4 research outputs found
Some new constructions of optimal linear codes and alphabet-optimal -locally repairable codes
In distributed storage systems, locally repairable codes (LRCs) are designed
to reduce disk I/O and repair costs by enabling recovery of each code symbol
from a small number of other symbols. To handle multiple node failures,
-LRCs are introduced to enable local recovery in the event of up to
failed nodes. Constructing optimal -LRCs has been a
significant research topic over the past decade. In \cite{Luo2022}, Luo
\emph{et al.} proposed a construction of linear codes by using unions of some
projective subspaces within a projective space. Several new classes of Griesmer
codes and distance-optimal codes were constructed, and some of them were proved
to be alphabet-optimal -LRCs.
In this paper, we first modify the method of constructing linear codes in
\cite{Luo2022} by considering a more general situation of intersecting
projective subspaces. This modification enables us to construct good codes with
more flexible parameters. Additionally, we present the conditions for the
constructed linear codes to qualify as Griesmer codes or achieve distance
optimality. Next, we explore the locality of linear codes constructed by
eliminating elements from a complete projective space. The novelty of our work
lies in establishing the locality as , , or -locality,
in contrast to the previous literature that only considered -locality.
Moreover, by combining analysis of code parameters and the C-M like bound for
-LRCs, we construct some alphabet-optimal -LRCs which
may be either Griesmer codes or not Griesmer codes. Finally, we investigate the
availability and alphabet-optimality of -LRCs constructed from our
modified framework.Comment: 25 page