Disjunct matrices, also known as cover-free families and superimposed codes, are combinatorial arrays widely used in group testing. Among their variants, those that satisfy an additional combinatorial property called inclusiveness form a special class suitable for computationally efficient and highly error-tolerant group testing under the general inhibitor complex model, a broad framework that subsumes practical settings such as DNA screening. Despite this relevance, the asymptotic behavior of the inclusive variant of disjunct matrices has remained largely unexplored. In particular, it was not previously known whether this variant can achieve an asymptotically positive rate, a requirement for scalable group testing designs. In this work, we establish the first nontrivial asymptotic lower bound on the maximum achievable rate of the inclusive variant, which matches the strongest known upper bound up to a logarithmic factor. Our proof is based on the probabilistic method and yields a simple and efficient randomized construction. Furthermore, we derandomize this construction to obtain a deterministic polynomial-time construction. These results clarify the asymptotic potential of robust and scalable group testing under the general inhibitor complex model.

We settle the problem of determining the asymptotic behavior of the parameters of optimal difference systems of sets, or DSSes for short, which were originally introduced for computationally efficient frame synchronization under the presence of additive noise. We prove that the lowest achievable redundancy of a DSS asymptotically attains Levenshtein's lower bound for any alphabet size and relative index, answering the question of Levenshtein posed in 1971. Our proof is probabilistic and gives a linear-time randomized algorithm for constructing asymptotically optimal DSSes with high probability for any alphabet size and information rate. This provides efficient self-synchronizing codes with strong noise resilience. We also point out an application of DSSes to phase detection.

Weak superimposed codes are combinatorial structures related closely to generalized cover-free families, superimposed codes, and disjunct matrices in that they are only required to satisfy similar but less stringent conditions. This class of codes may also be seen as a stricter variant of what are known as locally thin families in combinatorics. Originally, weak superimposed codes were introduced in the context of multimedia content protection against illegal distribution of copies under the assumption that a coalition of malicious users may employ the averaging attack with adversarial noise. As in many other kinds of codes in information theory, it is of interest and importance in the study of weak superimposed codes to find the highest achievable rate in the asymptotic regime and give an efficient construction that produces an infinite sequence of codes that achieve it. Here, we prove a tighter lower bound than the sharpest known one on the rate of optimal weak superimposed codes and give a polynomial-time randomized construction algorithm for codes that asymptotically attain our improved bound with high probability. Our probabilistic approach is versatile and applicable to many other related codes and arrays.

X-codes form a special class of linear maps which were originally introduced for data compression in VLSI testing and are also known to give special parity-check matrices for linear codes suitable for error-erasure channels. In the context of circuit testing, an (m, n, d, x) X-code compresses n-bit output data R from the circuit under test into m bits, while allowing for detecting the existence of an up to d-bit-wise anomaly in R even if up to x bits of the original uncompressed R are unknowable to the tester. Using probabilistic combinatorics, we give a nontrivial lower bound for any d ≥ 2 on the maximum number n of codewords such that an (m, n, d, 2) X-code of constant weight 3 exists. This is the first result that shows the existence of an infinite sequence of X-codes whose compaction ratio tends to infinity for any fixed d under severe weight restrictions. We also give a deterministic polynomial-time algorithm that produces X-codes that achieve our bound.

It is known that redundant parity-check equations can improve the performance of an LDPC code by reducing the number of harmful substructures in the parity-check matrix. However, it is a difficult problem to design a parity-check matrix in such a way that it avoids substructures that are known to be harmful to iterative decoding while keeping the number of redundant parity-check equations moderate and ensuring other desirable properties. We explicitly give redundant parity-check matrices for cyclic regular LDPC codes of length n and minimum distance d ~ √n in which there are only n parity-check equations but no stopping sets of size d+1 or smaller except for those that correspond to the nonzero codewords of the smallest weight. We do this by showing that the well-known projective LDPC codes from the incidence matrices of projective planes PG(2, q) with q even have this property. This result may give insight into how the small number of redundant parity-check equations in the geometric LDPC codes may be contributing to the good performance reported in the literature. We also give a slightly improved upper bound on the size of a smallest generic erasure correcting set.

X-codes are special linear maps for a compression technique, called X-compact. In the context of integrated circuit testing, an (m, n, d, x) X-code is an m×n binary matrix which compresses n-bit output from the circuit under test into m bits while allowing for detecting the existence of up to d erroneous output bits even if up to x bits of the correct behavior are unknowable. In particular, X-codes of constant column weight x+1 are of special interest for practical reasons. While it is known that there exist infinite series of (m, n, 1, 2) X-codes of constant weight 3 with n=Θ(m2), here we show that for d ≥ 4 the largest possible n is o(m2). This is an application of extremal graph theory and the first asymptotic improvement on this problem. We also investigate a special class of X-codes of constant weight 3 with a design theoretic property that boosts test quality when there are fewer unknowable bits than anticipated. We improve the tightest known lower bound on the largest number n of columns of an X-code of this kind through probabilistic combinatorics. A deterministic algorithm is also given that produces X-codes of this type attaining the improved bound and runs in time polynomial in m.

