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In this paper, we consider encoding strategies for the Z-channel with noiseless feedback. We analyze the combinatorial setting where the maximum number of errors inflicted by an adversary is proportional to the number of transmissions, which goes to infinity. Without feedback, it is known that the rate of optimal asymmetric-error-correcting codes for the error fraction $\tau \ge 1/4$ vanishes as the blocklength grows. In this paper, we give an efficient feedback encoding scheme with $n$ transmissions that achieves a positive rate for any fraction of errors $\tau < 1$ and $n\to \infty $. Additionally, we state an upper bound on the rate of asymptotically long feedback asymmetric error-correcting codes. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the capacity of a new class of discrete memoryless multicast networks having a tree topology. For achievability, a novel coding scheme is constructed where some relays employ a combination of decode-and-forward and compress-and-forward and the other relays perform a random binning such that codebook constructions and relay operations are independent for each node and do not depend on the network topology. For converse, a new technique of iteratively manipulating inequalities exploiting the tree topology is used. This class of multicast tree networks includes the class of diamond networks studied by Kang and Ulukus as a special case. [ABSTRACT FROM PUBLISHER]

In this paper, we extend the localized error correction code introduced by L. A. Bassalygo and coworkers from Hamming space to projective space. For constant dimensional localized error correction codes in projective space, we have a lower bound and an upper bound of the capacity, which are asymptotically tight when z< x\leq n-z\over 2, where x, z, and n are dimensions of codewords, error configurations, and the ground space, respectively. We determine the capacity of nonconstant dimensional localized error correction codes when z < n\over 3. [ABSTRACT FROM PUBLISHER]

In this paper, the sequence families of which maximum correlation is determined by the Weil bound of exponential sums are revisited. Using the same approach, two new constructions with large family sizes and low maximum correlation are given. The first construction is an analog of one recent result derived from the interleaved structure of Sidel'nikov sequences. For a prime p and an integer M\vert (p-1), the new M-ary sequence families of period p are obtained from irreducible quadratic polynomials and known power residue-based sequence families. The new sequence families increase family sizes of the known power residue-based sequence families, but keep the maximum correlation unchanged. In the second construction, the sequences derived from the Weil representation are generalized, where each new sequence is the elementwise product of a modulated Sidel'nikov sequence and a modulated trace sequence. For positive integers d

We consider a variable-length source coding problem subject to local decodability constraints. In particular, we investigate the blocklength scaling behavior attainable by encodings of $r$ -sparse binary sequences, under the constraint that any source bit can be correctly decoded upon probing at most $d$ codeword bits. We consider both adaptive and non-adaptive access models, and derive upper and lower bounds that often coincide up to constant factors. Such a characterization for the fixed-blocklength analog of our problem, known as the bit probe complexity of static membership, remains unknown despite considerable attention from researchers over the last few decades. We also show that locally decodable schemes for sparse sequences are able to decode 0s (frequent source symbols) of the source with far fewer probes on average than they can decode 1s (infrequent source symbols), thus rigorizing the notion that infrequent symbols require high probe complexity, even on average. Connections to the fixed-blocklength model and to communication complexity are also briefly discussed. [ABSTRACT FROM AUTHOR]

The problem of universal source coding for binary trees is considered. Zhang, Yang, and Kieffer derived upper bounds on the average-case redundancy of codes based on directed acyclic graph (DAG) compression for binary tree sources with certain properties. In this paper, a natural class of binary tree sources is presented such that the demanded properties are fulfilled. Moreover, for both subclasses considered in the paper of Zhang, Yang, and Kieffer, their result is improved by deriving bounds on the maximal pointwise redundancy (or worst-case redundancy) instead of the average-case redundancy. Finally, using context-free tree grammars instead of DAGs, upper bounds on the maximal pointwise redundancy for certain binary tree sources are derived. This yields universal codes for new classes of binary tree sources. [ABSTRACT FROM AUTHOR]

List decoding of insertions and deletions in the Levenshtein metric is considered. The Levenshtein distance between two sequences is the minimum number of insertions and deletions needed to turn one of the sequences into the other. In this paper, a Johnson-like upper bound on the maximum list size when list decoding in the Levenshtein metric is derived. This bound depends only on the length and minimum Levenshtein distance of the code, the length of the received word, and the alphabet size. It shows that polynomial-time list decoding beyond half the Levenshtein distance is possible for many parameters. Further, we also prove a lower bound on list decoding of deletions with the well-known binary Varshamov–Tenengolts codes, which shows that the maximum list size grows exponentially with the number of deletions. Finally, an efficient list decoding algorithm for two insertions/deletions with VT codes is given. This decoder can be modified to a polynomial-time list decoder of any constant number of insertions/deletions. [ABSTRACT FROM AUTHOR]

We investigate the state-dependent Gaussian wiretap channel with noncausal channel state information at the transmitter (GWTC-N-CSIT), and explore whether three strategies (i.e., taking action on the state, legitimate receiver’s channel output feedback, and combining the former two strategies together) help to enhance the secrecy capacity of the GWTC-N-CSIT. To be specific, we first determine the secrecy capacity of the GWTC-N-CSIT with noiseless feedback. Next, we derive lower and upper bounds on the secrecy capacity of the GWTC-N-CSIT with action-dependent state. Finally, we derive lower and upper bounds on the secrecy capacity of the GWTC-N-CSIT with both action-dependent state and noiseless feedback, and show that these bounds meet for a special case. Numerical results of this paper indicate that all three strategies enhance the secrecy capacity of the GWTC-N-CSIT. The study of this paper offers new options for enhancing the secrecy rates of the state-dependent wiretap channel models. [ABSTRACT FROM AUTHOR]

