21 research outputs found
Bounds for complexity of syndrome decoding for poset metrics
In this work we show how to decompose a linear code relatively to any given
poset metric. We prove that the complexity of syndrome decoding is determined
by a maximal (primary) such decomposition and then show that a refinement of a
partial order leads to a refinement of the primary decomposition. Using this
and considering already known results about hierarchical posets, we can
establish upper and lower bounds for the complexity of syndrome decoding
relatively to a poset metric.Comment: Submitted to ITW 201
Applications of ordered weights in information transmission
This dissertation is devoted to a study of a class of linear codes related to a particular metric space that generalizes the Hamming space in that the metric function is defined by a partial order on the set of coordinates of the vector.
We begin with developing combinatorial and linear-algebraic aspects of linear ordered codes. In particular, we define multivariate rank enumerators for linear codes and show that they form a natural set of invariants in the study of the duality of linear codes. The rank enumerators are further shown to be connected to the shape distributions of linear codes, and enable us to give a simple proof of a MacWilliams-like theorem for the ordered case. We also pursue the connection between linear codes and matroids in the ordered case and show that the rank enumerator can be thought of as an instance of the classical matroid invariant called the Tutte polynomial. Finally, we consider the distributions of support weights of ordered codes and their expression via the rank enumerator. Altogether, these results generalize a group of well-known results for codes in the Hamming space to the ordered case.
Extending the research in the first part, we define simple probabilistic channel models that are in a certain sense matched to the ordered distance, and prove several results related to performance of linear codes on such channels. In particular, we define ordered wire-tap channels and establish several results related to the use of linear codes for reliable and secure transmission in such channel models.
In the third part of this dissertation we study polar coding schemes for channels with nonbinary input alphabets. We construct a family of linear codes that achieve the capacity of a nonbinary symmetric discrete memoryless channel with input alphabet of size q=2^r, r=2,3,.... A new feature of the coding scheme that arises in the nonbinary case is related to the emergence of several extremal configurations for the polarized data symbols. We establish monotonicity properties of the configurations and use them to show that total transmission rate approaches the symmetric capacity of the channel. We develop these results to include the case of ``controlled polarization'' under which the data symbols polarize to any predefined set of extremal configurations. We also outline an application of this construction to data encoding in video sequences of the MPEG-2 and H.264/MPEG-4 standards
The Packing Radius of a Code and Partitioning Problems: the Case for Poset Metrics
Until this work, the packing radius of a poset code was only known in the
cases where the poset was a chain, a hierarchy, a union of disjoint chains of
the same size, and for some families of codes. Our objective is to approach the
general case of any poset. To do this, we will divide the problem into two
parts.
The first part consists in finding the packing radius of a single vector. We
will show that this is equivalent to a generalization of a famous NP-hard
problem known as "the partition problem". Then, we will review the main results
known about this problem giving special attention to the algorithms to solve
it. The main ingredient to these algorithms is what is known as the
differentiating method, and therefore, we will extend it to the general case.
The second part consists in finding the vector that determines the packing
radius of the code. For this, we will show how it is sometimes possible to
compare the packing radius of two vectors without calculating them explicitly
On the fast computation of the weight enumerator polynomial and the value of digital nets over finite abelian groups
In this paper we introduce digital nets over finite abelian groups which
contain digital nets over finite fields and certain rings as a special case. We
prove a MacWilliams type identity for such digital nets. This identity can be
used to compute the strict -value of a digital net over finite abelian
groups. If the digital net has points in the dimensional unit cube
, then the -value can be computed in
operations and the weight enumerator polynomial can be computed in
operations, where operations mean arithmetic of
integers. By precomputing some values the number of operations of computing the
weight enumerator polynomial can be reduced further
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