|
| | | | |
| | | |
| Stat |
Members: 3669
Articles: 2'599'751
Articles rated: 2609
24 March 2025 |
|
| | | | |
|
Article overview
| |
|
|
Constructions of Optimal Cyclic $(r,delta)$ Locally Repairable Codes | | Bin Chen
; Shu-Tao Xia
; Jie Hao
; Fang-Wei Fu
; | | Date: |
5 Sep 2016 | | Abstract: | A code is said to be a $r$-local locally repairable code (LRC) if each of its
coordinates can be repaired by accessing at most $r$ other coordinates. When
some of the $r$ coordinates are also erased, the $r$-local LRC can not
accomplish the local repair, which leads to the concept of
$(r,delta)$-locality. A $q$-ary $[n, k]$ linear code $cC$ is said to have
$(r, delta)$-locality ($deltage 2$) if for each coordinate $i$, there exists
a punctured subcode of $cC$ with support containing $i$, whose length is at
most $r + delta - 1$, and whose minimum distance is at least $delta$. The
$(r, delta)$-LRC can tolerate $delta-1$ erasures in total, which degenerates
to a $r$-local LRC when $delta=2$. A $q$-ary $(r,delta)$ LRC is called
optimal if it meets the Singleton-like bound for $(r,delta)$-LRCs. A class of
optimal $q$-ary cyclic $r$-local LRCs with lengths $nmid q-1$ were constructed
by Tamo, Barg, Goparaju and Calderbank based on the $q$-ary Reed-Solomon codes.
In this paper, we construct a class of optimal $q$-ary cyclic $(r,delta)$-LRCs
($deltage 2$) with length $nmid q-1$, which generalizes the results of Tamo
emph{et al.} Moreover, we construct a new class of optimal $q$-ary cyclic
$r$-local LRCs with lengths $nmid q+1$ and a new class of optimal $q$-ary
cyclic $(r,delta)$-LRCs ($deltage 2$) with lengths $nmid q+1$. The
constructed optimal LRCs with length $n=q+1$ have the best-known length $q+1$
for the given finite field with size $q$ when the minimum distance is larger
than $4$. | | Source: | arXiv, 1609.1136 | | Services: | Forum | Review | PDF | Favorites | |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
|
| |
|
|
|
REGISTER