arXiv subjects

Chaoping Xing

Publications and source records attributed to Chaoping Xing.

At least 55 records · Page 3Linked to original sources

arXiv2019

Explicit Constructions of Two-Dimensional Reed-Solomon Codes in High Insertion and Deletion Noise Regime

Insertion and deletion (insdel for short) errors are synchronization errors in communication systems caused by the loss of positional information in the message. Reed-Solomon codes have gained a lot of interest due to its encoding simplicity, well structuredness and list-decoding capability in the classical setting. This interest also translates to the insdel metric setting, as the Guruswami-Sudan decoding algorithm can be utilized to provide a deletion correcting algorithm in the insdel metric. Nevertheless, there have been few studies on the insdel error-correcting capability of Reed-Solomon codes. Our main contributions in this paper are explicit constructions of two families of 2-dimensional Reed-Solomon codes with insdel error-correcting capabilities asymptotically reaching those provided by the Singleton bound. The first construction gives a family of Reed-Solomon codes with insdel error-correcting capability asymptotic to its length. The second construction provides a family of Reed Solomon codes with an exact insdel error-correcting capability up to its length. Both our constructions improve the previously known construction of 2-dimensional Reed-Solomon codes whose insdel error-correcting capability is only logarithmic on the code length.

cs.IT↗

arXiv2019

Optimal repairing schemes for Reed-Solomon codes with alphabet sizes linear in lengths under the rack-aware model

In modern practical data centers, storage nodes are usually organized into equally sized groups, which is called racks. The cost of cross-rack communication is much more expensive compared with the intra-rack communication cost. The codes for this system are called rack-aware regenerating codes. Similar to standard minimum storage regenerating (MSR) codes, it is a challenging task to construct minimum storage rack-aware regenerating (MSRR) codes achieving the cut-set bound. The known constructions of MSRR codes achieving the cut-set bound give codes with alphabet size $q$ exponential in the code length $n$, more precisely, $q=Ω(\exp(n^n))$. The main contribution of this paper is to provide explicit construction of MSRR codes achieving the cut-set bound with the alphabet size linear in $n$. To achieve this goal, we first present a general framework to repair Reed-Solomon codes. It turns out that the known repairing schemes of Reed-Solomon codes can be realized under our general framework. Several techniques are used in this paper. In particular, we use the degree decent method to repair failure node. This technique allows us to get Reed-Solomon codes with the alphabet size linear in $n$. The other techniques include choice of good polynomials. Note that good polynomials are used for construction of locally repairable code in literature. To the best of our knowledge, it is the first time in this paper to make use of good polynomials for constructions of regenerating codes.

cs.IT↗

arXiv2019

Constructive asymptotic bounds of locally repairable codes via function fields

Locally repairable codes have been investigated extensively in recent years due to practical applications in distributed and cloud storage systems. However, there are few asymptotical constructions of locally repairable codes in the literature. In this paper, we provide an explicit asymptotic construction of locally repairable codes over arbitrary finite fields from local expansions of functions at a rational place. This construction gives a Tsfasman-Vladut-Zink type bound for locally repairable codes. Its main advantage is that there are no constraints on both locality and alphabet size. Furthermore, we show that the Gilbert-Varshamov type bound on locally repairable codes over non-prime finite fields can be exceeded for sufficiently large alphabet size.

