8 research outputs found
The State of Lexicodes and Ferrers Diagram Rank-Metric Codes
In coding theory we wish to find as many codewords as possible, while simultaneously maintaining high distance between codewords to ease the detection and correction of errors. For linear codes, this translates to finding high-dimensional subspaces of a given metric space, where the induced distance between vectors stays above a specified minimum. In this work I describe the recent advances of this problem in the contexts of lexicodes and Ferrers diagram rank-metric codes.
In the first chapter, we study lexicodes. For a ring R, we describe a lexicographic ordering of the left R-module Rn. With this ordering we set up a greedy algorithm which sequentially selects vectors for which all linear combinations satisfy a given property. The resulting output is called a lexicode. This process was discussed earlier in the literature for fields and chain rings. We describe a generalization of the algorithm to finite principal ideal rings.
In the second chapter, we investigate Ferrers diagram rank-metric codes, which play a role in the construction of subspace codes. A well-known upper bound for dimension of these codes is conjectured to be sharp. We describe several solved cases of the conjecture, and further contribute new ones. In addition, probabilities for maximal Ferrers diagram codes and MRD codes are investigated in a new light. It is shown that for growing field size, the limiting probability depends highly on the Ferrers diagram
On the Geometry of Balls in the Grassmannian and List Decoding of Lifted Gabidulin Codes
The finite Grassmannian is defined as the set of all
-dimensional subspaces of the ambient space . Subsets of
the finite Grassmannian are called constant dimension codes and have recently
found an application in random network coding. In this setting codewords from
are sent through a network channel and, since errors may
occur during transmission, the received words can possible lie in
, where . In this paper, we study the balls in
with center that is not necessarily in
. We describe the balls with respect to two different
metrics, namely the subspace and the injection metric. Moreover, we use two
different techniques for describing these balls, one is the Pl\"ucker embedding
of , and the second one is a rational parametrization of
the matrix representation of the codewords.
With these results, we consider the problem of list decoding a certain family
of constant dimension codes, called lifted Gabidulin codes. We describe a way
of representing these codes by linear equations in either the matrix
representation or a subset of the Pl\"ucker coordinates. The union of these
equations and the equations which arise from the description of the ball of a
given radius in the Grassmannian describe the list of codewords with distance
less than or equal to the given radius from the received word.Comment: To be published in Designs, Codes and Cryptography (Springer
Codes and Designs Related to Lifted MRD Codes
Lifted maximum rank distance (MRD) codes, which are constant dimension codes,
are considered. It is shown that a lifted MRD code can be represented in such a
way that it forms a block design known as a transversal design. A slightly
different representation of this design makes it similar to a analog of a
transversal design. The structure of these designs is used to obtain upper
bounds on the sizes of constant dimension codes which contain a lifted MRD
code. Codes which attain these bounds are constructed. These codes are the
largest known codes for the given parameters. These transversal designs can be
also used to derive a new family of linear codes in the Hamming space. Bounds
on the minimum distance and the dimension of such codes are given.Comment: Submitted to IEEE Transactions on Information Theory. The material in
this paper was presented in part in the 2011 IEEE International Symposium on
Information Theory, Saint Petersburg, Russia, August 201
Bounds on edit metric codes with combinatorial DNA constraints
The design of a large and reliable DNA codeword library is a key problem in DNA based
computing. DNA codes, namely sets of fixed length edit metric codewords over the alphabet
{A, C, G, T}, satisfy certain combinatorial constraints with respect to biological and
chemical restrictions of DNA strands. The primary constraints that we consider are the
reverse--complement constraint and the fixed GC--content constraint, as well as the basic
edit distance constraint between codewords.
We focus on exploring the theory underlying DNA codes and discuss several approaches to
searching for optimal DNA codes. We use Conway's lexicode algorithm and an exhaustive
search algorithm to produce provably optimal DNA codes for codes with small parameter
values. And a genetic algorithm is proposed to search for some sub--optimal DNA codes
with relatively large parameter values, where we can consider their sizes as reasonable
lower bounds of DNA codes. Furthermore, we provide tables of bounds on sizes of DNA
codes with length from 1 to 9 and minimum distance from 1 to 9