Enumeration of codes of fixed cardinality up to isomorphism.

Proceedings of ALCOMA'05, Bayreuther Mathematische Schriften 74 (2005), 256-265.

We say that sequence $(a_n)$ is {\em quasi-polynomial} in $n$ if there are polynomials $P_0,...,P_{s-1}$ such that $a_n=P_i(n)$ where $i\equiv n\pmod{s}$. A $q$-ary non-linear code with block length $n$ and $r$ codewords is a subset of $A^n$ of cardinality $r$, where $A$ is an alphabet of $q$ symbols. We consider the isomorphism relation on the set of all codes induced by the group action of the wreath product $S_A \wr S_n$ on $A^n$. Let $c_{q,r,n}$ denote the number of isomorphism classes of $q$-ary codes with block length $n$ and $r$ codewords. We prove that, when the values of $q$ and $r$ are fixed, the sequence $(c_{q,r,n})$ is quasi-polynomial in $n$. We also discuss strategies for computing closed forms for the generating functions $f_{q,r}(z)=\sum_{n\ge 0} c_{q,r,n}z^n$.