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Motivated by applications to DNA storage, we study reconstruction and list-reconstruction schemes for integer vectors that suffer from limited-magnitude errors. We characterize the asymptotic size of the intersection of error balls in relation to the code’s minimum distance. We also devise efficient reconstruction algorithms for various limited-magnitude error parameter ranges. We then extend these algorithms to the list-reconstruction scheme, and show the trade-off between the asymptotic list size and the number of required channel outputs. These results apply to all codes, without any assumptions on the code structure. Finally, we also study linear reconstruction codes with small intersection, as well as show a connection to list-reconstruction codes for the tandem-duplication channel. [ABSTRACT FROM AUTHOR]

We study generalized covering radii, a fundamental property of linear codes that characterizes the trade-off between storage, latency, and access in linear data-query protocols such as PIR. We prove lower and upper bounds on the generalized covering radii of Reed-Muller codes, as well as finding their exact value in certain extreme cases. With the application to linear data-query protocols in mind, we also construct a covering algorithm that gets as input a set of points in space, and find a corresponding set of codewords from the Reed-Muller code that are jointly not farther away from the input than the upper bound on the generalized covering radius of the code. We prove that the algorithm runs in time that is polynomial in the code parameters. [ABSTRACT FROM AUTHOR]

Error-correcting codes over sets, with applications to DNA storage, are studied. The DNA-storage channel receives a set of sequences, and produces a corrupted version of the set, including sequence loss, symbol substitution, symbol insertion/deletion, and limited-magnitude errors in symbols. Various parameter regimes are studied. New bounds on code parameters are provided, which improve upon known bounds. New codes are constructed, at times matching the bounds up to lower-order terms or small constant factors. [ABSTRACT FROM AUTHOR]

We construct integer error-correcting codes and covering codes for the limited-magnitude error channel with more than one error. The codes are lattices that pack or cover the space with the appropriate error ball. Some of the constructions attain an asymptotic packing/covering density that is constant. The results are obtained via various methods, including the use of codes in the Hamming metric, modular Bt-sequences, 2-fold Sidon sets, and sets avoiding arithmetic progression. [ABSTRACT FROM AUTHOR]

We study scalar-linear and vector-linear solutions of the generalized combination network. We derive new upper and lower bounds on the maximum number of nodes in the middle layer, depending on the network parameters and the alphabet size. These bounds improve and extend the parameter range of known bounds. Using these new bounds we present a lower bound and an upper bound on the gap in the alphabet size between optimal scalar-linear and optimal vector-linear network coding solutions. For a fixed network structure, while varying the number of middle-layer nodes r, the asymptotic behavior of the upper and lower bounds shows that the gap is in Θ (log (r)). [ABSTRACT FROM AUTHOR]

Abstract: Group divisible covering designs (GDCDs) were introduced by Heinrich and Yin as a natural generalization of both covering designs and group divisible designs. They have applications in software testing and universal data compression. The minimum number of blocks in a k‐GDCD of type g u is a covering number denoted by C ( k , g u ). When k = 3, the values of C ( 3 , g u ) have been determined completely for all possible pairs ( g , u ). When k = 4, Francetić et al. constructed many families of optimal GDCDs, but the determination remained far from complete. In this paper, two specific 4‐IGDDs are constructed, thereby completing the existence problem for 4‐IGDDs of type ( g , h ) u. Then, additional families of optimal 4‐GDCDs are constructed. Consequently the cases for ( g , u ) whose status remains undetermined arise when g ≡ 7 mod 12 and u ≡ 3 mod 6, when g ≡ 11 , 14 , 17 , 23 mod 24 and u ≡ 5 mod 6, and in several small families for which one of g and u is fixed. [ABSTRACT FROM AUTHOR]

