Optimal (2,δ)(2,\delta) Locally Repairable Codes via Punctured Simplex Codes

Locally repairable codes (LRCs) have attracted a lot of attention due to their applications in distributed storage systems. In this paper, we provide new constructions of optimal $(2, \delta)$-LRCs. Firstly, by the techniques of finite geometry, we present a sufficient condition to guarantee a punctured simplex code to be a $(2, \delta)$-LRC. Secondly, by using characteristic sums over finite fields and Krawtchouk polynomials, we construct several families of LRCs with new parameters. All of our new LRCs are optimal with respect to the generalized Cadambe-Mazumdar bound.Comment: Accepted for publication in ISIT202

Singleton-Optimal LRCs and Perfect LRCs via Cyclic and Constacyclic Codes

Locally repairable codes (LRCs) have emerged as an important coding scheme in distributed storage systems (DSSs) with relatively low repair cost by accessing fewer non-failure nodes. Theoretical bounds and optimal constructions of LRCs have been widely investigated. Optimal LRCs via cyclic and constacyclic codes provide significant benefit of elegant algebraic structure and efficient encoding procedure. In this paper, we continue to consider the constructions of optimal LRCs via cyclic and constacyclic codes with long code length. Specifically, we first obtain two classes of $q$-ary cyclic Singleton-optimal $(n, k, d=6;r=2)$-LRCs with length $n=3(q+1)$ when $3 \mid (q-1)$ and $q$ is even, and length $n=\frac{3}{2}(q+1)$ when $3 \mid (q-1)$ and $q \equiv 1(\bmod~4)$, respectively. To the best of our knowledge, this is the first construction of $q$-ary cyclic Singleton-optimal LRCs with length $n>q+1$ and minimum distance $d \geq 5$. On the other hand, an LRC acheiving the Hamming-type bound is called a perfect LRC. By using cyclic and constacyclic codes, we construct two new families of $q$-ary perfect LRCs with length $n=\frac{q^m-1}{q-1}$, minimum distance $d=5$ and locality $r=2$