An error-erasure channel is a simple noise model that introduces both errors and erasures. While the two types of errors can be corrected simultaneously with error-correcting codes, it is also known that any linear code allows for first correcting errors and then erasures in two-step decoding. In particular, a carefully designed parity-check matrix not only allows for separating erasures from errors but also makes it possible to efficiently correct erasures. The separating redundancy of a linear code is the number of parity-check equations in a smallest parity-check matrix that has the required property for this error-erasure separation. In a sense, it is a parameter of a linear code that represents the minimum overhead for efficiently separating erasures from errors. While several bounds on separating redundancy are known, there still remains a wide gap between upper and lower bounds except for a few limited cases. In this paper, using probabilistic combinatorics and design theory, we improve both upper and lower bounds on separating redundancy. We also show a relation between parity-check matrices for error-erasure separation and special matrices, called X-codes, for data compaction circuits in VLSI testing. This leads to an exponentially improved bound on the size of an optimal X-code.

X-codes are linear maps with a special combinatorial property that generalizes superimposed codes, disjunct matrices, and cover-free families. In the context of circuit testing, a (t, n, d, x) X-code compresses n-bit output from the circuit under test into t bits while allowing for detecting the existence of up to d erroneous output bits even if up to x bits of the correct behavior are unknowable. A simple counting argument shows that a (t, n, d, x) X-code with t = O (log n) exists, where the coefficient of the logarithmic term when the base is 2 is at most 2x+1(d + x)ln 2 with In being the natural logarithm to base e. While there are also known constructions that provide X-codes with smaller t for given n and some specific d and x, no stronger general upper bounds on the smallest possible t that work for any d and x are available in the literature. Here, we derive general upper bounds in closed form that reduce the coefficient of the basic general bound to (x + 1)(d + x - 1)e ln 2. In terms of the highest achievable rate, our results exponentially improve the known asymptotic lower bound 1/(2x+1(d + x) ln2) to 1/((x + 1)(d + x - 1)e ln 2).

The trapping redundancy of a linear code is the number of rows of a smallest parity-check matrix such that no submatrix forms an (a, b)-trapping set. This concept was first introduced in the context of low-density parity-check (LDPC) codes in an attempt to estimate the number of redundant rows in a parity-check matrix suitable for iterative decoding. Essentially the same concepts appear in other contexts as well such as robust syndrome extraction for quantum error correction. Among the known upper bounds on the trapping redundancy, the strongest one was proposed by employing a powerful tool in probabilistic combinatorics, called the Lovász Local Lemma. Unfortunately, the proposed proof invoked this tool in a situation where an assumption made in the lemma does not necessarily hold. Hence, although we do not doubt that nonetheless the proposed bound actually holds, for it to be a mathematical theorem, a more rigorous proof is desired. Another disadvantage of the proposed bound is that it is only applicable to (a, b)-trapping sets with rather small a. Here, we give a more general and sharper upper bound on trapping redundancy by making mathematically more rigorous use of probabilistic combinatorics without relying on the lemma. Our bound is applicable to all potentially avoidable (a, b)-trapping sets with a smaller than the minimum distance of a given linear code, while being generally much sharper than the bound through the Lovász Local Lemma. In fact, our upper bound is sharp enough to exactly determine the trapping redundancy for many cases, thereby providing precise knowledge in the form of a more general bound with mathematical rigor.

Stabilizer codes are a fundamental class of error-correcting codes for quantum information that allow for syndrome decoding in the quantum domain. One of the substantial challenges in quantum error correction is that the quantum gates that perform error correction themselves are faulty in practice, which makes fault-tolerant implementation a vital component. Recently, a coding theoretic technique is proposed that makes it possible for syndrome decoding to help correct imperfectly extracted syndromes instead of fully relying on an external fault-tolerant mechanism. In particular, it was proved that any single-error-correcting stabilizer code can be made robust against single errors on either a data qubit or a syndrome bit by using at most one more stabilizer operator than necessary for standard syndrome decoding, while analogous overhead for making double-error-correcting stabilizer codes robust against double errors involving data qubits and/or syndrome bits was shown to be at most logarithmic. We generalize this result to t-error-correcting stabilizer codes and show that the overhead for achieving analogously defined global t-error correction for data qubits and/or syndrome bits is also at most logarithmic. The proof exploits combinatorial arrays that may be seen as parity-check matrices that detect errors of even weight but may overlook errors of odd weight.