cs.IT↗

arXiv2019

Efficient Multi-Point Local Decoding of Reed-Muller Codes via Interleaved Codex

Reed-Muller codes are among the most important classes of locally correctable codes. Currently local decoding of Reed-Muller codes is based on decoding on lines or quadratic curves to recover one single coordinate. To recover multiple coordinates simultaneously, the naive way is to repeat the local decoding for recovery of a single coordinate. This decoding algorithm might be more expensive, i.e., require higher query complexity. In this paper, we focus on Reed-Muller codes with usual parameter regime, namely, the total degree of evaluation polynomials is $d=Θ({q})$, where $q$ is the code alphabet size (in fact, $d$ can be as big as $q/4$ in our setting). By introducing a novel variation of codex, i.e., interleaved codex (the concept of codex has been used for arithmetic secret sharing \cite{C11,CCX12}), we are able to locally recover arbitrarily large number $k$ of coordinates of a Reed-Muller code simultaneously at the cost of querying $O(q^2k)$ coordinates. It turns out that our local decoding of Reed-Muller codes shows ({\it perhaps surprisingly}) that accessing $k$ locations is in fact cheaper than repeating the procedure for accessing a single location for $k$ times. Our estimation of success error probability is based on error probability bound for $t$-wise linearly independent variables given in \cite{BR94}.

cs.IT↗

arXiv2018

A Construction of Optimal Frequency Hopping Sequence Set via Combination of Multiplicative and Additive Groups of Finite Fields

In literatures, there are various constructions of frequency hopping sequence (FHS for short) sets with good Hamming correlations. Some papers employed only multiplicative groups of finite fields to construct FHS sets, while other papers implicitly used only additive groups of finite fields for construction of FHS sets. In this paper, we make use of both multiplicative and additive groups of finite fields simultaneously to present a construction of optimal FHS sets. The construction provides a new family of optimal $\left(q^m-1,\frac{q^{m-t}-1}{r},rq^t;\frac{q^{m-t}-1}{r}+1\right)$ frequency hopping sequence sets archiving the Peng-Fan bound. Thus, the FHS sets constructed in literatures using either multiplicative groups or additive groups of finite fields are all included in our family. In addition, some other FHS sets can be obtained via the well-known recursive constructions through one-coincidence sequence set.

cs.IT↗

arXiv2018

Construction of optimal locally recoverable codes and connection with hypergraph

Recently, it was discovered by several authors that a $q$-ary optimal locally recoverable code, i.e., a locally recoverable code archiving the Singleton-type bound, can have length much bigger than $q+1$. This is quite different from the classical $q$-ary MDS codes where it is conjectured that the code length is upper bounded by $q+1$ (or $q+2$ for some special case). This discovery inspired some recent studies on length of an optimal locally recoverable code. It was shown in \cite{LXY} that a $q$-ary optimal locally recoverable code is unbounded for $d=3,4$. Soon after, it was proved that a $q$-ary optimal locally recoverable code with distance $d$ and locality $r$ can have length $Ω_{d,r}(q^{1 + 1/\lfloor(d-3)/2\rfloor})$. Recently, an explicit construction of $q$-ary optimal locally recoverable codes for distance $d=5,6$ was given in \cite{J18} and \cite{BCGLP}. In this paper, we further investigate construction optimal locally recoverable codes along the line of using parity-check matrices. Inspired by classical Reed-Solomon codes and \cite{J18}, we equip parity-check matrices with the Vandermond structure. It is turns out that a parity-check matrix with the Vandermond structure produces an optimal locally recoverable code must obey certain disjoint property for subsets of $\mathbb{F}_q$. To our surprise, this disjoint condition is equivalent to a well-studied problem in extremal graph theory. With the help of extremal graph theory, we succeed to improve all of the best known results in \cite{GXY} for $d\geq 7$. In addition, for $d=6$, we are able to remove the constraint required in \cite{J18} that $q$ is even.