Let K = { k 1 , k 2 , … , k r } and L = { l 1 , l 2 , … , l s } be disjoint subsets of { 0 , 1 , … , p − 1 } , where p is a prime and A = { A 1 , A 2 , … , A m } is a family of subsets of [ n ] = { 1 , 2 , … , n } such that | A i | ( mod p ) ∈ K for all A i ∈ A and | A i ∩ A j | ( mod p ) ∈ L for i ≠ j . In 1991, Alon, Babai and Suzuki conjectured that if n ≥ s + max 1 ≤ i ≤ r k i , then | A | ≤ n s + n s − 1 + ⋯ + n s − r + 1 . In 2000, Qian and Ray-Chaudhuri proved the conjecture under the condition n ≥ 2 s − r . In 2015, Hwang and Kim verified this conjecture. In this paper, we will prove that if n ≥ 2 s − 2 r + 1 or n ≥ s + max 1 ≤ i ≤ r k i , then | A | ≤ n − 1 s + n − 1 s − 1 + ⋯ + n − 1 s − 2 r + 1 . This result strengthens both the upper bound of Alon–Babai–Suzuki’s conjecture and Qian and Ray-Chaudhuri’s result, when n ≥ 2 s − 2 . [ABSTRACT FROM AUTHOR]

The research on directed PBDs is motivated by the construction of t-deletion/insertion-correcting codes. Fuji-Hara, Miao, Wang, and Yin have determined the existence of directed PBDs with block sizes from the set [ABSTRACT FROM AUTHOR]

A uniformly resolvable design (URD) is a resolvable design in which each parallel class contains blocks of only one block size k . Such a class is denoted k -pc and for a given k the number of k -pcs is denoted r k . The number of points of the URD is denoted by v . In the literature, the existence of URDs of block sizes k 1 and k 2 with { k 1 , k 2 } ∈ { { 2 , 3 } , { 2 , 4 } , { 3 , 4 } } has been studied with much efforts. For { k 1 , k 2 } = { 2 , 3 } or { 3 , 4 } , the existence spectra have been determined completely, while for { k 1 , k 2 } = { 2 , 4 } there are still 1338 undetermined pairs of ( v , r 2 ) . In this paper we continue the study on the existence of URDs of block sizes 2 and 4. We improve the known results by reducing the number of open cases from 1338 to 16 . In addition, we prove that there exist 4 -RGDDs of types 2 178 , 2 250 and 2 334 . [ABSTRACT FROM AUTHOR]

Multiply constant-weight codes (MCWCs) were introduced recently to improve the reliability of certain physically unclonable function response. In this paper, the bounds of MCWCs and the constructions of optimal MCWCs are studied. First, we derive three different types of upper bounds which improve the Johnson-type bounds given by Chee et al. for some parameters. The asymptotic lower bound of MCWCs is also examined. Then, we obtain the asymptotic existence of two classes of optimal MCWCs, which shows that the Johnson-type bounds for MCWCs with distances 2\sum i=1^{m}wi-2 or $2mw-2w$ are asymptotically exact. Finally, we construct a class of optimal MCWCs with total weight four and distance six by establishing the connection between such MCWCs and a new kind of combinatorial structures. As a consequence, the maximum sizes of MCWCs with total weight less than or equal to four are determined almost completely. [ABSTRACT FROM AUTHOR]

A uniformly resolvable design (URD) is a resolvable design in which each parallel class contains blocks of only one block size k . Such a class is denoted k -pc and for a given k the number of k -pcs is denoted r k . Let v denote the number of points of the URD. For the case of block sizes 3 and 4 (both existing), the necessary conditions imply that v ≡ 0 ( mod 12 ) . It has been shown that almost all URDs with permissible r 3 and r 4 exist for v ≡ 0 ( mod 24 ) , v ≡ 0 ( mod 60 ) , v ≡ 36 ( mod 144 ) or v ≡ 36 ( mod 108 ) . In this paper, we prove that the necessary conditions for the existence of a URD with block sizes 3 and 4 are also sufficient, except when v = 12 , r 3 = 1 and r 4 = 3 . [ABSTRACT FROM AUTHOR]

A triple system is a collection of b blocks, each of size three, on a set of v points. It is j-balanced when every two j -sets of points appear in numbers of blocks that are as nearly equal as possible, and well balanced when it is j-balanced for each j ∈ {1, 2, 3}.Well-balanced systems arise in the minimization of variance in file availability in distributed file systems. It is shown that when a triple system that is 2-balanced and 3-balanced exists, so does one that is well balanced. Using known and new results on variants of group divisible designs, constructions for well-balanced triple systems are developed. Using these, the spectrum of pairs (v, b) for which such a well-balanced triple system exists is determined completely. [ABSTRACT FROM AUTHOR]

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 ternary constant-composition codes with Hamming weight four and minimum distance six. The problem is solved with a small number of lengths undetermined. The previously known results are those with code length no greater than 10. [ABSTRACT FROM AUTHOR]

