Singleton-like Bound Research Articles

Maximum flag-rank distance codes

In this paper we extend the study of linear spaces of upper triangular matrices endowed with the flag-rank metric. Such metric spaces are isometric to certain spaces of degenerate flags and have been suggested as suitable framework for network coding. In this setting we provide a Singleton-like bound which relates the parameters of a flag-rank-metric code. This allows us to introduce the family of maximum flag-rank distance codes, that are flag-rank-metric codes meeting the Singleton-like bound with equality. Finally, we provide several constructions of maximum flag-rank distance codes.

Optimal (<i>r</i>, δ)-Locally Repairable Codes from Reed-Solomon Codes

For an [n, k, d] (r, δ)-locally repairable codes ((r, δ)-LRCs) , its minimum distance d satisfies the Singleton-like bound. The construction of optimal (r, δ)-LRC, attaining this Singleton-like bound, is an important research problem in recent years for thier applications in distributed storage systems. In this letter, we use Reed-Solomon codes to construct two classes of optimal (r, δ)-LRCs. The optimal LRCs are given by the evaluations of multiple polynomials of degree at most r - 1 at some points in 𝔽q. The first class gives the [(r + δ - 1)t, rt - s, δ + s] optimal (r, δ)-LRC over 𝔽q provided that r + δ + s - 1 ≤ q, s ≤ δ, s < r, and any positive t. The code length is unbounded. The second class gives the [r +r' + d + δ - 2, r + r', d] optimal (r, δ)-LRC over 𝔽q provided that r - r' ≥ d - δ and r + d - 1 ≤ q + 1, which will produce optimal (r, δ)-LRCs with large minimum distance.

A characterization of optimal locally repairable codes

It is significant to investigate locally repairable codes (LRCs) since they have abundant applications to distributed storage systems. A locally repairable code is called optimal if it achieves the Singleton-like bound. In [27], I. Tamo et al. presented sufficient and necessary conditions for optimal LRCs. However, it is difficult to check these conditions for many parameters. In this work, we show that construction of optimal LRCs is equivalent to construction of a class of matrices with some conditions. With the help of these conditions, we completely determine all the optimal LRCs for some small parameters. In addition, by shortening technique for the matrices with these conditions, we can easily obtain many new optimal LRCs remaining the minimum distance.

An Improved Bound and Singleton-Optimal Constructions of Fractional Repetition Codes

Fractional repetition (FR) codes are a class of repair efficient erasure codes that can recover a failed storage node with both optimal repair bandwidth and complexity. In this paper, we study the minimum distance of FR codes, which is the smallest number of nodes whose failure leads to the unrecoverable loss of the stored file. We derive a new upper bound on the minimum distance of FR codes, which is tighter than the Singleton bound and a Singleton-like bound that takes locality into account. Based on regular graphs and combinatorial designs, several families of FR codes with optimal minimum distance are obtained.

Equivalence, group of automorphism and invariants of a family of rank metric codes arising from linearized polynomials

Let Fq denote the finite field with q=pλ elements. Maximum Rank metric codes (MRD for short) are subsets of Mm×n(Fq) whose number of elements attains the Singleton-like bound. The first MRD codes known were found by Delsarte (1978) and Gabidulin (1985). Sheekey (2015) presented a new class of MRD codes over Fq called twisted Gabidulin codes and also proposed a generalization of the twisted Gabidulin codes to the codes Hk,s(L1,L2). The equivalence and duality of twisted Gabidulin codes were discussed by Lunardon, Trombetti, and Zhou (2016). A new class of MRD codes in M2n×2n(Fq) was found by Trombetti-Zhou (2018). In this work, we characterize the equivalence of the class of codes proposed by Sheekey, generalizing the results known for twisted Gabidulin codes and Trombetti-Zhou codes. In the second part of the paper, we restrict ourselves to the case L1(x)=x, where we present its right nucleus, middle nucleus, Delsarte dual and adjoint codes. In the last section, we present the automorphism group of Hk,s(x,L(x)) and compute its cardinality. In particular, we obtain the number of elements in the automorphism group of some twisted Gabidulin codes.

New Constructions of Optimal Cyclic (r, δ) Locally Repairable Codes From Their Zeros

An (r, δ)-locally repairable code ((r, δ)-LRC for short) was introduced by Prakash et al. [14] for tolerating multiple failed nodes in distributed storage systems, which was a generalization of the concept of r-LRCs produced by Gopalan et al. [5]. An (r, δ)-LRC is said to be optimal if it achieves the Singleton-like bound. Recently, Chen et al. [2] generalized the construction of cyclic r-LRCs proposed by Tamo et al. [19], [20] and constructed several classes of optimal (r, δ)-LRCs of length n for n (q-1) or n (q+1), respectively in terms of a union of the set of zeros controlling the minimum distance and the set of zeros ensuring the locality. Following the work of [2], [3], this paper first characterizes (r, δ)-locality of a cyclic code via its zeros. Then we construct several classes of optimal cyclic (r, δ)-LRCs of length n for n (q - 1) or n (q+1), respectively from the product of two sets of zeros. Our constructions include all optimal cyclic (r, δ)-LRCs proposed in [2], [3], and our method seems more convenient to obtain optimal cyclic (r, δ)-LRCs with flexible parameters. Moreover, many optimal cyclic (r, δ)-LRCs of length n for n (q - 1) or n (q + 1), respectively with (r + δ - 1) n can be obtained from our method.

