Encoding of an APS of groups

Definition with symbols
Let $$(G,\Phi)$$ be an APS of groups, in other words, we have, for every $$n$$, a group $$G_n$$, and we also have block concatenation maps $$\Phi_{m,n}: G_m \times G_n \to G_{m+n}$$.

An encoding of this APS is defined as a choice of encoding $$C_n$$ for each $$G_n$$ (all over the same alphabet) along with an algorithm that can take as input the encodings of $$g \in G_m, h \in G_n$$, and output the encoding of $$gh \in G_{m+n}$$.

Dense encoding
An encoding of an APS of groups $$(G,\Phi)$$ is termed dense if there is a constant $$c$$ such that the maximum codeword length for any element of $$G_n$$ is bounded from above by $$c\log|G_n|$$.

Polynomial-time encoding
An encoding of an APS of groups $$(G,\Phi)$$ is termed polynomial-time if:


 * There is a univariate polynomial $$q$$ such that the time taken for group operations inside $$G_n$$ is bounded by $$q(a)$$ where $$a$$ is the maximum length of a code-word in $$G_n$$.
 * There is a bivariate polynomial $$p$$ such that the time taken for the block concatenation map $$\Phi_{m,n}$$ is bounded from above by $$p(a,b)$$ where $$a$$ and $$b$$ are respectively the lengths of the maximum code-words in $$G_m$$ and $$G_n$$.

All in one
We shall typically be interested in APSes of exponential growth, viz an APS where the order of the $$n^{th}$$ member is bounded from above by $$\exp {p(n)}$$ where $$p$$ is a fixed polynomial. For such APSes, finding a dense polynomial-time encoding means finding a very efficient encoding, in the sense that:


 * The maximum length of code-word for $$n$$ is a polynomial in $$n$$
 * The time taken for group operations within $$G_n$$ is polynomial in $$n$$.
 * The time taken for a block concatenation over $$G_m$$ and $$G_n$$ is polynomial in $$m$$ and $$n$$.