cs.IT↗

arXiv2018

Asymptotic Gilbert-Varshamov bound on Frequency Hopping Sequences

Given a $q$-ary frequency hopping sequence set of length $n$ and size $M$ with Hamming correlation $H$, one can obtain a $q$-ary (nonlinear) cyclic code of length $n$ and size $nM$ with Hamming distance $n-H$. Thus, every upper bound on the size of a code from coding theory gives an upper bound on the size of a frequency hopping sequence set. Indeed, all upper bounds from coding theory have been converted to upper bounds on frequency hopping sequence sets (\cite{Ding09}). On the other hand, a lower bound from coding theory does not automatically produce a lower bound for frequency hopping sequence sets. In particular, the most important lower bound--the Gilbert-Varshamov bound in coding theory has not been transformed to frequency hopping sequence sets. The purpose of this paper is to convert the Gilbert-Varshamov bound in coding theory to frequency hopping sequence sets by establishing a connection between a special family of cyclic codes (which are called hopping cyclic codes in this paper) and frequency hopping sequence sets. We provide two proofs of the Gilbert-Varshamov bound. One is based on probabilistic method that requires advanced tool--martingale. This proof covers the whole rate region. The other proof is purely elementary but only covers part of the rate region.

cs.IT↗

arXiv2018

Constructions of maximally recoverable local reconstruction codes via function fields

Local Reconstruction Codes (LRCs) allow for recovery from a small number of erasures in a local manner based on just a few other codeword symbols. A maximally recoverable (MR) LRC offers the best possible blend of such local and global fault tolerance, guaranteeing recovery from all erasure patterns which are information-theoretically correctable given the presence of local recovery groups. In an $(n,r,h,a)$-LRC, the $n$ codeword symbols are partitioned into $r$ disjoint groups each of which include $a$ local parity checks capable of locally correcting $a$ erasures. MR LRCs have received much attention recently, with many explicit constructions covering different regimes of parameters. Unfortunately, all known constructions require a large field size that exponential in $h$ or $a$, and it is of interest to obtain MR LRCs of minimal possible field size. In this work, we develop an approach based on function fields to construct MR LRCs. Our method recovers, and in most parameter regimes improves, the field size of previous approaches. For instance, for the case of small $r \ll ε\log n$ and large $h \ge Ω(n^{1-ε})$, we improve the field size from roughly $n^h$ to $n^{εh}$. For the case of $a=1$ (one local parity check), we improve the field size quadratically from $r^{h(h+1)}$ to $r^{h \lfloor (h+1)/2 \rfloor}$ for some range of $r$. The improvements are modest, but more importantly are obtained in a unified manner via a promising new idea.

cs.IT↗

arXiv2018

How long can optimal locally repairable codes be?

A locally repairable code (LRC) with locality $r$ allows for the recovery of any erased codeword symbol using only $r$ other codeword symbols. A Singleton-type bound dictates the best possible trade-off between the dimension and distance of LRCs --- an LRC attaining this trade-off is deemed \emph{optimal}. Such optimal LRCs have been constructed over alphabets growing linearly in the block length. Unlike the classical Singleton bound, however, it was not known if such a linear growth in the alphabet size is necessary, or for that matter even if the alphabet needs to grow at all with the block length. Indeed, for small code distances $3,4$, arbitrarily long optimal LRCs were known over fixed alphabets. Here, we prove that for distances $d \ge 5$, the code length $n$ of an optimal LRC over an alphabet of size $q$ must be at most roughly $O(d q^3)$. For the case $d=5$, our upper bound is $O(q^2)$. We complement these bounds by showing the existence of optimal LRCs of length $Ω_{d,r}(q^{1+1/\lfloor(d-3)/2\rfloor})$ when $d \le r+2$. These bounds match when $d=5$, thus pinning down $n=Θ(q^2)$ as the asymptotically largest length of an optimal LRC for this case.

cs.IT↗

arXiv2018

New extension constructions of optimal frequency hopping sequence sets

In this paper, a general framework of constructing optimal frequency hopping sequence (FHS) sets is presented based on the designated direct product. Under the framework, we obtain infinitely many new optimal FHS sets by combining a family of sequences that are newly constructed in this paper with some known optimal FHS sets. Our constructions of optimal FHS sets are also based on extension method. However, our constructions remove the constraint requiring that the extension factor is co-prime with the length of original FHSs and get new parameters. In literature, most of the extension constructions suffer from this constraint. As a result, our constructions allow a great flexibility of choosing parameters of FHS sets for a given frequency-hopping spread spectrum system.

