Maximum Distance Separable Codes Research Articles

A survey on optimal locally repairable codes

Locally repairable codes, or locally recoverable codes (LRC for short) are designed for application in distributed and cloud storage systems. Similar to the classical block codes, there is an important bound called the Singleton-type bound for locally repairable codes. In this paper, an optimal locally repairable code refers to a block code achieving this Singleton-type bound. Like the classical MDS (maximum distance separable) codes, the optimal locally repairable codes have some very nice combinatorial structures. Since the introduction of Singleton-type bound for locally repairable codes, people have put tremendous effort on constructions of optimal locally repairable codes. In this paper, we give a survey on optimal locally repairable codes,including bounds and constructions.

On Grid Quorums for Erasure Coded Data.

We consider the problem of designing grid quorum systems for maximum distance separable (MDS) erasure code based distributed storage systems. Quorums are used as a mechanism to maintain consistency in replication based storage systems, for which grid quorums have been shown to produce optimal load characteristics. This motivates the study of grid quorums in the context of erasure code based distributed storage systems. We show how grid quorums can be built for erasure coded data, investigate the load characteristics of these quorum systems, and demonstrate how sequential consistency is achieved even in the presence of storage node failures.

Near MDS codes from oval polynomials

A linear code with parameters of the form [n,k,n−k+1] is referred to as an MDS (maximum distance separable) code. A linear code with parameters of the form [n,k,n−k] is said to be almost MDS (i.e., almost maximum distance separable) or AMDS for short. A code is said to be near maximum distance separable (in short, near MDS or NMDS) if both the code and its dual are almost maximum distance separable. Near MDS codes correspond to interesting objects in finite geometry and have nice applications in combinatorics and cryptography. There are many unsolved problems about near MDS codes. It is hard to construct an infinite family of near MDS codes whose weight distributions can be settled. In this paper, seven infinite families of [2m+1,3,2m−2] near MDS codes over GF(2m) and seven infinite families of [2m+2,3,2m−1] near MDS codes over GF(2m) are constructed with special oval polynomials for odd m. In addition, nine infinite families of optimal [2m+3,3,2m] near MDS codes over GF(2m) are constructed with oval polynomials in general. The weight distributions of these near MDS codes are settled.

Secure Codes With Accessibility for Distributed Storage

A distributed storage system must support efficient access to stored data while ensuring recovery of temporally unavailable nodes. Another important aspect of a distributed storage system is security. In this paper, we bring these features together and investigate the problem of efficient access to stored data in presence of a passive eavesdropper with access to limited number of nodes. The access efficiency is measured in two different terms, namely, the number of accessed nodes and the volume of generated network traffic. These quantities possess a natural connection to locality and repair bandwidth in distributed storage system. For each of them we derive bounds on parameters and provide explicit constructions based on maximum distance separable codes. Motivated by practical perspectives we propose the techniques to ensure the same workload on each node as well as constructions over small fields based on subfield subcodes, Euclidean geometry codes and Reed-Muller codes. Finally, we derive an asymptotic random coding bound on parameters of a secure distributed storage system and propose further research directions.

Sparse MDS Matrices over Small Fields: A Proof of the GM-MDS Conjecture

A $k \times n$ matrix over a field is called an MDS (maximum distance separable) matrix if it satisfies the following property: Any $k$ columns of it are linearly independent. Equivalently, its rows span an MDS code. A question arising in coding theory is what zero patterns MDS matrices can have. There is a natural combinatorial condition, called the rectangle condition, which is necessary over any field, and sufficient over exponentially large fields, concretely of size ${n-1 \choose k-1}$. The GM-MDS conjecture of Dau, Song, and Yuen [On the existence of MDS codes over small fields with constrained generator matrices, in 2014 IEEE International Symposium on Information Theory (ISIT), pp. 1787--1791] speculated that whenever the rectangle condition holds, there exist algebraic constructions over much smaller fields of size $n+k-1$, and gave an algebraic conjecture that implies this. In this work, we prove this algebraic conjecture. In an independent and parallel work, Yildiz and Hassibi [Optimum linear codes with support constraints over small fields, in 2018 IEEE Information Theory Workshop (ITW), pp. 1--5] found an alternative proof for the algebraic conjecture.