There has been much recent interest in private information retrieval (PIR) in models where a database is stored across several servers using coding techniques from distributed storage, rather than being a simply replicated. In particular, a recent breakthrough result of Fazelli, Vardy, and Yaakobi introduces the notion of a PIR code and a PIR array code, and uses this notion to produce efficient PIR protocols. In this paper, we are interested in designing PIR array codes. We consider the case when we have $m$ servers, with each server storing a fraction $(1/s)$ of the bits of the database; here $s$ is a fixed rational number with $s > 1$. A PIR array code with the $k$ -PIR property enables a $k$ -server PIR protocol (with $k\leq m$) to be emulated on $m$ servers, with the overall storage requirements of the protocol being reduced. The communication complexity of a PIR protocol reduces as $k$ grows, so the virtual server rate, defined to be $k/m$ , is an important parameter. We study the maximum virtual server rate of a PIR array code with the $k$ -PIR property. We present upper bounds on the achievable virtual server rate, some constructions, and ideas how to obtain the PIR array codes with the highest possible virtual server rate. In particular, we present constructions that asymptotically meet our upper bounds and the exact largest virtual server rate is obtained when $1 < s \leq 2$. A $k$ -PIR code (and similarly a $k$ -PIR array code) is also a locally repairable code with symbol availability $k-1$. Such a code ensures $k$ parallel reads for each information symbol. So the virtual server rate is very closely related to the symbol availability of the code when used as a locally repairable code. The results of this paper are discussed also in this context where subspace codes also have an important role. [ABSTRACT FROM AUTHOR]

A common method for reading information stored in DNA molecules is shotgun sequencing. This method outputs a histogram of the frequencies of all the molecules’ substrings of a given length $\ell $. To protect against noisy readings, the rank-modulation scheme encodes the information in the relative ranking of the substring frequencies, instead of their absolute values. However, the best rank-modulation codes for shotgun sequencing have low rates which are asymptotically vanishing. In this paper we propose new constructions of rank-modulation codes for shotgun sequencing. The first code construction is systematic, allowing the user to arbitrarily set the frequencies of a large subset of the substrings, which the encoder then completes to a permutation that may be realized by a DNA molecule. The construction is then improved by allowing the user to set the frequencies of additional substrings, at the cost of imposing constraints on the frequencies. The resulting codes have higher, non-vanishing rates, compared with previously known codes. As an example, for histograms of substrings of length $\ell =2$ , and an alphabet of size 4 (as in DNA molecules), we are able to construct a code with rate $\approx 0.909$ , whereas previously, the best construction resulted in a code with rate $\approx 0.654$. Additionally, the encoded information in our construction may be written to shorter DNA molecules than possible before. We also prove that the systematic codes constructed in this paper are the largest possible among all systematic codes. [ABSTRACT FROM AUTHOR]

In a paper by Guruswami et al., it was shown that the length $n$ of a $q$ -ary linear locally recoverable code with distance $d \geqslant 5$ is upper bounded by $O(dq^{3})$. Thus, it is a challenging problem to construct $q$ -ary locally recoverable codes with distance $d \geqslant 5$ and length approaching the upper bound. The same paper also gave an algorithmic construction of $q$ -ary locally recoverable codes with locality $r$ and length $n=\Omega _{r}(q^{2})$ for $d=5$ and 6, where $\Omega _{r}$ means that the implicit constant depends on locality $r$. In this paper, we present an explicit construction of $q$ -ary locally recoverable codes of distance $d= 5$ and 6 via binary constant weight codes. It turns out that 1) our construction is simpler and more explicit and 2) the length of our codes is greater than previously known. [ABSTRACT FROM AUTHOR]

This paper studies the capacity of an $n$ -dimensional vector Gaussian noise channel subject to the constraint that an input must lie in the ball of radius $R$ centered at the origin. It is known that in this setting, the optimizing input distribution is supported on a finite number of concentric spheres. However, the number, the positions, and the probabilities of the spheres are generally unknown. This paper characterizes necessary and sufficient conditions on the constraint $R$ , such that the input distribution supported on a single sphere is optimal. The maximum $\bar {R}_{n}$ , such that using only a single sphere is optimal, is shown to be a solution of an integral equation. Moreover, it is shown that $\bar {R}_{n}$ scales as $\sqrt {n}$ and the exact limit of $\frac {\bar {R}_{n}}{\sqrt {n}}$ is found. [ABSTRACT FROM AUTHOR]

The zero-error capacity of a channel is defined as the maximum rate at which it is possible to transmit information with zero probability of error. In this paper, we settle all previously unsolved cases for the zero-error capacity of binary channels with one memory. [ABSTRACT FROM AUTHOR]

We revisit and take a closer look at a (not so well known) result of a 2017 paper, showing that the differential uniformity of any vectorial function is bounded from below by an expression depending on the size of its image set. We make explicit the resulting tight lower bound on the image set size of differentially $\delta $ -uniform functions (which is the only currently known non-trivial lower bound on the image set size of such functions). We also significantly improve an upper bound on the nonlinearity of vectorial functions obtained in the same reference and involving their image set size. We study when the resulting bound is sharper than the covering radius bound. We obtain as a by-product a lower bound on the Hamming distance between differentially $\delta $ -uniform functions and affine functions, which we improve significantly with a second bound. This leads us to study what can be the maximum Hamming distance between vectorial functions and affine functions. We provide an upper bound which is slightly sharper than a bound by Liu, Mesnager and Chen when $m < n$ , and a second upper bound, which is much stronger in the case (happening in practice) where $m$ is near $n$ ; we study the tightness of this latter bound; this leads to an interesting question on APN functions, which we address (negatively). We finally derive an upper bound on the nonlinearity of vectorial functions by means of their Hamming distance to affine functions and make more precise the bound on the differential uniformity which was the starting point of the paper. [ABSTRACT FROM AUTHOR]

Volume estimates of metric balls in manifolds find diverse applications in information and coding theory. In this paper, new results for the volume of a metric ball in unitary group are derived via tools from random matrix theory. The first result is an integral representation of the exact volume, which involves a Toeplitz determinant of Bessel functions. A simple but accurate limiting volume formula is then obtained by invoking Szegő’s strong limit theorem for large Toeplitz matrices. The derived asymptotic volume formula enables analytical evaluation of some coding-theoretic bounds of unitary codes. In particular, the Gilbert–Varshamov lower bound and the Hamming upper bound on the cardinality as well as the resulting bounds on code rate and minimum distance are derived. Moreover, bounds on the scaling law of code rate are found. Finally, a closed-form bound on the diversity sum relevant to unitary space-time codes is obtained, which was only computed numerically in the literature. [ABSTRACT FROM AUTHOR]

