercube satisfying given conditions

$\mathcal{C}$ is a collection of binary vectors of length $n$, i.e. $\mathcal{C}\subseteq\{0,1\}^n$. For arbitrary $x,y,z\in\mathcal{C}$ and $x\neq z$, $y\neq z$, there always holds that the Euclidean ...
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### How can I decode efficiently a triple-error-correcting binary BCH code?

In a given $\mathrm{BCH}(N,K)$, $T=3$ code over $\mathrm{GF}(2^m)$, there are ways to find the error locations in a given $N$-bit codeword directly from the syndromes without going through the normal ...
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### irregular LDPC code construction algorithm

I want to construct a sparse random binary matrix ${{\bf{H}}_{m \times n}}$ that has the following properties 1- Faction of columns of weight $i$ is ${v_i}$ . 2- Fraction of rows of weight $i$...
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### Binary Codes and its distance [closed]

Find n binary codes of length n such that distance between each pair is n/2 , where n is a even number , if possible?How to generate all codes? for example if n=4 we have 1110,1101,1011,0111 each pair ...
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### A question on representation of graphs

Take a complete graph $K_n$. You want to assign a vectors from $\Bbb F_2^d$ to every edge such that sum of vectors in every simple cycle does not sum to $0$ vector. The question is what is minimum $d$ ...
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### What odd-length binary codes have Hamming weights restricted to be multiples of eight?

Let $G$ be a $k$ by $n$ binary matrix with row vectors $\lbrace \vec{x}_j {\rbrace} _{j=1}^k$. We can interpret $G$ as a generator matrix of a linear $[n,k]$ code $\cal{C}$ whose codewords consist of ...
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### Coding for channels with concentrated error

Can we implement a reduced-error transmission over a channel with error frequency having $\liminf<\frac{1}{2}$ and $\limsup>\frac{1}{2}$? We know that if a channel with error flips (in the ...
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### How do you use the bits you get back from Bits Back Coding?

Bits Back coding is a scheme to transmit an observation x. You can read about it here [1]. To my understanding, it works like this: The encoder samples a message z from a distribution Q(z|x) that it ...
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### Picking codewords that are close

I posted this question in http://math.stackexchange.com/questions/1142698/picking-codewords-that-are-close a week back. Let $[n,k,d]$ be a linear code over $\Bbb F_q$ with minimum distance $d$ and ...
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### Block error-correcting codes over inhomogeneous alphabets

For $n := (n_1,\dots,n_N) \in \mathbb{N}_{>1}^N$, let $X_n := \prod_{j=1}^N [n_j]$, where as usual $[m] := \{1,\dots,m\}$. Are there any known generic constructions for (Hamming) sphere ...
Consider two continous-time stochastic processes $\{A(t)\}_{t \ge 0}$ and $\{B(t)\}_{t \ge 0}$ with $A(t)=t$ and $B(t)=t$. Each process starts at $t=0$ and emits "ticks" at increasing time slots. For ...
Denote $\mathcal{F_n}$ as collection of multiaffine polynomials $f\in\Bbb F_2[x_1,\dots,x_n]$. Denote total degree of $f\in\mathcal{F_n}$ as $deg(f)$ (note $deg(f)\leq n$). Denote \$e_i=(0,\dots,0,\...