A construction of MDS involutory matrices using MDS self-dual codes: a preliminary result

The Maximum Distance Separable (MDS) matrices have an important application in cryptography, for example see [1]. In order to reduce the implementation complexities, it is important to consider MDS matrices which are involutory. Some constructions of involutory MDS matrices can be found in [2, 3]. In this paper, we give some properties related to a construction of involutory MDS matrices using orthogonal matrices derived from Euclidean self-dual MDS codes. Then, we do a computer search to find involutory MDS matrices with small sizes using classical simulated annealing algorithm. The proposed construction is different compared to the ones in [2, 3].

Dynamic Coded Caching in Wireless Networks

We consider distributed and dynamic caching of coded content at small base stations (SBSs) in an area served by a macro base station (MBS). Specifically, content is encoded using a maximum distance separable code and cached according to a time-to-live (TTL) cache eviction policy, which allows coded packets to be removed from the caches at periodic times. Mobile users requesting a particular content download coded packets from SBSs within communication range. If additional packets are required to decode the file, these are downloaded from the MBS. We formulate an optimization problem that is efficiently solved numerically, providing TTL caching policies minimizing the overall network load. We demonstrate that distributed coded caching using TTL caching policies can offer significant reductions in terms of network load when request arrivals are bursty. We show how the distributed coded caching problem utilizing TTL caching policies can be analyzed as a specific single cache, convex optimization problem. Our problem encompasses static caching and the single cache as special cases. We prove that, interestingly, static caching is optimal under a Poisson request process, and that for a single cache the optimization problem has a surprisingly simple solution.

On Hamming and b-symbol distance distributions of repeated-root constacyclic codes of length $$4p^s$$ over $${\pmb {\mathbb {F}}}_{p^m}+u {\pmb {\mathbb {F}}}_{p^m}$$

Let p be a prime such that $$p^m \equiv 1\pmod {4}$$ , and $$\mathcal{R}=\mathbb F_{p^m}+u\mathbb F_{p^m}$$ . For any non-square unit $$\lambda $$ of $$\mathcal{R}$$ , the Hamming and b-symbol distances of all $$\lambda $$ -constacyclic codes of length $$4p^s$$ over $$\mathcal{R}$$ are completely determined. As examples, several good codes with new parameters are constructed. We also identified all Maximum Distance Separable constacyclic codes of length $$4p^s$$ over $$\mathcal{R}$$ with respect to the Hamming distance as well as the b-symbol distance. Also, we got some non-trivial MDS b-symbol Type 3, $$\gamma $$ -constacyclic codes of length $$4p^s$$ codes over $$\mathcal{R}$$ with respect to b-symbol distance for $$b=4$$ .

Repeated-Root Constacyclic Codes With Pair-Metric

Symbol-pair codes are block codes when considering the pair-metric, and are designed to protect against pair-errors occur in the pair-read vector which consisting of overlapping pairs of symbols. A code that achieves the analog of Singleton bound for symbol-pair codes is called a maximum distance separable (MDS) symbol-pair code. The contribution of this letter is twofold. First, we give a lower bound of minimum pair-distance of repeated-root constacyclic codes in terms of minimum Hamming distance, which extends the lower bound given by Chen, Lin and Liu. Second, we give three new classes of MDS symbol-pair codes. As we know, for an MDS symbol-pair code with a fixed minimum pair-distance, the larger the code length of the code has, the higher the code rate is. The first class of MDS symbol-pair codes we show has unbounded code length and the other two classes of MDS symbol-pair codes have larger code lengths than the code lengths of existing MDS symbol-pair codes under the same minimum pair-distance 6.

MDS coding is better than replication for job completion times

If queue-states are unknown to a dispatcher, a replication strategy has been proposed where job copies are sent randomly to servers. When the service of any copy is completed, redundant copies are removed. For jobs that can be algebraically manipulated and have linear servers, to get the response-time performance of sending d copies of n jobs via the replication strategy, with Maximum Distance Separable (MDS) codes we show that n+d jobs suffice. That is, while replication is multiplicative, MDS is linear.

