An efficient algorithm for constructing minimal trellises for codes over finite abelian groups

Abstract

We present an efficient algorithm for computing the minimal trellis for a group code over a finite abelian group, given a generator matrix for the code. We also show how to compute a succinct representation of the minimal trellis for such a code, and present algorithms that use this information to compute efficiently local descriptions of the minimal trellis. This extends the work of Kschischang and Sorokine (see ibid., vol.41, no.6, p.1926-37, 1995), who treated the case of linear codes over fields. An important application of our algorithms is to the construction of minimal trellises for lattices. A key step in our work is handling codes over cyclic groups C_p^α, where p is a prime. Such a code can be viewed as a module over the ring Z_p^α. Because of the presence of zero divisors in the ring, modules do not share the useful properties of vector spaces. We get around this difficulty by restricting the notion of linear combination to a p-linear combination, and by introducing the notion of a p-generator sequence, which enjoys properties similar to those of a generator matrix for a vector space.