# Encoding of an IAPS of groups

From Groupprops

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## Definition

Let be an IAPS of groups. In other words, for every , there is a group , and there is a map .

An encoding of this IAPS over a binary (or constant-sized) alphabet is the following.

- For every , we specify an encoding of over that alphabet.
- For every , we specify an algorithm that takes as input the code for and , and outputs the code for .

## Properties

### Dense encoding of an IAPS of groups

`Further information: dense encoding of an IAPS of groups`

An encoding of an IAPS of groups is said to be *dense* if the maximum possible length of a code-word for is bounded by some constant times the logarithm of the size of .

### Polynomial-time encoding of an IAPS of groups

`Further information: polynomial-time encoding of an IAPS of groups`

An encoding of an IAPS of groups is said to be polynomial-time if:

- The time taken for all the operations (outputting the identity element, computing the product, computing the inverse, testing for membership) is polynomial in the length of the maximum code-word.
- The time taken for is polynomial in the lengths of the maximum code-words for and .

## Examples

### Encoding of symmetric groups

`Further information: Encoding of symmetric groups`

### Encoding of genereal linear groups

`Further information: Encoding of general linear groups`