1.) Good Network Codes---By Ferrers Diagram Rank-Metric Codes
In random linear network coding (RLNC), each node
forwards random linear combinations of all packets received so
far. The data packets can be seen as vectors over a finite field and the internal network structure is assumed to
be unknown. However, due to the linear combinations, one
single erroneous packet can propagate widely throughout the whole network and can render the whole transmission
useless. This makes error-correcting codes in RLNC essential to guarantee reliability.
It was shown in by Kötter and Kschischang that
subspace codes are highly suitable for this purpose.
The multi-level construction by Etzion and Silberstein is one of the
constructions providing codes with the largest known cardinality
for subspace codes. This construction is based on the union of several lifted rank-metric codes,
which are constructed in Ferrers diagrams.
In this project, we have investigated and constructed optimal rank-metric codes in Ferrers diagrams.
First, we have considered rank-metric anticodes and proved a code-anticode bound for Ferrers diagram rank-metric codes.
Four techniques and constructions of Ferrers diagram rank-metric
codes were presented, each providing optimal codes for different
diagrams and parameters for which no optimal solution was known before.
2.) Good Network Codes---Investigation of List Decodability of Gabidulin Codes
Subspace codes can be applied for error-correction in network coding.
A special class of almost-optimal subspace codes can be constructed
by lifting rank-metric codes.
Gabidulin codes can be seen as the rank-metric
equivalent of Reed–Solomon codes.
In this project, subspace codes were used to prove two bounds
on the list size in decoding certain Gabidulin codes. The first
bound is an existential one, showing that exponentially-sized
lists exist for codes with specific parameters. The second bound
presents exponentially-sized lists explicitly, for a different set of
parameters. Both bounds rule out the possibility of efficiently
list decoding several families of Gabidulin codes for any radius
beyond half the minimum distance. Such a result was known so
far only for non-linear rank-metric codes, and not for Gabidulin
codes.
These results reveal a significant difference in list decoding
Gabidulin and Reed–Solomon codes, although the definitions
of these code classes strongly resemble each other.
3.) Codes for Partially Stuck-at Memory Cells
In this project, we have studied a new model of defect
memory cells, called partially stuck-at memory cells, which is
motivated by the behavior of multi-level cells in non-volatile
memories such as flash memories and phase change memories.
Our main contribution in the project is the study of codes for
masking u partially stuck-at cells. We have first derived lower and upper
bounds on the redundancy of such codes. We have then presented three code
constructions over an alphabet of size q.
Furthermore, we have studied the dual defect model in
which cells cannot reach higher levels, and shown that codes for
partially stuck-at cells can be used to mask this type of defects
as well. Lastly, we have analyzed the capacity of the partially stuck-at
memory channel.
