Optimal locally repairable codes of distance 33 and 44 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

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 $d=\Theta({q})$, 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}

Subspace Designs Based on Algebraic Function Fields

Subspace designs are a (large) collection of high-dimensional subspaces {H_i} of F_q^m such that for any low-dimensional subspace W, only a small number of subspaces from the collection have non-trivial intersection with W; more precisely, the sum of dimensions of W cap H_i is at most some parameter L. The notion was put forth by Guruswami and Xing (STOC\u2713) with applications to list decoding variants of Reed-Solomon and algebraic-geometric codes, and later also used for explicit rank-metric codes with optimal list decoding radius. Guruswami and Kopparty (FOCS\u2713, Combinatorica\u2716) gave an explicit construction of subspace designs with near-optimal parameters. This construction was based on polynomials and has close connections to folded Reed-Solomon codes, and required large field size (specifically q >= m). Forbes and Guruswami (RANDOM\u2715) used this construction to give explicit constant degree "dimension expanders" over large fields, and noted that subspace designs are a powerful tool in linear-algebraic pseudorandomness. Here, we construct subspace designs over any field, at the expense of a modest worsening of the bound $L$ on total intersection dimension. Our approach is based on a (non-trivial) extension of the polynomial-based construction to algebraic function fields, and instantiating the approach with cyclotomic function fields. Plugging in our new subspace designs in the construction of Forbes and Guruswami yields dimension expanders over F^n for any field F, with logarithmic degree and expansion guarantee for subspaces of dimension Omega(n/(log(log(n))))

The HIV-1 late domain-2 S40A polymorphism in antiretroviral (or ART)-exposed individuals influences protease inhibitor susceptibility.

BackgroundThe p6 region of the HIV-1 structural precursor polyprotein, Gag, contains two motifs, P7TAP11 and L35YPLXSL41, designated as late (L) domain-1 and -2, respectively. These motifs bind the ESCRT-I factor Tsg101 and the ESCRT adaptor Alix, respectively, and are critical for efficient budding of virus particles from the plasma membrane. L domain-2 is thought to be functionally redundant to PTAP. To identify possible other functions of L domain-2, we examined this motif in dominant viruses that emerged in a group of 14 women who had detectable levels of HIV-1 in both plasma and genital tract despite a history of current or previous antiretroviral therapy.ResultsRemarkably, variants possessing mutations or rare polymorphisms in the highly conserved L domain-2 were identified in seven of these women. A mutation in a conserved residue (S40A) that does not reduce Gag interaction with Alix and therefore did not reduce budding efficiency was further investigated. This mutation causes a simultaneous change in the Pol reading frame but exhibits little deficiency in Gag processing and virion maturation. Whether introduced into the HIV-1 NL4-3 strain genome or a model protease (PR) precursor, S40A reduced production of mature PR. This same mutation also led to high level detection of two extended forms of PR that were fairly stable compared to the WT in the presence of IDV at various concentrations; one of the extended forms was effective in trans processing even at micromolar IDV.ConclusionsOur results indicate that L domain-2, considered redundant in vitro, can undergo mutations in vivo that significantly alter PR function. These may contribute fitness benefits in both the absence and presence of PR inhibitor

Beating the probabilistic lower bound on perfect hashing

For an integer $q\ge 2$, a perfect $q$-hash code $C$ is a block code over $[q]:=\{1,\ldots,q\}$ of length $n$ in which every subset $\{\mathbf{c}_1,\mathbf{c}_2,\dots,\mathbf{c}_q\}$ of $q$ elements is separated, i.e., there exists $i\in[n]$ such that $\{\mathrm{proj}_i(\mathbf{c}_1),\dots,\mathrm{proj}_i(\mathbf{c}_q)\}=[q]$, where $\mathrm{proj}_i(\mathbf{c}_j)$ denotes the $i$th position of $\mathbf{c}_j$. Finding the maximum size $M(n,q)$ of perfect $q$-hash codes of length $n$, for given $q$ and $n$, is a fundamental problem in combinatorics, information theory, and computer science. In this paper, we are interested in asymptotical behavior of this problem. More precisely speaking, we will focus on the quantity $R_q:=\limsup_{n\rightarrow\infty}\frac{\log_2 M(n,q)}n$. A well-known probabilistic argument shows an existence lower bound on $R_q$, namely $R_q\ge\frac1{q-1}\log_2\left(\frac1{1-q!/q^q}\right)$ \cite{FK,K86}. This is still the best-known lower bound till now except for the case $q=3$ for which K\"{o}rner and Matron \cite{KM} found that the concatenation technique could lead to perfect $3$-hash codes beating this the probabilistic lower bound. The improvement on the lower bound on $R_3$ was discovered in 1988 and there has been no any progress on lower bound on $R_q$ for more than 30 years despite of some work on upper bounds on $R_q$. In this paper we show that this probabilistic lower bound can be improved for $q=4,8$ and all odd integers between $3$ and $25$, and \emph{all sufficiently large} $q$ with $q \pmod 4\neq 2$.Comment: arXiv admin note: text overlap with arXiv:1010.5764 by other author

Construction of optimal locally recoverable codes and connection with hypergraph

Locally recoverable codes are a class of block codes with an additional property called locality. A locally recoverable code with locality r can recover a symbol by reading at most r other symbols. Recently, it was discovered by several authors that a q-ary optimal locally recoverable code, i.e., a locally recoverable code achieving the Singleton-type bound, can have length much bigger than q + 1. In this paper, we present both the upper bound and the lower bound on the length of optimal locally recoverable codes. Our lower bound improves the best known result in [12] for all distance d \xe2\x89\xa5 7. This result is built on the observation of the parity-check matrix equipped with the Vandermonde structure. It turns out that a parity-check matrix with the Vandermonde structure produces an optimal locally recoverable code if it satisfies a certain expansion property for subsets of Fq. To our surprise, this expansion property is then shown to be equivalent to a well-studied problem in extremal graph theory. Our upper bound is derived by an refined analysis of the arguments of Theorem 3.3 in [6]