In this work, we consider the problem of variable packet-error coding, which emerges in network communication scenarios where a source transmits information to a destination through multiple disjoint paths. The objective is to design codes with dynamic error-correcting capabilities that adapt to a varying number of errors. Specifically, we first provide a bound on the rate-distortion trade-off for general variable packet-error coding schemes. Then, we present a construction that uses higherorder MDS codes and provides a variable packet-error coding scheme that achieves a better rate-distortion trade-off compared to known results for general parameter regimes.

In nonadaptive group testing, the main research objective is to design an efficient algorithm to identify a set of up to$t$positive elements among$n$samples with as few tests as possible. Disjunct matrices and separable matrices are two classical combinatorial structures while one provides a more efficient decoding algorithm and the other needs fewer tests, i.e., larger rate. Recently, a notion of strongly separable matrix has been introduced, which has the same identifying ability as a disjunct matrix, but has larger rate. In this paper, we use a modified probabilistic method to improve the lower bounds for the rate of strongly separable matrices. Using this method, we also improve the lower bounds for some well-known combinatorial structures, including locally thin set families and cancellative set families.

Grassmannian$\mathcal {G}_{q}(n,k)$is the set of all$k$-dimensional subspaces of the vector space$\mathbb {F}_{q}^{n}$. Recently, Etzion and Zhang introduced a new notion called covering Grassmannian code which can be used in network coding solutions for generalized combination networks. An$\alpha $-$(n,k,\delta)_{q}^{c}$covering Grassmannian code$\mathcal {C}$is a subset of$\mathcal {G}_{q}(n,k)$such that every set of$\alpha $codewords of$\mathcal {C}$spans a subspace of dimension at least$\delta +k$in$\mathbb {F}_{q}^{n}$. In this paper, we derive new upper and lower bounds on the size of covering Grassmannian codes. These bounds improve and extend the parameter range of known bounds.

As a crucial technique for integrated circuits (IC) test response compaction, X-compact employs a special kind of codes called X-codes for reliable compressions of the test response in the presence of unknown logic values (Xs). From a combinatorial view point, Fujiwara and Colbourn introduced an equivalent definition of X-codes and studied X-codes of small weights that have good detectability and X-tolerance. An (m,n,d,x) X-code is an m× n binary matrix with column vectors as its codewords. The parametersd,xcorrespond to the test quality of the code. In this paper, bounds and constructions for constant weighted X-codes are investigated. First, we obtain a general result on the maximum number of codewords n for an (m,n,d,x) X-code of weight w, and we further improve this lower bound for the case with x=2 and w=3 through the probabilistic method. Then, using tools from additive combinatorics and finite fields, we present some explicit constructions for constant weighted X-codes with d=3,7 and x=2, which are optimal for the case when d=3, w=4 and nearly optimal for the case when d=3, w=3. We also consider a special class of X-codes introduced by Fujiwara and Colbourn and improve the best known lower bound on the maximum number of codewords for this kind of X-codes.

As an important coding scheme in modern distributed storage systems, locally repairable codes (LRCs) have attracted a lot of attentions from perspectives of both practical applications and theoretical research. As a major topic in the research of LRCs, bounds and constructions of the corresponding optimal codes are of particular concerns. In this work, codes with (r,δ)-locality which have optimal minimal distance w.r.t. the bound given by Prakash et al. are considered. Through parity-check matrix approach, constructions of both optimal (r,δ)-LRCs with all symbol locality ( (r,δ)a-LRCs) and optimal (r,δ)-LRCs with information locality ( (r,δ)i-LRCs) are provided. As a generalization of a work of Xing and Yuan, these constructions are built on a connection between sparse hypergraphs and optimal (r,δ)-LRCs. With the help of constructions of large sparse hypergraphs, the lengths of codes obtained from our construction can be super-linear in the alphabet size. This improves upon previous constructions when the minimal distance of the code is at least 3δ+1. As two applications, optimal H-LRCs with super-linear lengths and GSD codes with unbounded lengths are also constructed.

Constant-composition codes are a subclass of constant weight codes. When$d\geq 4$, the problem of determining the maximum size of a constant-composition code for general parameters is much less understood. In this correspondence, we give a new construction for constant-composition codes with techniques in additive number theory. Moreover, we provide a new connection between constant-composition codes and linear block codes with certain properties. It turns out that when$d\geq 4$our new lower bounds improve the one by Ding (2008) substantially.

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.

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 12, 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.

Optical orthogonal signature pattern codes (OOSPCs) have played an important role in a novel type of optical code-division multiple-access network for 2-D image transmission. In this paper, we give four direct constructions for OOSPCs based on polynomials and rational functions over finite fields. We also use r -simple matrices to present a recursive construction for OOSPCs. These constructions yield new families of asymptotically optimal OOSPCs.

Frameproof codes are used to fingerprint digital data. They can prevent copyrighted materials from unauthorized use. In this paper, we study upper and lower bounds for$w$-frameproof codes of length$N$over an alphabet of size$q$. The upper bound is based on a combinatorial approach and the lower bound is based on a probabilistic construction. Both bounds can improve one of the previous results when$q$is small compared with$w$, say$cq\leq w$for some constant$c\leq q$. Furthermore, we pay special attention to binary frameproof codes. We show a binary$w$-frameproof code of length$N$cannot have more than$N$codewords if$N<\binom {w+1}{2}$.

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 Cheeet 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$2\sum _{i=1}^{m}w_{i}-2$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.