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Register a new userMDS Code Constructions with Small Sub-packetization and Near-optimal Repair Bandwidth
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Ankit Singh Rawat
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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This paper addresses the problem of constructing MDS codes that enable exact repair of each code block with small repair bandwidth, which refers to the total amount of information flow from the remaining code blocks during the repair process. This problem naturally arises in the context of distributed storage systems as the node repair problem [7]. The constructions of exact-repairable MDS codes with optimal repair-bandwidth require working with large sub-packetization levels, which restricts their employment in practice. This paper presents constructions for MDS codes that simultaneously provide both small repair bandwidth and small sub-packetization level. In particular, this paper presents two general approaches to construct exact-repairable MDS codes that aim at significantly reducing the required sub-packetization level at the cost of slightly sub-optimal repair bandwidth. The first approach gives MDS codes that have repair bandwidth at most twice the optimal repair-bandwidth. Additionally, these codes also have the smallest possible sub-packetization level $ell = O(r)$, where $r$ denotes the number of parity blocks. This approach is then generalized to design codes that have their repair bandwidth approaching the optimal repair-bandwidth at the cost of graceful increment in the required sub-packetization level. The second approach transforms an MDS code with optimal repair-bandwidth and large sub-packetization level into a longer MDS code with small sub-packetization level and near-optimal repair bandwidth. For a given $r$, the obtained codes have their sub-packetization level scaling logarithmically with the code length. In addition, the obtained codes require field size only linear in the code length and ensure load balancing among the intact code blocks in terms of the information downloaded from these blocks during the exact reconstruction of a code block.
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An $(n, M)$ vector code $mathcal{C} subseteq mathbb{F}^n$ is a collection of $M$ codewords where $n$ elements (from the field $mathbb{F}$) in each of the codewords are referred to as code blocks. Assuming that $mathbb{F} cong mathbb{B}^{ell}$, the code blocks are treated as $ell$-length vectors over the base field $mathbb{B}$. Equivalently, the code is said to have the sub-packetization level $ell$. This paper addresses the problem of constructing MDS vector codes which enable exact reconstruction of each code block by downloading small amount of information from the remaining code blocks. The repair bandwidth of a code measures the information flow from the remaining code blocks during the reconstruction of a single code block. This problem naturally arises in the context of distributed storage systems as the node repair problem [4]. Assuming that $M = |mathbb{B}|^{kell}$, the repair bandwidth of an MDS vector code is lower bounded by $big(frac{n - 1}{n - k}big)cdot ell$ symbols (over the base field $mathbb{B}$) which is also referred to as the cut-set bound [4]. For all values of $n$ and $k$, the MDS vector codes that attain the cut-set bound with the sub-packetization level $ell = (n-k)^{lceil{{n}/{(n-k)}}rceil}$ are known in the literature [23, 35]. This paper presents a construction for MDS vector codes which simultaneously ensures both small repair bandwidth and small sub-packetization level. The obtained codes have the smallest possible sub-packetization level $ell = O(n - k)$ for an MDS vector code and the repair bandwidth which is at most twice the cut-set bound. The paper then generalizes this code construction so that the repair bandwidth of the obtained codes approach the cut-set bound at the cost of increased sub-packetization level. The constructions presented in this paper give MDS vector codes which are linear over the base field $mathbb{B}$.
Minimum storage regenerating (MSR) codes are MDS codes which allow for recovery of any single erased symbol with optimal repair bandwidth, based on the smallest possible fraction of the contents downloaded from each of the other symbols. Recently, certain Reed-Solomon codes were constructed which are MSR. However, the sub-packetization of these codes is exponentially large, growing like $n^{Omega(n)}$ in the constant-rate regime. In this work, we study the relaxed notion of $epsilon$-MSR codes, which incur a factor of $(1+epsilon)$ higher than the optimal repair bandwidth, in the context of Reed-Solomon codes. We give constructions of constant-rate $epsilon$-MSR Reed-Solomon codes with polynomial sub-packetization of $n^{O(1/epsilon)}$ and thereby giving an explicit tradeoff between the repair bandwidth and sub-packetization.
An $(m,n,a,b)$-tensor code consists of $mtimes n$ matrices whose columns satisfy `$a$ parity checks and rows satisfy `$b$ parity checks (i.e., a tensor code is the tensor product of a column code and row code). Tensor codes are useful in distributed storage because a single erasure can be corrected quickly either by reading its row or column. Maximally Recoverable (MR) Tensor Codes, introduced by Gopalan et al., are tensor codes which can correct every erasure pattern that is information theoretically possible to correct. The main questions about MR Tensor Codes are characterizing which erasure patterns are correctable and obtaining explicit constructions over small fields. In this paper, we study the important special case when $a=1$, i.e., the columns satisfy a single parity check equation. We introduce the notion of higher order MDS codes (MDS$(ell)$ codes) which is an interesting generalization of the well-known MDS codes, where $ell$ captures the order of genericity of points in a low-dimensional space. We then prove that a tensor code with $a=1$ is MR iff the row code is an MDS$(m)$ code. We then show that MDS$(m)$ codes satisfy some weak duality. Using this characterization and duality, we prove that $(m,n,a=1,b)$-MR tensor codes require fields of size $q=Omega_{m,b}(n^{min{b,m}-1})$. Our lower bound also extends to the setting of $a>1$. We also give a deterministic polynomial time algorithm to check if a given erasure pattern is correctable by the MR tensor code (when $a=1$).
This paper presents the construction of an explicit, optimal-access, high-rate MSR code for any $(n,k,d=k+1,k+2,k+3)$ parameters over the finite field $fQ$ having sub-packetization $alpha = q^{lceilfrac{n}{q}rceil}$, where $q=d-k+1$ and $Q = O(n)$. The sub-packetization of the current construction meets the lower bound proven in a recent work by Balaji et al. in cite{BalKum}. To our understanding the codes presented in this paper are the first explicit constructions of MSR codes with $d<(n-1)$ having optimal sub-packetization, optimal access and small field size.
Maximum distance separable (MDS) codes are optimal where the minimum distance cannot be improved for a given length and code size. Twisted Reed-Solomon codes over finite fields were introduced in 2017, which are generalization of Reed-Solomon codes. Twisted Reed-Solomon codes can be applied in cryptography which prefer the codes with large minimum distance. MDS codes can be constructed from twisted Reed-Solomon codes, and most of them are not equivalent to Reed-Solomon codes. In this paper, we first generalize twisted Reed-Solomon codes to generalized twisted Reed-Solomon codes, then we give some new explicit constructions of MDS (generalized) twisted Reed-Solomon codes. In some cases, our constructions can get MDS codes with the length longer than the constructions of previous works. Linear complementary dual (LCD) codes are linear codes that intersect with their duals trivially. LCD codes can be applied in cryptography. This application of LCD codes renewed the interest in the construction of LCD codes having a large minimum distance. We also provide new constructions of LCD MDS codes from generalized twisted Reed-Solomon codes.
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