The state-dependent point-to-point Gaussian channel with a helper is first studied, in which a transmitter communicates with a receiver via a state-corrupted channel. The state is not known to the transmitter nor to the receiver, but known to a helper noncausally, which then wishes to assist the receiver to cancel the state. Differently from the previous work that characterized the capacity only in the infinite state power regime, this paper explores the general case with arbitrary state power. A lower bound on the capacity is derived based on an achievable scheme that integrates direct state subtraction and single-bin dirty paper coding. By analyzing this lower bound and further comparing it with the existing upper bounds, the capacity of the channel is characterized for a wide range of channel parameters. Such an idea of characterizing the capacity is further extended to study the two-user state-dependent multiple access channel with a helper. By comparing the derived inner and outer bounds, the channel parameters are partitioned into appropriate cases, and for each case, either segments on the capacity region boundary or the full capacity region are characterized. [ABSTRACT FROM PUBLISHER]

This paper focuses on the problem of separately modulating and jointly estimating two independent continuous-valued parameters sent over a Gaussian multiple-access channel (MAC) under the mean square error (MSE) criterion without bandwidth constraints. To this end, we first improve an existing lower bound on the MSE that is obtained using the parameter modulation-estimation techniques for the single-user additive white Gaussian noise (AWGN) channel. As for the main contribution of this paper, this improved modulation-estimation analysis is generalized to the model of the two-user Gaussian MAC. We present outer bounds to the achievable region in the plane of the MSE’s of the two user parameters, which provides a trade-off between the MSE’s, where we used zero-rate lower bounds on the error probability of Gaussian channels by Shannon and Polyanskiy et al. Numerical results showed that, the multi-user adaptation of the zero-rate lower bound by Polyanskiy et al. provides a tighter overall lower bound on the MSE pairs than the classical Shannon bound. In addition, we introduced upper bounds on the MSE exponents, namely, the exponential decay rates of these MSE’s in the asymptotic regime of long blocks that could make use of any bound on the error exponent of a single-user AWGN channel. The obtained results are numerically evaluated for three different bounds on the reliability function of the Gaussian channel. It is shown that the adaptation of the reliability function by Ashikhmin et al. to the MAC provides a significantly tighter characterization than Shannon’s sphere-packing bound and the divergence bound. [ABSTRACT FROM AUTHOR]

Using some recent results on asymptotically equivalent sequences of matrices, we present in this paper, a new derivation of the capacity formula given by Brandenburg and Wyner for a discrete-time Gaussian multiple-input-multiple-output channel with memory. In this paper, we tackle not only the case considered by them, where the number of channel inputs and the number of channel outputs are the same, but also when both numbers are different. [ABSTRACT FROM AUTHOR]

This paper studies low-latency streaming codes for the multi-hop network. The source transmits a sequence of messages to a destination through a chain of relays, and requires the destination to reconstruct each message by its deadline. We assume that each communication link is subjected to a certain maximum number of packet erasures. The case of a single relay (a three-node network) was considered in Fong et al. (2020). A coding scheme known as symbol-wise decode and forward was proposed. In the present work, we propose an alternative scheme that is different from Fong et al. (2020) and still achieves the same rate as in Fong et al. (2020) for the one hop case as the field-size goes to infinity. Furthermore, our proposed scheme naturally generalizes to the case of multiple-relay nodes yielding new achievable rates for this setting. The main difference with Fong et al. (2020) is that our proposed scheme exploits the ability of the relay nodes to adapt the transmission based on the erasures on the previous link. Hence, we refer to our scheme as “state-dependent” and contrast it with the scheme in Fong et al. (2020) that is state-independent. Our scheme requires the relay nodes to append a header to the transmitted packets, and we show that the size of the header does not depend on the field-size of the code. We also derive an upper bound on the maximal streaming rate achievable over a network with an arbitrary number of relays. We show that this upper bound matches our achievable rate in the special case when the maximal number of erasures on the first link is greater than or equal to the maximal number of erasures on each of the following links, and the field size goes to infinity. [ABSTRACT FROM AUTHOR]

In this paper, we consider variable-length coding over the memoryless binary asymmetric channel (BAC) with full noiseless feedback, including the binary symmetric channel (BSC) as a special case. In 2012, Naghshvar et al. introduced a coding scheme, which we refer to as the small-enough-difference (SED) coding scheme. For symmetric binary-input channels, the deterministic variable-length feedback (VLF) code constructed with the SED coding scheme asymptotically achieves both capacity and Burnashev’s optimal error exponent. Building on the work of Naghshvar et al., this paper extends the SED coding scheme to the BAC and develops a non-asymptotic VLF achievability bound that is shown to achieve both capacity and the optimal error exponent. For the specific case of the BSC, we develop an additional non-asymptotic VLF achievability bound using a two-phase analysis that leverages both a submartingale synthesis and a Markov chain time of first passage analysis. Numerical evaluations show that both new VLF achievability bounds outperform Polyanskiy’s achievability bound for variable-length stop-feedback codes. [ABSTRACT FROM AUTHOR]

This paper presents a unifying framework to construct low-density parity-check (LDPC) codes with associated Tanner graphs of desired girth. Towards this goal, we highlight the role that a certain square matrix that appears in the product of the parity-check matrix with its transpose has in the construction of codes with graphs of desired girth and further explore it in order to generate the set of necessary and sufficient conditions for a Tanner graph to have a given girth between 6 and 12. For each such girth, we present algorithms to construct codes of the desired girth and we show how to use them to compute the minimum necessary value of the lifting factor. For girth larger than 12, we show how to use multi-step graph lifting methods to deterministically modify codes in order to increase their girth. We also give a new perspective on LDPC protograph-based parity-check matrices by viewing them as rows of a parity-check matrix equal to the sum of certain permutation matrices and obtain an important connection between all protographs and those with variable nodes of degree 2. We also show that the results and methodology that we develop for the all-one protograph can be used and adapted to analyze the girth of the Tanner graph of any parity-check matrix and demonstrate how this can be done using a well-known irregular, multi-edge protograph specified by the NASA Consultative Committee for Space Data Systems (CCSDS). Throughout the paper, we exemplify our theoretical results with constructions of LDPC codes with Tanner graphs of any girth between 6 and 14 and give sufficient conditions for a multi-step lifted parity-check matrix to have girth between 14 and 22. [ABSTRACT FROM AUTHOR]