cs.IT↗

arXiv2018

List Decodability of Symbol-Pair Codes

We investigate the list decodability of symbol-pair codes in the present paper. Firstly, we show that list decodability of every symbol-pair code does not exceed the Gilbert-Varshamov bound. On the other hand, we are able to prove that with high probability, a random symbol-pair code can be list decoded up to the Gilbert-Varshamov bound. Our second result of this paper is to derive the Johnson-type bound, i.e., a lower bound on list decoding radius in terms of minimum distance. Finally, we present a list decoding algorithm of Reed-Solomon codes beyond the Johnson-type bound.

cs.IT↗

arXiv2018

Optimal locally repairable codes of distance $3$ and $4$ via cyclic codes

Like classical block codes, a locally repairable code also obeys the Singleton-type bound (we call a locally repairable code {\it optimal} if it achieves the Singleton-type bound). In the breakthrough work of \cite{TB14}, several classes of optimal locally repairable codes were constructed via subcodes of Reed-Solomon codes. Thus, the lengths of the codes given in \cite{TB14} are upper bounded by the code alphabet size $q$. Recently, it was proved through extension of construction in \cite{TB14} that length of $q$-ary optimal locally repairable codes can be $q+1$ in \cite{JMX17}. Surprisingly, \cite{BHHMV16} presented a few examples of $q$-ary optimal locally repairable codes of small distance and locality with code length achieving roughly $q^2$. Very recently, it was further shown in \cite{LMX17} that there exist $q$-ary optimal locally repairable codes with length bigger than $q+1$ and distance propositional to $n$. Thus, it becomes an interesting and challenging problem to construct new families of $q$-ary optimal locally repairable codes of length bigger than $q+1$. In this paper, we construct a class of optimal locally repairable codes of distance $3$ and $4$ with unbounded length (i.e., length of the codes is independent of the code alphabet size). Our technique is through cyclic codes with particular generator and parity-check polynomials that are carefully chosen.

cs.IT↗

arXiv2017

Optimal locally repairable codes via elliptic curves

Constructing locally repairable codes achieving Singleton-type bound (we call them optimal codes in this paper) is a challenging task and has attracted great attention in the last few years. Tamo and Barg \cite{TB14} first gave a breakthrough result in this topic by cleverly considering subcodes of Reed-Solomon codes. Thus, $q$-ary optimal locally repairable codes from subcodes of Reed-Solomon codes given in \cite{TB14} have length upper bounded by $q$. Recently, it was shown through extension of construction in \cite{TB14} that length of $q$-ary optimal locally repairable codes can be $q+1$ in \cite{JMX17}. Surprisingly it was shown in \cite{BHHMV16} that, unlike classical MDS codes, $q$-ary optimal locally repairable codes could have length bigger than $q+1$. Thus, it becomes an interesting and challenging problem to construct $q$-ary optimal locally repairable codes of length bigger than $q+1$. In the present paper, we make use of rich algebraic structures of elliptic curves to construct a family of $q$-ary optimal locally repairable codes of length up to $q+2\sqrt{q}$. It turns out that locality of our codes can be as big as $23$ and distance can be linear in length.

cs.IT↗

arXiv2017

Construction of asymptotically good locally repairable codes via automorphism groups of function fields

Locally repairable codes have been investigated extensively in recent years due to practical application in distributed storage as well as theoretical interest. However, not much work on asymptotical behavior of locally repairable codes has been done until now. In particular, there is a little result on constructive lower bound on asymptotical behavior of locally repairable codes. In this paper, we extend the construction given in \cite{BTV17} via automorphism groups of function field towers. The main advantage of our construction is to allow more flexibility of locality. Furthermore, we show that the Gilbert-Varshamov type bound on locally repairable codes can be improved for all sufficiently large alphabet size $q$.

cs.IT↗

arXiv2017

Construction of optimal locally repairable codes via automorphism groups of rational function fields

