Questions tagged [coding-theory]

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

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39
votes
13answers
3k views

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 ...
11
votes
1answer
394 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 ...
3
votes
2answers
204 views

Minimum distance of a code

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 ...
2
votes
0answers
72 views

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 $\#P-Complete$, ...
6
votes
4answers
555 views

Optimal encoding of $k$-subsets of $n$

A colleague asks the following question: Let $S_k$ denote the subsets of $n$ of size $k$. Is there an optimal and efficient encoding of these subsets ? Namely, is there an function $f$ from $[0,...
5
votes
0answers
298 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\...
5
votes
2answers
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 ...
2
votes
1answer
121 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/...