The group testing problem consists of determining a small set of defective items from a larger set of items based on tests on groups of items, and is relevant in applications such as medical testing, communication protocols, pattern matching, and many more. While rigorous group testing algorithms have long been known with runtime at least linear in the number of items, a recent line of works has sought to reduce the runtime to ${\mathrm{ poly}}({k} \log {n})$ , where n is the number of items and k is the number of defectives. In this paper, we present such an algorithm for non-adaptive group testing termed bit mixing coding (BMC), which builds on techniques that encode item indices in the test matrix, while incorporating novel ideas based on erasure-correction coding. We show that BMC achieves asymptotically vanishing error probability with ${O}({k} \log {n})$ tests and ${O}({k}^{2} \cdot \log {k} \cdot \log {n})$ runtime, in the limit as ${n} \to \infty $ (with k having an arbitrary dependence on n). This closes an open problem of simultaneously achieving ${\mathrm{ poly}}({k} \log {n})$ decoding time using ${O}({k} \log {n})$ tests without any assumptions on k. In addition, we show that the same scaling laws can be attained in a commonly-considered noisy setting, in which each test outcome is flipped with constant probability. [ABSTRACT FROM AUTHOR]

Sidelnikov sequences, a kind of cyclotomic sequences with many desired properties such as low correlation and variable alphabet sizes, can be employed to construct a polyphase sequence family that has many applications in high-speed data communications. Recently, cyclotomic numbers have been used to investigate the linear complexity of Sidelnikov sequences, mainly about binary ones, although the limitation on the orders of the available cyclotomic numbers makes it difficult. This paper continues to study the linear complexity over $\mathbb {F}_{M}$ of $M$ -ary Sidelnikov sequence of period $q-1$ using Hasse derivative, which implies $q=p^{m}$ , $m\geq 1$ and $M|(q-1)$. The $t$ th Hasse derivative formulas are presented in terms of cyclotomic numbers, and some upper bounds on the linear complexity for $M=2$ and 3 are obtained only with some additional restrictions on $q$. Furthermore, concrete illustrations for several families of these sequences, such as $q\equiv 1\pmod {2}$ and $q\equiv 1\pmod {3}$ , show these upper bounds are tight and reachable; especially for $q=2\times 3^{\lambda }+1 (1\leq \lambda \leq 20)$ , the exact linear complexities over $\mathbb {F}_{3}$ of the ternary Sidelnikov sequences are determined; and it turns out that all the linear complexities of the sequences considered are very close to their periods. [ABSTRACT FROM AUTHOR]

For fixed integers $n$ and $b\geq k$ , let $A(b,k,n)$ the largest size of a subset of $\{1,2,\ldots,b\}^{n}$ such that, for any $k$ distinct elements in the set, there is a coordinate where they all differ. Bounding $A(b,k,n)$ is a problem of relevant interest in information theory and computer science, relating to the zero-error capacity with list decoding and to the study of $(b, k)$ -hash families of functions. It is known that, for fixed $b$ and $k$ , $A(b,k,n)$ grows exponentially in $n$. In this paper, we determine new exponential upper bounds for different values of $b$ and $k$. A first bound on $A(b,k,n)$ for general $b$ and $k$ was derived by Fredman and Komlós in the ’80s and improved for certain $b\neq k$ by Körner and Marton and by Arikan. Only very recently better bounds were derived for general $b$ and $k$ by Guruswami and Riazanov, while stronger results for small values of $b=k$ were obtained by Arikan, by Dalai, Guruswami and Radhakrishnan, and by Costa and Dalai. In this paper, we strengthen the bounds for some specific values of $b$ and $k$. Our contribution is a new computational method for obtaining upper bounds on the values of a quadratic form defined over discrete probability distributions in arbitrary dimensions, which emerged as a central ingredient in recent works. The proposed method reduces an infinite-dimensional problem to a finite one, which we manage to further simplify by means of a series of optimality conditions. [ABSTRACT FROM AUTHOR]

The family of Goppa codes is one of the most interesting subclasses of linear codes. As the McEliece cryptosystem often chooses a random Goppa code as its key, knowledge of the number of inequivalent Goppa codes for fixed parameters may facilitate in the evaluation of the security of such a cryptosystem. In this paper we present a new approach to give an upper bound on the number of inequivalent extended irreducible binary Goppa codes. To be more specific, let $n>3$ be an odd prime number and $q=2^{n}$ ; let $r\geq 3$ be a positive integer satisfying $\gcd (r,n)=1$ and $\gcd \big (r,q(q^{2}-1)\big)=1$. We obtain an upper bound for the number of inequivalent extended irreducible binary Goppa codes of length $q+1$ and degree $r$. [ABSTRACT FROM AUTHOR]

The strong secrecy transmission problem of the arbitrarily varying wiretap channel (AVWC) with input and state constraints is investigated in this paper. First, a stochastic-encoder code lower bound of the strong secrecy capacity is established by applying the type argument and Csiszár’s almost independent coloring lemma. Then, a superposition stochastic-encoder code lower bound of the secrecy capacity is provided. The superposition stochastic-encoder code lower bound can be larger than the ordinary stochastic-encoder code lower bound. Random code lower and upper bounds of the secrecy capacity of the AVWC with constraints are further provided. Based on these results, we further consider a special case of the model, namely severely less noisy AVWC, and give the stochastic-encoder code and random code capacities. It is proved that the stochastic-encoder code capacity of the AVWC with constraints is either equal to or strictly smaller than the corresponding random code capacity, which is consistent with the property of the ordinary AVC. Finally, some numerical examples are presented to better illustrate our capacity results. Compared to the soft covering lemma that requires the codewords to be generated i.i.d., our method has more relaxed requirements regarding codebooks. It is proved that the good codebooks for secure transmission can be generated by choosing codewords randomly from a given type set, which is critical when considering the AVWC with constraints. [ABSTRACT FROM AUTHOR]

This paper describes new, simple, recursive methods of construction for orientable sequences, i.e. periodic binary sequences in which any $n$ -tuple occurs at most once in a period in either direction. As has been previously described, such sequences have potential applications in automatic position-location systems, where the sequence is encoded onto a surface and a reader needs only examine $n$ consecutive encoded bits to determine its location and orientation on the surface. The only previously described method of construction (due to Dai et al.) is somewhat complex, whereas the new techniques are simple to both describe and implement. The methods of construction cover both the standard ‘infinite periodic’ case, and also the aperiodic, finite sequence, case. Both the new methods build on the Lempel homomorphism, first introduced as a means of recursively generating de Bruijn sequences. [ABSTRACT FROM AUTHOR]

