Questions tagged [coding-theory]

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

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39 votes
13 answers
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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 ...
4 votes
1 answer
436 views

Approximation algorithms for min vector subset-sum over GF(2)

In this question vzn asked about the following problem, which I'll call Vector-Subset-Sum. Given a set of vectors $v_i$ over GF(2) and a target vector $y$, is there a subset of the $v_i$ summing to ...
Jeremy Kun's user avatar
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4 votes
1 answer
324 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/...
Geoffrey Irving's user avatar
11 votes
1 answer
476 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 ...
Dimitris's user avatar
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7 votes
4 answers
738 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,...
Suresh Venkat's user avatar
5 votes
0 answers
330 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\...
Pavithran Iyer's user avatar
5 votes
2 answers
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 ...
Roman Yankovsky's user avatar
3 votes
2 answers
403 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 ...
Root's user avatar
  • 387
2 votes
0 answers
75 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 $\#P-Complete$, ...
Pavithran Iyer's user avatar