More on Codes Over Finite Quotients of Polynomial Rings

Let $q=p^{r}$ be a prime power, ${\mathbb {F}}_{q}$ be the finite field of order q and $f(x)$ be a monic polynomial in ${\mathbb {F}}_{q}[x]$ . Set ${\mathbb {A}}:={\mathbb {F}}_{q}[x]/{\left \lt {{ f(x) }}\right \gt }$ . In this paper we continue the study (started by T. P. Berger and N. El Amrani) of $\mathbb {A}$ -codes of length l over $\mathbb {A}$ , i.e. $\mathbb {A}$ -submodules of ${\mathbb {A}}^{l}$ . We introduce two types of unique generating sets, called type I and type II basis of divisors, for an $\mathbb {A}$ -code. Using this, we present a building-up construction so that one can obtain all distinct $\mathbb {A}$ -codes of length l, with their basis of divisors. We complete the classification for the special case $l=2$ and enumerate all the $\mathbb {A}$ -codes of length 2. As an example, we list all binary index-2 quasi-cyclic codes of lengths 16 and 32, and all ternary index-2 quasi-cyclic codes of lengths 6 and 18, which are best-known codes.