A block code is called a locally recoverable code (LRC for short) with $(r,\delta)$ -locality, if subject to any $\delta -1$ erasure failures, every symbol in the encoding can still be recovered by accessing at most $r$ other symbols. Recently, it was discovered by several authors that a $q$ -ary optimal $(r,\delta)$ -LRC, i.e., an LRC achieving the generalized 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 cases). In this paper, we further investigate constructions of optimal $(r,\delta)$ -LRCs along the line of using parity-check matrices. Inspired by classical Reed-Solomon codes and the work in Jin (2019), we equip parity-check matrices with the Vandermonde structure. It turns out that a parity-check matrix with the Vandermonde structure that produces an optimal LRC must obey certain disjoint property for subsets of $\mathbb {F}_{q}$. We manage to show the existence of these disjoint subsets. Thus, this yields an optimal $(r,\delta)$ -LRC with code length $\Omega \left({q^{\frac {\delta }{2}\left({1+\frac {1}{\lfloor \frac {d-1}{\delta }\rfloor -1}}\right)}}\right)\vphantom {\bigg)_{j}}$ , where $d$ is the minimum distance. In particular, to our surprise, for $\delta =2$ 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 Guruswami et al. (2018) for $d\geq 7$. In addition, for $d=6$ , we are able to remove the constraint required in Jin (2019) that $q$ is even. [ABSTRACT FROM AUTHOR]

We investigate the list decodability of symbol-pair codes in this paper. First, we show that the 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 in the pair metric. 1 A symbol-pair code is referred to a code in the pair metric. [ABSTRACT FROM AUTHOR]

This paper derives upper and lower bounds on the secrecy capacity-memory tradeoff of a wiretap erasure broadcast channel (BC) with ${\mathsf{K}}_{w} $ weak receivers and ${\mathsf {K}}_{s} $ strong receivers, where weak receivers and strong receivers have the same erasure probabilities and cache sizes, respectively. The lower bounds are achieved by the schemes that meticulously combine joint cache-channel coding with wiretap coding and key-aided one-time pads. The presented upper bound holds more generally for arbitrary degraded BCs and arbitrary cache sizes. When only weak receivers have cache memories, upper and lower bounds coincide for small and large cache memories, thus providing the exact secrecy capacity-memory tradeoff for this setup. The derived bounds further allow us to conclude that the secrecy capacity is positive even when the eavesdropper is stronger than all the legitimate receivers with cache memories. Moreover, they show that the secrecy capacity-memory tradeoff can be significantly smaller than its non-secure counterpart, but it grows much faster when cache memories are small. This paper also presents a lower bound on the global secrecy capacity-memory tradeoff where one is allowed to optimize the cache assignment subject to a total cache budget. It is close to the best known lower bound without secrecy constraint. For small total cache budget, the global secrecy capacity-memory tradeoff is achieved by assigning all the available cache memory uniformly over all the receivers if the eavesdropper is stronger than all the legitimate receivers, and it is achieved by assigning the cache memory uniformly only over the weak receivers if the eavesdropper is weaker than the strong receivers. [ABSTRACT FROM AUTHOR]

In the $q$ -ary online (or “causal”) channel coding model, a sender wishes to communicate a message to a receiver by transmitting a codeword $\mathbf {x} =(x_{1},\ldots,x_{n}) \in \{0,1,\ldots,q-1\}^{n}$ symbol-by-symbol via a channel limited to at most $pn$ errors and $p^{*} n$ erasures. The channel is “online” in the sense that at the $i$ th step of communication the channel decides whether to corrupt the $i$ th symbol or not based on its view so far, i.e., its decision depends only on the transmitted symbols $(x_{1},\ldots,x_{i})$. This is in contrast to the classical adversarial channel in which the corruption is chosen by a channel that has full knowledge of the sent codeword $\mathbf {x}$. In this paper, we study the capacity of $q$ -ary online channels for a combined corruption model, in which the channel may impose at most $pn$ errors and at most $p^{*} n$ erasures on the transmitted codeword. The online channel (in both the error and erasure case) has seen a number of recent studies, which present both upper and lower bounds on its capacity. In this paper, we give a full characterization of the capacity as a function of $q,p$ , and $p^{*}$. [ABSTRACT FROM AUTHOR]

The problem of network function computation over a directed acyclic network is investigated in this paper. In such a network, a sink node desires to compute with zero error a target function, of which the inputs are generated at multiple source nodes. The edges in the network are assumed to be error-free and have limited capacity. The nodes in the network are assumed to have unbounded computing capability and be able to perform network coding. The computing rate of a network code that can compute the target function over the network is the average number of times that the target function is computed with zero error for one use of the network. In this paper, we obtain an improved upper bound on the computing capacity, which is applicable to arbitrary target functions and arbitrary network topologies. This improved upper bound not only is an enhancement of the previous upper bounds but also is the first tight upper bound on the computing capacity for computing an arithmetic sum over a certain non-tree network, which has been widely studied in the literature. We also introduce a multi-dimensional array approach that facilitates evaluation of the improved upper bound. Furthermore, we apply this bound to the problem of computing a vector-linear function over a network. With this bound, we are not only able to enhance a previous result on computing a vector-linear function over a network but also simplify the proof significantly. Finally, we prove that for computing the binary maximum function over the reverse butterfly network, our improved upper bound is not achievable. This result establishes that in general our improved upper bound is non-achievable, but whether it is asymptotically achievable or not remains open. [ABSTRACT FROM AUTHOR]