MDS and I-Perfect Codes in Pomset Metric

Recently pomset metric was introduced to accommodate Lee metric, thereby generalizing poset metrics for codes over $\mathbb {Z}_{{m}}$ . This work mainly studies results on Maximum distance separable (MDS) pomset codes and perfect codes over $\mathbb {Z}_{{m}}$ . If I is an ideal in a poset, the concept of I -perfect codes that was introduced to study MDS poset codes is now extended to the study of MDS pomset codes as well. But, unlike the ideals in posets, here the ideals in pomsets are considered as those with full count and those with partial count. In fact, the I -balls are no more linear when the ideal I is with partial count. This makes the investigation of the relationship of I -perfect codes with MDS pomset codes little different from that with MDS poset codes. We thoroughly study the I -balls where I is with partial count. Singleton type bound is established for codes with pomset metric and the connection of MDS codes with I -perfect codes is investigated upon. Then, we prove the duality theorem for codes with chain pomset. Finally, we determine the weight distribution of MDS pomset codes when the pomset is a chain.

Two Classes of Binary MDS Array Codes With Asymptotically Optimal Repair for Any Single Column

An $m\times (k+r)$ binary maximum distance separable (MDS) array code contains $k$ information columns and $r$ parity columns with each entry being a bit, where any $k$ out of $k+r$ columns can recover the $k$ information columns. When there is a failed column, it is critical to minimize the repair bandwidth that is the total number of bits downloaded from $d$ out of $k+r-1$ surviving columns in repairing the failed column. In this article, we first propose two explicit constructions of binary MDS array codes that have asymptotically optimal repair bandwidth for any information column, where $r\geq 2$ and $d=k+r-1$ for the first construction, and $r\geq 4$ is an even number and $d=k+\frac {r}{2}-1$ for the second construction. By applying a generic transformation for the proposed two classes of binary MDS array codes, we then obtain two classes of new binary MDS array codes that also have optimal repair bandwidth for any parity column and asymptotically optimal repair bandwidth for any information column.

Network-Coded Cooperative Systems With Generalized User-Relay Selection

We consider a network-coded cooperative (NCC) system that consists of N ≥ 2 sources, M ≥ 1 decode- and-forward (DF) relays, and a single destination. The relays perform network coding (NC) on the received sources' symbols using maximum distance separable (MDS) codes. For this system, we propose the most generalized user-relay selection (GURS) scheme in the literature that selects any arbitrary subsets of K users and any arbitrary subsets of L relays subject to practical constraints such as load balancing conditions and scheduling policy. Our analytical results and design guidelines generalize and subsume all existing results as special cases. To this end, we derive a new closed-form outage probability (OP) expression, assuming non-identically and independently distributed (n.i.i.d.) Rayleigh fading channels. The asymptotic outage expression at high signal-to-noise ratio (SNR) regime is further derived, based on which, the achievable diversity order and coding gain are quantified. The theoretical derivations are also validated through Monte-Carlo simulation.

ϵ-MSR Codes: Contacting Fewer Code Blocks for Exact Repair

$\epsilon $ -Minimum Storage Regenerating ( $\epsilon $ -MSR) codes form a special class of Maximum Distance Separable (MDS) codes, providing mechanisms for exact regeneration of a single code block in their codewords by downloading slightly sub-optimal amount of information from the remaining code blocks. The key advantage of these codes is a significantly lower sub-packetization that grows only logarithmically with the code length, while providing optimality in storage and error-correcting capacity. However, existing constructions of $\epsilon $ -MSR codes require each remaining code block to be available for the repair of any failed code block. In this paper, we construct $\epsilon $ -MSR codes that can repair any failed code block by contacting fewer number of available code blocks. When a code block fails, our repair procedure needs to contact a few compulsory code blocks and is free to choose any subset of available code blocks for the remaining choices. Our construction requires a field size linear in code length and ensures load balancing (in terms of information downloaded) among the contacted code blocks for repairing a failed code block.