In this paper, we continue to investigate the existence of 5 -GDDs, 4-frames and 4-RGDDs, which have been studied by numerous researchers in the past two decades due to their vital applications in the constructions for other types of designs. Much progress has been made for the existence problems of these designs, while the problems are still open. Our new results mainly lie in the cases with relatively large group size. We also construct some smaller designs, including 4 -RGDDs of types 2 184 , 9 44 , 14 10 , 18 18 , 22 10 and 36 11 . [ABSTRACT FROM AUTHOR]

Abstract: Non-uniform group divisible designs (GDDs) and non-uniform Kirkman frames are useful in the constructions for other types of designs. In this paper, we consider the existence problems for -GDDs of type with and Kirkman frames of type . First, we determine completely the spectrum for uniform -GDDs of type . Then, we consider the entire existence problem for non-uniform -GDDs of type with . We show that, for each given , up to a small number of undetermined cases of , the necessary conditions on for the existence of a -GDD of type are also sufficient, except possibly when for and when for . Finally, a similar result for Kirkman frames of type is obtained. We show that, for each given , up to a small number of undetermined cases of , the necessary conditions on for the existence of a Kirkman frame of type are also sufficient, except possibly when for and when for . [Copyright &y& Elsevier]

We study whether an asymmetric limited-magnitude ball may tile Z n. This ball generalizes previously studied shapes: crosses, semi-crosses, and quasi-crosses. Such tilings act as perfect error-correcting codes in a channel which changes a transmitted integer vector in a bounded number of entries by limited-magnitude errors. A construction of lattice tilings based on perfect codes in the Hamming metric is given. Several non-existence results are proved, both for general tilings, and lattice tilings. A complete classification of lattice tilings for two certain cases is proved. [ABSTRACT FROM AUTHOR]

Nonuniform group divisible designs (GDDs) have been studied by numerous researchers for the past two decades due to their essential role in the constructions for other types of designs. In this paper, we investigate the existence problem of [ABSTRACT FROM AUTHOR]

Abstract: Non-uniform group divisible designs are instrumental in the constructions for other types of designs. Most of the progress for the existence of -GDDs of type is on the case when is even, where the existence for small has played a key role. In order to determine the spectrum for -GDDs of type with being odd, we continue to investigate the small cases with in this paper. We show that, for each , the necessary conditions for the existence of a -GDD of type are also sufficient. As the applications of these GDDs, we obtain a few pairwise balanced designs with minimum block size 4. Meanwhile, we also improve the existence result for frame self-orthogonal Mendelsohn triple systems of type by reducing an infinite class of possible exceptions, namely and , to eight undetermined cases. [Copyright &y& Elsevier]

To equip DNA-based data storage with random-access capabilities, Yazdi et al. (2018) prepended DNA strands with specially chosen address sequences called primers and provided certain design criteria for these primers. We provide explicit constructions of error-correcting codes that are suitable as primer addresses and equip these constructions with efficient encoding algorithms. Specifically, our constructions take cyclic or linear codes as inputs and produce sets of primers with similar error-correcting capabilities. Using certain classes of BCH codes, we obtain infinite families of primer sets of length $n$ , minimum distance $d$ with $(d+1) \log _{4}\,\,n +O(1)$ redundant symbols. Our techniques involve reversible cyclic codes (1964), an encoding method of Tavares et al. (1971) and Knuth’s balancing technique (1986). In our investigation, we also construct efficient and explicit binary balanced error-correcting codes and codes for DNA computing. [ABSTRACT FROM AUTHOR]

Given a database, the private information retrieval (PIR) protocol allows a user to make queries to several servers and retrieve a certain item of the database via the feedbacks without revealing the identity of the specific item to any single server. Classic $k$ -server PIR protocols work on replicated databases, i.e., each of the $k$ servers stores a whole copy of the database. Recently, new PIR models were proposed with coding techniques arising from the distributed storage system. In these new models, each server only stores a fraction $1/s$ of the whole database, where $s>1$ is the given rational number. The PIR array codes are recently proposed by Fazeli, Vardy, and Yaakobi to characterize the new models. The central problem in designing a PIR array code with $m$ servers and the $k$ -PIR property (which indicates that these $m$ servers may emulate a classic $k$ -server PIR protocol) is to maximize $k/m$ , known as the virtual server rate. Our main contribution to this problem is twofold. First, for the case $1 < s\le 2$ , although the PIR array codes with optimal rate have been constructed recently by Blackburn and Etzion, the number of servers in their construction is rather large. We determine the minimum number of servers admitting the existence of a PIR array code with an optimal rate for a certain range of parameters. Second, for the case $s>2$ , a new upper bound on the rate of a PIR array code is presented. Besides, we also have some discussions on an asymptotically optimal construction by Blackburn and Etzion. [ABSTRACT FROM AUTHOR]