Consider a channel with a given input alphabet size and input distribution. Our aim is to degrade or upgrade it to a channel with at most L output letters. Channel Q is degraded with respect to a channel W if Q can be obtained from W by processing the output of W. Upgrading is the inverse relation. This paper contains four main results. The first result, from which the paper title is derived, deals with the so called “greedy–merge” algorithm. We derive an upper bound on the reduction in mutual information between input and output, as a function of L. This upper bound is within a constant factor of an algorithm-independent lower bound. Thus, we establish that greedy–merge is optimal in the power-law sense (i.e. the power of L). The other main results deal with upgrading. The second result shows that a certain sequence of channels, which was previously shown to be “hard” for degrading, displays the same hardness in the context of upgrading. That is, suppose we are given such a channel and a corresponding input distribution. If we upgrade (degrade) to a new channel with L output letters, we incur an increase (decrease) in mutual information between input and output. We show that a previously derived bound on the decrease in mutual information for the degrading case is also a lower bound on the increase for the upgrading case. The third result is an efficient algorithm for optimal upgrading, in the binary-input case. That is, we are given a channel and an input distribution. We must find an upgraded channel with L output letters, for which the increase in mutual information is minimal. We give a simple characterization of such a channel, which implies an efficient algorithm. The fourth result is an analog of the first result for the upgrading case when the input is binary. That is, we first present a sub-optimal algorithm for the setting considered in the third result. The main advantage of the sub-optimal algorithm is that it is amenable to analysis. We carry out the analysis and show that the increase incurred in mutual information is within a constant factor of the lower bound derived in the second result. [ABSTRACT FROM AUTHOR]

Tietäväinen’s upper and lower bounds assert that for block-length- $n$ linear codes with dual distance $d$ , the covering radius $R$ is at most $({n}/{2})-(({1}/{2})-o(1))\sqrt {dn}$ and typically at least $({n}/{2})-\Theta (({dn\log {({n}/{d})}})^{1/2})$. The gap between those bounds on $R -({n}/{2})$ is a $\Theta (({\log {({n}/{d})}})^{1/2})$ factor related to the gap between the worst covering radius given $d$ and the sphere-covering bound. Our focus in this paper is on the case when $d = o(n)$ , i.e., when the code size is subexponential and the gap is $w(1)$. We show that up to a constant, the gap can be eliminated by relaxing the covering requirement to allow for missing $o(1)$ fraction of points. Namely, if the dual distance $d = o(n)$ , then for sufficiently large $d$ , almost all points can be covered with radius $R\leq ({n}/{2})-\Theta (({dn\log {({n}/{d})}})^{1/2})$. Compared with random linear codes, our bound on $R-({n}/{2})$ is asymptotically tight up to a factor less than 3. We give applications to dual-BCH codes. The proof builds on the author’s previous work on the weight distribution of cosets of linear codes, which we simplify in this paper and extend from codes to probability distributions on $\{0,1\}^{n}$ , thus enabling the extension of the earlier result to $(d-1)$ -wise independent distributions. [ABSTRACT FROM AUTHOR]

The rank modulation scheme has been proposed for efficient writing and storing data in nonvolatile memory storage. Error correction in the rank modulation scheme is done by considering permutation codes. In this paper, we consider codes in the set of all permutations on n elements, S_{n} , using the Kendall \tau -metric. The main goal of this paper is to derive new bounds on the size of such codes. For this purpose, we also consider perfect codes, diameter perfect codes, and the size of optimal anticodes in the Kendall \tau -metric, structures which have their own considerable interest. We prove that there are no perfect single-error-correcting codes in S_{n} , where n>4 is a prime or 4\leq n\leq 10 . We present lower bounds on the size of optimal anticodes with odd diameter. As a consequence, we obtain a new upper bound on the size of codes in S_{n} with even minimum Kendall \tau -distance. We present larger single-error-correcting codes than the known ones in S5 and S7 . [ABSTRACT FROM AUTHOR]

A function computation problem over a directed acyclic network has been considered in the literature, where a sink node is required to compute a target function correctly with the inputs arbitrarily generated at multiple source nodes. The network links are error free but capacity limited, and the intermediate nodes perform network coding. The computing rate of a network code is the average number of times that the target function is computed for one use of the network, i.e., each link in the network is used at most once. In the existing papers, two cut-set bounds were proposed on the computing rate. However, we in this paper show that these bounds are not valid for general network function computation problems. We analyze the reason of the invalidity and propose a general cut-set bound by using a new equivalence relation associated with the inputs of the target function. Moreover, some results in the existing papers were proved by applying the invalid upper bound. We also justify the validity of these results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the filtering and smoothing recursions in nonparametric finite state space hidden Markov models (HMMs) when the parameters of the model are unknown and replaced by estimators. We provide an explicit and time uniform control of the filtering and smoothing errors in total variation norm as a function of the parameter estimation errors. We prove that the risk for the filtering and smoothing errors may be uniformly upper bounded by the \mathrm L^1 -risk of the estimators. It has been proved very recently that statistical inference for finite state space nonparametric HMMs is possible. We study how the recent spectral methods developed in the parametric setting may be extended to the nonparametric framework and we give explicit upper bounds for the \mathrm L^2 -risk of the nonparametric spectral estimators. In the case where the observation space is compact, this provides explicit rates for the filtering and smoothing errors in total variation norm. The performance of the spectral method is assessed with simulated data for both the estimation of the (nonparametric) conditional distribution of the observations and the estimation of the marginal smoothing distributions. [ABSTRACT FROM AUTHOR]

This paper deals with an abstraction of a unified problem of drug discovery and pathogen identification. Pathogen identification involves the identification of disease-causing biomolecules. Drug discovery involves finding chemical compounds, called lead compounds, that bind to pathogenic proteins and eventually inhibit the function of the protein. In this paper, the lead compounds are abstracted as inhibitors, pathogenic proteins as defectives, and the mixture of “ineffective” chemical compounds and non-pathogenic proteins as normal items. A defective could be immune to the presence of an inhibitor in a test. So, a test containing a defective is positive if it does not contain its “associated” inhibitor. The goal of this paper is to identify the defectives, inhibitors, and their “associations” with high probability, or in other words, learn the immune defectives graph (IDG) efficiently through group tests. We propose a probabilistic non-adaptive pooling design, a probabilistic two-stage adaptive pooling design, and decoding algorithms for learning the IDG. For the two-stage adaptive-pooling design, we show that the sample complexity of the number of tests required to guarantee recovery of the inhibitors, defectives, and their associations with high probability, i.e., the upper bound, exceeds the proposed lower bound by a logarithmic multiplicative factor in the number of items. To be precise, lower and upper bounds of $\Omega ((r+d)\log n + rd)$ and O(rd \log n)$ tests, respectively, are identified for classifying r$ inhibitors and d$ defectives amongst n tests) exceeds the proposed lower bound (given by \max \{\Omega ((r+d)\log n + rd),\Omega (({r^{2}}/{\log r})\log n),\Omega (d^{2})\} tests) by at most a logarithmic multiplicative factor in the number of items. [ABSTRACT FROM AUTHOR]

