62 research outputs found
Fourier-Reflexive Partitions and MacWilliams Identities for Additive Codes
A partition of a finite abelian group gives rise to a dual partition on the
character group via the Fourier transform. Properties of the dual partitions
are investigated and a convenient test is given for the case that the bidual
partition coincides the primal partition. Such partitions permit MacWilliams
identities for the partition enumerators of additive codes. It is shown that
dualization commutes with taking products and symmetrized products of
partitions on cartesian powers of the given group. After translating the
results to Frobenius rings, which are identified with their character module,
the approach is applied to partitions that arise from poset structures
Partitions of Matrix Spaces With an Application to -Rook Polynomials
We study the row-space partition and the pivot partition on the matrix space
. We show that both these partitions are reflexive
and that the row-space partition is self-dual. Moreover, using various
combinatorial methods, we explicitly compute the Krawtchouk coefficients
associated with these partitions. This establishes MacWilliams-type identities
for the row-space and pivot enumerators of linear rank-metric codes. We then
generalize the Singleton-like bound for rank-metric codes, and introduce two
new concepts of code extremality. Both of them generalize the notion of MRD
codes and are preserved by trace-duality. Moreover, codes that are extremal
according to either notion satisfy strong rigidity properties analogous to
those of MRD codes. As an application of our results to combinatorics, we give
closed formulas for the -rook polynomials associated with Ferrers diagram
boards. Moreover, we exploit connections between matrices over finite fields
and rook placements to prove that the number of matrices of rank over
supported on a Ferrers diagram is a polynomial in , whose
degree is strictly increasing in . Finally, we investigate the natural
analogues of the MacWilliams Extension Theorem for the rank, the row-space, and
the pivot partitions
Linear tail-biting trellises: Characteristic generators and the BCJR-construction
We investigate the constructions of tail-biting trellises for linear block
codes introduced by Koetter/Vardy (2003) and Nori/Shankar (2006). For a given
code we will define the sets of characteristic generators more generally than
by Koetter/Vardy and we will investigate how the choice of characteristic
generators affects the set of resulting product trellises, called KV-trellises.
Furthermore, we will show that each KV-trellis is a BCJR-trellis, defined in a
slightly stronger sense than by Nori/Shankar, and that the latter are always
non-mergeable. Finally, we will address a duality conjecture of Koetter/Vardy
by making use of a dualization technique of BCJR-trellises and prove the
conjecture for minimal trellises.Comment: 28 page
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