Locally repairable codes, or locally recoverable codes (LRC for short) are designed for application in distributed and cloud storage systems. Similar to 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 classical MDS codes, optimal locally repairable codes carry some very nice combinatorial structures. Since introduction of the Singleton-type bound for locally repairable codes, people have put tremendous effort on constructions of optimal locally repairable codes. Due to hardness of this problem, there are few constructions of optimal locally repairable codes in literature. Most of these constructions are realized via either combinatorial or algebraic structures. In this paper, we employ automorphism groups of rational function fields to construct optimal locally repairable codes by considering the group action on the projective lines over finite fields. It turns out that we are able to construct optimal locally repairable codes with reflexibility of locality as well as smaller alphabet size comparable to the code length. In particular, we produce new families of $q$-ary locally repairable codes, including codes of length $q+1$ via cyclic groups and codes via dihedral groups.

cs.IT↗

arXiv2017

On the Closest Vector Problem for Lattices Constructed from Polynomials and Their Cryptographic Applications

In this paper, we propose new classes of trapdoor functions to solve the closest vector problem in lattices. Specifically, we construct lattices based on properties of polynomials for which the closest vector problem is hard to solve unless some trapdoor information is revealed. We thoroughly analyze the security of our proposed functions using state-of-the-art attacks and results on lattice reductions. Finally, we describe how our functions can be used to design quantum-safe encryption schemes with reasonable public key sizes. In particular, our scheme can offer around $106$ bits of security with a public key size of around $6.4$ $\texttt{KB}$. Our encryption schemes are efficient with respect to key generation, encryption and decryption.

cs.CR↗

arXiv2017

Efficiently repairing algebraic geometry codes

Minimum storage regenerating codes have minimum storage of data in each node and therefore are maximal distance separable (MDS for short) codes. Thus, the number of nodes is upper bounded by $2^{\fb}$, where $\fb$ is the bits of data stored in each node. From both theoretical and practical points of view (see the details in Section 1), it is natural to consider regenerating codes that nearly have minimum storage of data, and meanwhile the number of nodes is unbounded. One of the candidates for such regenerating codes is an algebraic geometry code. In this paper, we generalize the repairing algorithm of Reed-Solomon codes given in \cite[STOC2016]{GW16} to algebraic geometry codes and present an efficient repairing algorithm for arbitrary one-point algebraic geometry codes. By applying our repairing algorithm to the one-point algebraic geometry codes based on the Garcia-Stichtenoth tower, one can repair a code of rate $1-\Ge$ and length $n$ over $\F_{q}$ with bandwidth $(n-1)(1-\Gt)\log q$ for any $\Ge=2^{(\Gt-1/2)\log q}$ with a real $τ\in(0,1/2)$. In addition, storage in each node for an algebraic geometry code is close to the minimum storage. Due to nice structures of Hermitian curves, repairing of Hermitian codes is also investigated. As a result, we are able to show that algebraic geometry codes are regenerating codes with good parameters. An example reveals that Hermitian codes outperform Reed-Solomon codes for certain parameters.

cs.IT↗

arXiv2017

On subfields of the Hermitian function fields involving the involution automorphism

A function field over a finite field is called maximal if it achieves the Hasse-Weil bound. Finding possible genera that maximal function fields achieve has both theoretical interest and practical applications to coding theory and other topics. As a subfield of a maximal function field is also maximal, one way to find maximal function fields is to find all subfields of a maximal function field. Due to the large automorphism group of the Hermitian function field, it is natural to find as many subfields of the Hermitian function field as possible. In literature, most of papers studied subfields fixed by subgroups of the decomposition group at one point (usually the point at infinity). This is because it becomes much more complicated to study the subfield fixed by a subgroup that is not contained in the decomposition group at one point. In this paper, we study subfields of the Hermitian function field fixed by subgroups that are not contained in the decomposition group of any point except the cyclic subgroups. It turns out that some new maximal function fields are found.

math.NT↗