This paper studies codes that correct a burst of deletions or insertions. Namely, a code will be called a $b$ -burst-deletion/insertion-correcting code if it can correct a burst of deletions/insertions of any $b$ consecutive bits. While the lower bound on the redundancy of such codes was shown by Levenshtein to be asymptotically $\log (n)+b-1$ , the redundancy of the best code construction by Cheng et al. is $b(\log (n/b+1))$ . In this paper, we close on this gap and provide codes with redundancy at most $\log (n) + (b-1)\log (\log (n)) +b -\log (b)$ . We first show that the models of insertions and deletions are equivalent and thus it is enough to study codes correcting a burst of deletions. We then derive a non-asymptotic upper bound on the size of $b$ -burst-deletion-correcting codes and extend the burst deletion model to two more cases: 1) a deletion burst of at most $b$ consecutive bits and 2) a deletion burst of size at most $b$ (not necessarily consecutive). We extend our code construction for the first case and study the second case for $b=3,4$ . [ABSTRACT FROM PUBLISHER]

Insertion–deletion codes (insdel codes for short) play an important role in synchronization error correction. The higher the minimum insdel distance, the more insdel errors the code can correct. Haeupler and Shahrasbi established the Singleton bound for insdel codes: the minimum insdel distance of any $[n,k]$ linear code over $\mathbb {F}_{q}$ satisfies $d\leq 2n-2k+2$. There have been some constructions of insdel codes through Reed-Solomon codes with high capabilities, but none has come close to this bound. Recently, Do Duc et al. showed that the minimum insdel distance of any $[n,k]$ Reed-Solomon code is no more than $2n-2k$ if $q$ is large enough compared to the code length $n$ ; optimal codes that meet the new bound were also constructed explicitly. The contribution of this paper is twofold. We first show that the minimum insdel distance of any $[n,k]$ linear code over $\mathbb {F}_{q}$ satisfies $d\leq 2n-2k$ if $n > k > 1$. This result improves and generalizes the previously known results in the literature. We then give a sufficient condition under which the minimum insdel distance of a 2-D Reed-Solomon code of length $n$ over $\mathbb {F}_{q}$ is exactly equal to $2n-4$. As a consequence, we show that the sufficient condition is not hard to achieve; we explicitly construct an infinite family of optimal 2-D Reed-Somolom codes meeting the bound. [ABSTRACT FROM AUTHOR]

This paper studies an $n$ -dimensional additive Gaussian noise channel with a peak-power-constrained input. It is well known that, in this case, when $n=1$ the capacity-achieving input distribution is discrete with finitely many mass points, and when $n>1$ the capacity-achieving input distribution is supported on finitely many concentric shells. However, due to the previous proof technique, not even a bound on the exact number of mass points/shells was available. This paper provides an alternative proof of the finiteness of the number mass points/shells of the capacity-achieving input distribution while producing the first firm bounds on the number of mass points and shells, paving an alternative way for approaching many such problems. The first main result of this paper is an order tight implicit bound which shows that the number of mass points in the capacity-achieving input distribution is within a factor of two from the number of zeros of the downward shifted capacity-achieving output probability density function. Next, this implicit bound is utilized to provide a first firm upper on the support size of optimal input distribution, an $O(\mathsf {A}^{2})$ upper bound where $\mathsf {A}$ denotes the constraint on the input amplitude. The second main result of this paper generalizes the first one to the case when $n>1$ , showing that, for each and every dimension $n\ge 1$ , the number of shells that the optimal input distribution contains is $O(\mathsf {A}^{2})$. Finally, the third main result of this paper reconsiders the case $n=1$ with an additional average power constraint, demonstrating a similar $O(\mathsf {A}^{2})$ bound. [ABSTRACT FROM AUTHOR]

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. 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 a valid lower bound on frequency hopping sequence sets. The purpose of this paper is to transform the Gilbert-Varshamov bound from 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 a probabilistic method that requires advanced tool–martingale. This proof covers the whole rate region. Another proof is purely elementary but only covers part of the rate region. [ABSTRACT FROM AUTHOR]

Locally repairable codes with more than one recovering set are demanded in the application to distributed storage. For each failure node (or disk), it is desired to have as many recovering sets as possible. In this paper, we make use of automorphisms of rational function fields to construct locally repairable codes with multiple recovering sets. Although we focus on two recovering sets, our construction can be easily generalized to the case of multiple recovering sets. In particular, we obtain a class of locally repairable codes with minimum distance only 1 less than the upper bound. [ABSTRACT FROM AUTHOR]

Locally recoverable codes are very important due to their applications in distributed storage systems. In this paper, by using cyclic codes, we construct two classes of optimal q-ary cyclic $(\text {r}, \delta _{1})$ locally recoverable codes with parameters $[2(\text {q}+1), 2(\text {q}+1)-2\delta _{1}, \delta _{1}+2]_{\text {q}}$ , where $q$ is an odd prime power and $\text {r}+\delta _{1}-1=\text {q}+1$ , and $(\text {r}, \delta _{2})$ locally recoverable codes with parameters $[2(\text {q}+1), 2(\text {q}+1)-4\delta _{2}+2, \delta _{2}+2]_{q}$ , where $\text {q}\geq 7$ is an odd prime power, $\text {q}\equiv 3~ (mod~ 4)$ and $r+\delta _{2}-1=\frac {\text {q}+1}{2}$. Compared with the known cyclic and constacyclic $(\text {r}, \delta)$ locally recoverable codes, our construction yields new optimal cyclic $(\text {r}, \delta)$ locally recoverable codes. [ABSTRACT FROM AUTHOR]