Improved Bounds and Singleton-Optimal Constructions of Locally Repairable Codes With Minimum Distance 5 and 6

Repair locality has been an important metric in a distributed storage system (DSS). Erasure codes with small locality are more popular in a DSS, which means fewer available nodes participating in the repair process of failed nodes. Locally repairable codes (LRCs) as a new coding scheme have given more rise to the system performance and attracted a lot of interest in the theoretical research in coding theory. The particular concern among the research problems is the bounds and optimal constructions of LRCs. The problem of optimal constructions of LRCs includes the most important case of Singleton-optimal LRCs whose minimum distance achieves the Singleton-like bound, which is the core consideration in this paper. In this work, we first of all derive an improved and general upper bound on the code length of Singleton-optimal LRCs with minimum distance $d=5, 6$ , some known constructions are shown to exactly achieve our new bound, which verifies its tightness. For locality $r=2$ and distance $d=6$ , we construct three new Singleton-optimal LRCs whose code length $n=3(q+1)$ , $n=3(q+\sqrt {q}+1)$ and $n=3(2q-4)$ , respectively. Moreover, we obtain a complete characterization for Singleton-optimal LRCs with $r=2$ and $d=6$ . Such characterization has established an important connection between the existence of Singleton-optimal LRCs and that of a special subset of lines of finite projective plane $PG(2, q)$ , thus provides a methodology for constructing LRCs with longer length based on any advance on finite projective plane $PG(2, q)$ . In the end, we employ the well-known line-point incidence matrix and Johnson bounds for constant weight codes to derive tighter upper bounds on the code length. These new bounds further help us to prove that some of the previous Singleton-optimal constructions or their extensions achieve the longest possible code length for $q=3, 4, 5, 7$ . It’s worth noting that all of our Singleton-optimal constructions possess small locality $r=2$ , which are attractive in a DSS.

Sequence-Subset Distance and Coding for Error Control in DNA-Based Data Storage

The process of DNA-based data storage (DNA storage for short) can be mathematically modelled as a communication channel, termed DNA storage channel, whose inputs and outputs are sets of unordered sequences. To design error correcting codes for DNA storage channel, a new metric, termed the sequence-subset distance , is introduced, which generalizes the Hamming distance to a distance function defined between any two sets of unordered vectors and helps to establish a uniform framework to design error correcting codes for DNA storage channel. We further introduce a family of error correcting codes, referred to as sequence-subset codes , for DNA storage and show that the error-correcting ability of such codes is completely determined by their minimum distance. We derive some upper bounds on the size of the sequence-subset codes including a tight bound for a special case, a Singleton-like bound and a Plotkin-like bound. We also propose some constructions, including an optimal construction for that special case, which imply lower bounds on the size of such codes.

Bounds and Constructions of Locally Repairable Codes: Parity-Check Matrix Approach

A locally repairable code (LRC) is a linear code such that every code symbol can be recovered by accessing a small number of other code symbols. In this paper, we study bounds and constructions of LRCs from the viewpoint of parity-check matrices. Firstly, a simple and unified framework based on parity-check matrix to analyze the bounds of LRCs is proposed, and several new explicit bounds on the minimum distance of LRCs in terms of the field size are presented. In particular, we give an alternate proof of the Singleton-like bound for LRCs first proved by Gopalan et al. Some structural properties on optimal LRCs that achieve the Singleton-like bound are given. Then, we focus on constructions of optimal LRCs over the binary field. It is proved that there are only five classes of possible parameters with which optimal binary LRCs exist. Moreover, by employing the proposed parity-check matrix approach, we completely enumerate all these five classes of optimal binary LRCs attaining the Singleton-like bound in the sense of equivalence of linear codes.

Optimal Pliable Fractional Repetition Codes That are Locally Recoverable: A Bipartite Graph Approach

The main purpose of this paper is to construct pliable fractional repetition (FR) codes that are locally recoverable for distributed storage systems (DSSs). FR codes are integral in constructing a class of distributed storage codes with exact repair by transfer. Pliable FR codes are a new type of FR codes in which both the per-node storage and repetition degree can easily be adjusted simultaneously; thus, pliable FR codes are vital for DSSs in which parameters can dynamically change over time. However, the constructions of pliable FR codes with repair locality remain unknown. In addition, the tradeoffs between the code minimum distance of an FR code and its repair locality are not fully understood. To address these problems, this paper first presents general results regarding FR codes. Subsequently, this paper presents an improved Singleton-like bound for locally recoverable FR codes under an additional requirement that each node must be part of a local structure that, upon failure, allows it to be exactly recovered by a simple download process. Moreover, this paper proposes a construction of locally recoverable FR codes that can achieve the proposed Singleton-like bound; this construction is based on bipartite graphs with a given girth. In particular, this paper also proposes a general bipartite-graph-based approach to constructing optimal pliable FR codes with and without repair localities; in this approach, a new family of bipartite graphs, called matching-feasible graphs, is introduced. Finally, this paper proposes the explicit constructions of optimal pliable FR codes by using a family of matching-feasible graphs with arbitrary large girth. Notably, in addition to attaining a Singleton-like bound for FR codes, the explicit pliable FR codes are optimal locally recoverable FR codes from two perspectives of repair locality. The explicit pliable FR codes can also be used as FR batch codes to provide load balancing in DSSs.