Quantum Codes from Generalized Reed-Solomon Codes and Matrix-Product Codes

One of the central tasks in quantum error-correction is to construct quantum codes that have good parameters. In this paper, we construct three new classes of quantum MDS codes from classical Hermitian self-orthogonal generalized Reed-Solomon codes. We also present some classes of quantum codes from matrix-product codes. It turns out that many of our quantum codes are new in the sense that the parameters of quantum codes cannot be obtained from all previous constructions

A general private information retrieval scheme for MDS coded databases with colluding servers

The problem of private information retrieval gets renewed attentions in recent years due to its information-theoretic reformulation and applications in distributed storage systems. PIR capacity is the maximal number of bits privately retrieved per one bit of downloaded bit. The capacity has been fully solved for some degenerating cases. For a general case where the database is both coded and colluded, the exact capacity remains unknown. We build a general private information retrieval scheme for MDS coded databases with colluding servers. Our scheme achieves the rate $(1+R+R^2+\cdots+R^{M-1})$, where $R=1-\frac{{{N-T}\choose K}}{{N\choose K}}$. Compared to existing PIR schemes, our scheme performs better for a certain range of parameters and is suitable for any underlying MDS code used in the distributed storage system.Comment: Submitted to IEEE Transactions on Information Theor

Private Information Retrieval from MDS Coded Databases with Colluding Servers under Several Variant Models

Private information retrieval (PIR) gets renewed attentions due to its information-theoretic reformulation and its application in distributed storage system (DSS). The general PIR model considers a coded database containing $N$ servers storing $M$ files. Each file is stored independently via the same arbitrary $(N,K)$-MDS code. A user wants to retrieve a specific file from the database privately against an arbitrary set of $T$ colluding servers. A key problem is to analyze the PIR capacity, defined as the maximal number of bits privately retrieved per one downloaded bit. Several extensions for the general model appear by bringing in various additional constraints. In this paper, we propose a general PIR scheme for several variant PIR models including: PIR with robust servers, PIR with Byzantine servers, the multi-file PIR model and PIR with arbitrary collusion patterns.Comment: The current draft is extended by considering several PIR models. The original version named "Multi-file Private Information Retrieval from MDS Coded Databases with Colluding Servers" is abridged into a section within the current draft. arXiv admin note: text overlap with arXiv:1704.0678

Snake-in-the-Box Codes for Rank Modulation under Kendall's τ\tau-Metric

For a Gray code in the scheme of rank modulation for flash memories, the codewords are permutations and two consecutive codewords are obtained using a push-to-the-top operation. We consider snake-in-the-box codes under Kendall's $\tau$-metric, which is a Gray code capable of detecting one Kendall's $\tau$-error. We answer two open problems posed by Horovitz and Etzion. Firstly, we prove the validity of a construction given by them, resulting in a snake of size $M_{2n+1}=\frac{(2n+1)!}{2}-2n+1$. Secondly, we come up with a different construction aiming at a longer snake of size $M_{2n+1}=\frac{(2n+1)!}{2}-2n+3$. The construction is applied successfully to $S_7$.Comment: arXiv admin note: text overlap with arXiv:1311.4703 by other author

Optimal binary linear locally repairable codes with disjoint repair groups

In recent years, several classes of codes are introduced to provide some fault-tolerance and guarantee system reliability in distributed storage systems, among which locally repairable codes (LRCs for short) play an important role. However, most known constructions are over large fields with sizes close to the code length, which lead to the systems computationally expensive. Due to this, binary LRCs are of interest in practice. In this paper, we focus on binary linear LRCs with disjoint repair groups. We first derive an explicit bound for the dimension k of such codes, which can be served as a generalization of the bounds given in [11, 36, 37]. We also give several new constructions of binary LRCs with minimum distance $d = 6$ based on weakly independent sets and partial spreads, which are optimal with respect to our newly obtained bound. In particular, for locality $r\in \{2,3\}$ and minimum distance $d = 6$, we obtain the desired optimal binary linear LRCs with disjoint repair groups for almost all parameters