This paper studies the problem of information symbol availability in codes: we refer to a systematic code as code with $(r, t)$ -availability if every information (systematic) symbol can be reconstructed from $t$ disjoint groups of other code symbols, each of the sizes at most $r$ . This paper shows that it is possible to construct codes that can support a scaling number of parallel reads while keeping the rate to be an arbitrarily high constant. It further shows that this is possible with the minimum Hamming distance arbitrarily close to the Singleton bound. This paper also presents a bound demonstrating a tradeoff between rate, minimum Hamming distance, and availability parameters. Our codes match the aforementioned bound, and their constructions rely on certain combinatorial structures. Resolvable designs provide one way to realize these required combinatorial structures. The two constructions presented in this paper require field sizes, which are linear and exponential in the code length, respectively. From a practical standpoint, our codes are relevant for distributed storage applications involving hot data, i.e., the information, which is frequently accessed by multiple processes in parallel. [ABSTRACT FROM AUTHOR]

This paper studies networks that consist of $N$ half-duplex relays assisting the communication between a source and a destination. In ISIT’12 Brahma et al. conjectured that in Gaussian half-duplex diamond networks (i.e., without a direct link between the source and the destination, and with N$ non-interfering relays), an approximately optimal relay scheduling policy (i.e., achieving the cut-set upper bound to within a constant gap uniformly over all channel gains) has at most N+1$ active states (i.e., at most N+1 possible relay listen-transmit configurations have a strictly positive probability). Such relay scheduling policies were referred to as simple. In ITW’13, we conjectured that simple approximately optimal relay scheduling policies exist for any Gaussian half-duplex multi-relay network irrespectively of the topology. This paper formally proves this more general version of the conjecture and shows it holds beyond Gaussian noise networks. In particular, for any class of memoryless half-duplex $N$ -relay networks with independent noises and for which independent inputs are approximately optimal in the cut-set upper bound, an approximately optimal simple relay scheduling policy exists. The key step of the proof is to write the minimum of the submodular cut-set function by means of its Lovász extension and use the greedy algorithm for submodular polyhedra to highlight structural properties of the optimal solution. This, together with the saddle-point property of min–max problems and the existence of optimal basic feasible solutions for linear programs, proves the conjecture. As an example, for $N$ -relay Gaussian networks with independent noises, where each node is equipped with multiple antennas and where each antenna can be configured to listen or transmit irrespectively of the others, the existence of an approximately optimal simple relay scheduling policy with at most $N+1$ active states, irrespectively of the total number of antennas in the system, is proved. [ABSTRACT FROM PUBLISHER]

In this paper, we derive non-asymptotic achievability and converse bounds on the random number generation with/without side-information. Our bounds are efficiently computable in the sense that the computational complexity does not depend on the block length. We also characterize the asymptotic behaviors of the large deviation regime and the moderate deviation regime by using our bounds, which implies that our bounds are asymptotically tight in those regimes. We also show the second-order rates of those problems, and derive single letter forms of the variances characterizing the second-order rates. Furthermore, we address the relative entropy rate and the modified mutual information rate for these problems. [ABSTRACT FROM AUTHOR]

In this paper, we determine the entire secure degrees of freedom (s.d.o.f.) regions of the $K$ -user Gaussian multiple access (MAC) wiretap channel and the $K$ -user interference channel (IC) with secrecy constraints. For the IC, we consider three secrecy constraints: K-user IC with an external eavesdropper (IC-EE), K-user IC with confidential messages (IC-CM), and their combination Kuser IC with confidential messages and external eavesdropper (IC-CM-EE). The converse for the IC includes constraints both due to secrecy as well as due to interference. For the IC, although the portion of the region close to the optimum sum s.d.o.f. point is governed by the upper bounds due to secrecy constraints, the other portions of the region are governed by the upper bounds due to interference constraints. Different from the existing literature, in order to fully understand the characterization of the s.d.o.f. region of the IC, one has to study the four-user case, i.e., the two- or three-user cases do not illustrate the full generality of the problem. In order to prove the achievability, we use the polytope structure of the converse region. In both MAC and IC cases, we develop explicit schemes that achieve the extreme points of the polytope region given by the converse. In particular, the extreme points of the MAC region are achieved by an $m$ -user MAC wiretap channel with $K-m$ helpers, i.e., by setting $K-m$ users’ secure rates to zero and utilizing them as pure (structured) cooperative jammers. The extreme points of the IC region are achieved by a $(K-m)$ -user IC with confidential messages, $m$ helpers, and $N$ external eavesdroppers, for $m\ge 1$ and a finite $N$ . A byproduct of our results in this paper is that the sum s.d.o.f. is achieved only at one extreme point of the s.d.o.f. region, which is the symmetric-rate extreme point, for both MAC and IC channel models. [ABSTRACT FROM AUTHOR]

In this paper, we study a new model of defect memory cells, called partially stuck-at memory cells, which is motivated by the behavior of multi-level cells in non-volatile memories, such as flash memories and phase change memories. If a cell can store the q levels 0, 1, {\dots }, q-1 , we say that it is partially stuck-at level s$ , where $1 \leq s \leq q-1$ , if it can only store values, which are at least $s$ . We follow the common setup where the encoder knows the positions and levels of the partially stuck-at cells whereas the decoder does not. Our main contribution in this paper is the study of codes for masking $u$ partially stuck-at cells. We first derive lower and upper bounds on the redundancy of such codes. The upper bounds are based on two trivial constructions. We then present three code constructions over an alphabet of size $q$ , by first considering the case where the cells are partially stuck-at level $s=1$ . The first construction works for $u

Information retrieval in associative memories was studied in a recent paper by Yaakobi and Bruck (2012). Associations between memory entries give us the t -neighbourhood of an entry. In their model, an information unit is retrieved from the memory with the aid of input clues, which are chosen from a reference set. In this paper, we consider the situation where the information unit is found unambiguously using the associated t -neighbourhoods of the input clues. A varying number of input clues are allowed, but a limit mu on the maximum number of them is imposed. Of course, we would like mu to be as small as possible. We consider the problem over the binary Hamming space \mathbb F^n and focus on the minimum of mu , denoted by $\nu (n;t)$ . Using linear reference sets, we show that $\nu (n;2) \leq 5$ for any $n \geq 9$ . We also give infinite families of reference sets, which provide good bounds on $\nu (n;t)$ for $t=3$ . In addition, efficient methods are given to obtain bounds on $\nu (n;t)$ for any $t$ from known reference sets. We also discuss the applications of this model to the Levenshtein’s sequence reconstruction problem and the sensor network monitoring. [ABSTRACT FROM AUTHOR]