On symbol-pair weight distribution of MDS codes and simplex codes over finite fields

Symbol-pair code is a new coding framework proposed to combat pair-errors in symbol-pair read channels. Remarkably, a classical maximum distance separable (MDS) code is also an MDS symbol-pair code. In this paper, we investigate the symbol-pair weight distribution of MDS codes and the symbol b-weight distribution of simplex codes over finite fields respectively. Surprisingly, one can derive that all the nonzero codewords of simplex codes have the same symbol b-weight and rearrangement of coordinates in simplex codes may induce different symbol-pair weights. Moreover, the symbol b-weight distribution of variation simplex codes over certain finite fields is determined.

Twisted Reed–Solomon Codes With One-Dimensional Hull

The hull of a linear code is defined to be the intersection of the code and its dual. When the size of the hull is small, it has been proved that some algorithms for checking permutation equivalence of two linear codes and computing the automorphism group of a linear code are very effective in general. Maximum distance separable (MDS) codes are codes meeting the Singleton bound. Twisted Reed-Solomon codes is a generalization of Reed-Solomon codes, which is also a nice construction for MDS codes. In this short letter, we obtain some twisted Reed-Solomon MDS codes with one-dimensional hull. Moreover, these codes are not monomially equivalent to Reed-Solomon codes.

Constructions of k -uniform and absolutely maximally entangled states beyond maximum distance codes

Pure multipartite quantum states of n parties and local dimension q are called k-uniform if all reductions to k parties are maximally mixed. These states are relevant for our understanding of multipartite entanglement, quantum information protocols, and the construction of quantum error correction codes. To our knowledge, the only known systematic construction of these quantum states is based on classical error correction codes. We present a systematic method to construct other examples of k-uniform states and show that the states derived through our construction are not equivalent to any k-uniform state constructed from the so-called maximum distance separable error correction codes. Furthermore, we use our method to construct several examples of absolutely maximally entangled states whose existence was open so far.

On the Storage Cost of Private Information Retrieval

We consider the fundamental tradeoff between the storage cost and the download cost in private information retrieval systems, without any explicit structural restrictions on the storage codes, such as maximum distance separable codes or uncoded storage. Two novel outer bounds are provided, which have the following implications. When the messages are stored without any redundancy across the databases, the optimal PIR strategy is to download all the messages; on the other hand, for PIR capacity-achieving codes, each database can reduce the storage cost, from storing all the messages, by no more than one message on average. We then focus on the two-message two-database case, and show that a stronger outer bound can be derived through a novel pseudo-message technique. This stronger outer bound suggests that a precise characterization of the storage-download tradeoff may require non-Shannon type inequalities, or at least more sophisticated bounding techniques.

Quantum Private Information Retrieval From Coded and Colluding Servers

In the classical private information retrieval (PIR) setup, a user wants to retrieve a file from a database or a distributed storage system (DSS) without revealing the file identity to the servers holding the data. In the quantum PIR (QPIR) setting, a user privately retrieves a classical file by receiving quantum information from the servers. The QPIR problem has been treated by Song et al. in the case of replicated servers, both without collusion and with all but one servers colluding. In this paper, the QPIR setting is extended to account for maximum distance separable (MDS) coded servers. The proposed protocol works for any $[n,k]$ -MDS code and $t$ -collusion with $t=n-k$ . Similarly to the previous cases, the rates achieved are better than those known or conjectured in the classical counterparts. Further, it is demonstrated how the protocol can adapted to achieve significantly higher retrieval rates from DSSs encoded with a locally repairable code (LRC) with disjoint repair groups, each of which is an MDS code.

Capacity-Achieving Private Information Retrieval Codes From MDS-Coded Databases With Minimum Message Size

We consider constructing capacity-achieving linear codes with minimum message size for private information retrieval (PIR) from N non-colluding databases, where each message is coded using maximum distance separable (MDS) codes, such that it can be recovered from accessing the contents of any T databases. It is shown that the minimum message size (sometimes also referred to as the sub-packetization factor) is significantly, in fact exponentially, lower than previously believed. More precisely, when K > T/ gcd(N, T) where K is the total number of messages in the system and gcd(·, ·) means the greatest common divisor, we establish, by providing both novel code constructions and a matching converse, the minimum message size as lcm(N - T, T), where lcm(·, ·) means the least common multiple. On the other hand, when K is small, we show that it is in fact possible to design codes with a message size even smaller than lcm(N - T, T).