Multi-encoding of a group

Definition

Basic definition

Let $G$ be a group. A multi-encoding of $G$ over a binary alphabet (or equivalently, over any constant-sized alphabet) is the following data:

• A multivalued injective mapping from the group to the set of words in that alphabet. In other words, each element of the group is expressed as a word over the alphabet, in possibly more ways than one. This is called the code-word for that element of the group and the mapping itself is termed the encoding.
• An algorithm that takes in the codes for $g,h\in G$ and outputs the code for $gh$.
• An algorithm that takes in the code for $g\in G$ and outputs the code for $g^{-1}$.
• An algorithm that takes in any string and outputs whether or not it is a valid codeword (that is, whether it arises as the encoding of some group element).
• An algorithm that takes as unput two code-words and outputs whether they represent the same group element.

If the mapping is single-valued, viz every group element has a unique code-word, then we simply call it an encoding of the group.

Properties

Size of fibers

To measure the extent to which a multi-encoding is like an encoding, we look at the sizes of the image of every point. Ideally we want this size to be bounded by a small constant.

Passing from a multi-encoding to an encoding

An encoding is obtained as a section of a multi-encoding if, for every group element, it simply picks one of the

Subgroups and quotients

Subgroups

Given a group with a multi-encoding, any subgroup also gets the multi-encoding provided that we have a membership test for the subgroup inside the group.

We can also choose to describe the subgroup by means of its generating set.

Quotients

Given a group with a multi-encoding, we can obtain a multi-encoding for its quotient by a normal subgroup provided there is a membership test for that normal subgroup.