Separating hash families: A Johnson-type bound and new constructions

Separating hash families are useful combinatorial structures which are generalizations of many well-studied objects in combinatorics, cryptography and coding theory. In this paper, using tools from graph theory and additive number theory, we solve several open problems and conjectures concerning bounds and constructions for separating hash families. Firstly, we discover that the cardinality of a separating hash family satisfies a Johnson-type inequality. As a result, we obtain a new upper bound, which is superior to all previous ones. Secondly, we present a construction for an infinite class of perfect hash families. It is based on the Hamming graphs in coding theory and generalizes many constructions that appeared before. It provides an affirmative answer to both Bazrafshan-Trung's open problem on separating hash families and Alon-Stav's conjecture on parent-identifying codes. Thirdly, let $p_t(N,q)$ denote the maximal cardinality of a $t$-perfect hash family of length $N$ over an alphabet of size $q$. Walker II and Colbourn conjectured that $p_3(3,q)=o(q^2)$. We verify this conjecture by proving $q^{2-o(1)}<p_3(3,q)=o(q^2)$. Our proof can be viewed as an application of Ruzsa-Szemer{\'e}di's (6,3)-theorem. We also prove $q^{2-o(1)}<p_4(4,q)=o(q^2)$. Two new notions in graph theory and additive number theory, namely rainbow cycles and $R$-sum-free sets, are introduced to prove this result. These two bounds support a question of Blackburn, Etzion, Stinson and Zaverucha. Finally, we establish a bridge between perfect hash families and hypergraph Tur{\'a}n problems. This connection has not been noticed before. As a consequence, many new results and problems arise.Comment: 20 pages, accepted in SIAM Journal on Discrete Mathematic

Quantum Block and Synchronizable Codes Derived from Certain Classes of Polynomials

One central theme in quantum error-correction is to construct quantum codes that have a large minimum distance. In this paper, we first present a construction of classical codes based on certain class of polynomials. Through these classical codes, we are able to obtain some new quantum codes. It turns out that some of quantum codes exhibited here have better parameters than the ones available in the literature. Meanwhile, we give a new class of quantum synchronizable codes with highest possible tolerance against misalignment from duadic codes.Comment: 9 pages. arXiv admin note: text overlap with arXiv:1403.6192, arXiv:1311.3416 by other author

New bounds on the number of tests for disjunct matrices

Given $n$ items with at most $d$ of which being positive, instead of testing these items individually, the theory of combinatorial group testing aims to identify all positive items using as few tests as possible. This paper is devoted to a fundamental and thirty-year-old problem in the nonadaptive group testing theory. A binary matrix is called $d$-disjunct if the boolean sum of arbitrary $d$ columns does not contain another column not in this collection. Let $T(d)$ denote the minimal $t$ such that there exists a $t\times n$ $d$-disjunct matrix with $n>t$. $T(d)$ can also be viewed as the minimal $t$ such that there exists a nonadaptive group testing scheme which is better than the trivial one that tests each item individually. It was known that $T(d)\ge\binom{d+2}{2}$ and was conjectured that $T(d)\ge(d+1)^2$. In this paper we narrow the gap by proving $T(d)/d^2\ge(15+\sqrt{33})/24$, a quantity in [6/7,7/8].Comment: 4 pages, to appear in IEEE Transactions on Information Theor

Quaternary Constant-Composition Codes with Weight Four and Distances Five or Six

The sizes of optimal constant-composition codes of weight three have been determined by Chee, Ge and Ling with four cases in doubt. Group divisible codes played an important role in their constructions. In this paper, we study the problem of constructing optimal quaternary constant-composition codes with Hamming weight four and minimum distances five or six through group divisible codes and Room square approaches. The problem is solved leaving only five lengths undetermined. Previously, the results on the sizes of such quaternary constant-composition codes were scarce.Comment: 23 pages, 3 table

The Weight Hierarchy of Some Reducible Cyclic Codes

The generalized Hamming weights (GHWs) of linear codes are fundamental parameters, the knowledge of which is of great interest in many applications. However, to determine the GHWs of linear codes is difficult in general. In this paper, we study the GHWs for a family of reducible cyclic codes and obtain the complete weight hierarchy in several cases. This is achieved by extending the idea of \cite{YLFL} into higher dimension and by employing some interesting combinatorial arguments. It shall be noted that these cyclic codes may have arbitrary number of nonzeroes