The construction of group divisible designs (GDDs) is a basic problem in design theory. While there have been some methods concerning the constructions of uniform GDDs, the construction of nonuniform GDDs remains a challenging problem. In this paper, we present a new approach to the construction of nonuniform GDDs with group type [ABSTRACT FROM AUTHOR]

Grooming uniform all-to-all traffic in optical (SONET) rings with grooming ratio C requires the determination of a decomposition of the complete graph into subgraphs each having at most C edges. The drop cost of such a grooming is the total number of vertices of nonzero degree in these subgraphs, and the grooming is optimal when the drop cost is minimum. The determination of optimal C-groomings has been considered for [ABSTRACT FROM AUTHOR]

In a bus with $n$ wires, each wire has two states, ‘0’ or ‘1’, representing one bit of information. Whenever the state transitions from ‘0’ to ‘1’, or ‘1’ to ‘0’, joule heating causes the temperature to rise, and high temperatures have adverse effects on on-chip bus performance. Recently, the class of low-power cooling (LPC) codes was proposed to control such state transitions during each transmission. As suggested in earlier work, LPC codes may be used to control simultaneously both the peak temperature and the average power consumption of on-chip buses. Specifically, an $(n,t,w)$ -LPC code is a coding scheme over $n$ wires that (i) avoids state transitions on the $t$ hottest wires (thus preventing the peak temperature from rising); and (ii) allows at most $w$ state transitions in each transmission (thus reducing average power consumption). In this paper, for any fixed value of $w$ , several constructions are presented for large LPC codes that can be encoded and decoded in time $O\left ({n \log ^{2} (n/w)}\right)$ along with the corresponding encoding/decoding schemes. In particular, we construct LPC codes of size $(n/w)^{w-1}$ , which are asymptotically optimal. We then modify these LPC codes to also correct errors in time $O(n^{3})$. For the case where $w$ is proportional to $n$ , we further present a different construction of large LPC codes, based on a mapping from cooling codes to LPC codes. Using this construction, we obtain two families of LPC codes whose encoding and decoding complexities are $O(n^{3})$. [ABSTRACT FROM AUTHOR]

Permutation codes and multipermutation codes are widely studied due to various applications in information theory. Designing codes correcting deletion errors has been the main subject of works in the literature and to the best of our knowledge, there exist only optimal codes capable of correcting a single deletion in a permutation. In this paper, we construct several classes of permutation and multipermutation codes that are capable of correcting a burst deletion of length $s \geq 2$ , for both stable and unstable models. Efficient error decoders are provided to show the correctness of our constructions. [ABSTRACT FROM AUTHOR]

A G-design of order n is a decomposition of the complete graph on n vertices into edge-disjoint subgraphs isomorphic to G. Grooming uniform all-to-all traffic in optical ring networks with grooming ratio C requires the determination of graph decompositions of the complete graph on n vertices into subgraphs each having at most C edges. The drop cost of such a grooming is the total number of vertices of nonzero degree in these subgraphs, and the grooming is optimal when the drop cost is minimum. The existence spectrum problem of G-designs for five-vertex graphs is a long standing problem posed by Bermond, Huang, Rosa and Sotteau in 1980, which is closely related to traffic groomings in optical networks. Although considerable progress has been made over the past 30 years, the existence problems for such G-designs and their related traffic groomings in optical networks are far from complete. In this paper, we first give a complete solution to this spectrum problem for five-vertex graphs by eliminating all the undetermined possible exceptions. Then, we determine almost completely the minimum drop cost of 8-groomings for all orders n by reducing the 37 possible exceptions to 8. Finally, we show the minimum possible drop cost of 9-groomings for all orders n is realizable with 14 exceptions and 12 possible exceptions. [ABSTRACT FROM AUTHOR]