Quantum error correction is an important building block for reliable quantum information processing. A challenging hurdle in the theory of quantum error correction is that it is significantly more difficult to design error-correcting codes with desirable properties for quantum information processing than for traditional digital communications and computation. A typical obstacle to constructing a variety of strong quantum error-correcting codes is the complicated restrictions imposed on the structure of a code. Recently, promising solutions to this problem have been proposed in quantum information science, where in principle any binary linear code can be turned into a quantum error-correcting code by assuming a small number of reliable quantum bits. This paper studies how best to take advantage of these latest ideas to construct desirable quantum error-correcting codes of very high information rate. Our methods exploit structured high-rate low-density parity-check codes available in the classical domain and provide quantum analogues that inherit their characteristic low decoding complexity and high error correction performance even at moderate code lengths. Our approach to designing high-rate quantum error-correcting codes also allows for making direct use of other major syndrome decoding methods for linear codes, making it possible to deal with a situation where promising quantum analogues of low-density parity-check codes are difficult to find.

Quantum synchronizable codes are quantum error-correcting codes designed to correct the effects of both quantum noise and block synchronization errors. While it is known that quantum synchronizable codes can be constructed from cyclic codes that satisfy special properties, only a few classes of cyclic codes have been proved to give promising quantum synchronizable codes. In this paper, using quadratic residue codes and their supercodes, we give a simple construction for quantum synchronizable codes whose synchronization capabilities attain the upper bound. The method is applicable to cyclic codes of prime length.

Quantum synchronizable error-correcting codes are special quantum error-correcting codes that are designed to correct both the effect of quantum noise on qubits and misalignment in block synchronization. It is known that, in principle, such a code can be constructed through a combination of a classical linear code and its subcode if the two are both cyclic and dual-containing. However, finding such classical codes that lead to promising quantum synchronizable error-correcting codes is not a trivial task. In fact, although there are two families of classical codes that are proved to produce quantum synchronizable codes with good minimum distances and highest possible tolerance against misalignment, their code lengths have been restricted to primes and Mersenne numbers. In this paper, examining the incidence vectors of projective spaces over the finite fields of characteristic 2, we give quantum synchronizable codes from cyclic codes whose lengths are not primes or Mersenne numbers. These projective geometric codes achieve good performance in quantum error correction and possess the best possible ability to recover synchronization, thereby enriching the variety of good quantum synchronizable codes. We also extend the current knowledge of cyclic codes in classical coding theory by explicitly giving generator polynomials of the finite geometric codes and completely characterizing the minimum weight nonzero codewords. In addition to the codes based on projective spaces, we carry out a similar analysis on the well-known cyclic codes from Euclidean spaces that are known to be majority logic decodable and determine their exact minimum distances.

Pulse position modulation (PPM) is a popular signal modulation technique which converts signals into M-ary data by means of the position of a pulse within a time interval. While PPM and its variations have great advantages in many contexts, this type of modulation is vulnerable to loss of synchronization, potentially causing a severe error floor or throughput penalty even when little or no noise is assumed. Another disadvantage is that this type of modulation typically offers no error correction mechanism on its own, making them sensitive to intersymbol interference and environmental noise. In this paper, we propose a coding theoretic variation of PPM that allows for significantly more efficient symbol and frame synchronization as well as strong error correction. The proposed scheme can be divided into a synchronization layer and a modulation layer. This makes our technique compatible with major existing techniques such as standard PPM, multipulse PPM, and expurgated PPM as well in that the scheme can be realized by adding a simple synchronization layer to one of these standard techniques. We also develop a generalization of expurgated PPM suited for the modulation layer of the proposed self-synchronizing modulation scheme. This generalized PPM can also be used as stand-alone error-correcting PPM with a larger number of available symbols.

We develop a theoretical framework for frame synchronization, also known as block synchronization, in the quantum domain, which makes it possible to attach classical and quantum metadata to quantum information over a noisy channel even when the information source and sink are framewise asynchronous. This eliminates the need of frame synchronization at the hardware level and allows for parsing qubit sequences during quantum information processing. Our framework exploits binary constant-weight codes that are self-synchronizing. Possible applications may include asynchronous quantum communication such as a self-synchronizing quantum network where one can hop into the channel at any time, catch the next coming quantum information with a label indicating the sender, and reply by routing her quantum information with control qubits for quantum switches all without assuming prior frame synchronization between users.

Self-synchronization under the presence of additive noise can be achieved by allocating a certain number of bits of each codeword as markers for synchronization. Difference systems of sets are combinatorial designs which specify the positions of synchronization markers in codewords in such a way that the resulting error-tolerant self-synchronizing codes may be realized as cosets of linear codes. Ideally, difference systems of sets should sacrifice as few bits as possible for a given code length, alphabet size, and error-tolerance capability. However, it seems difficult to attain optimality with respect to known bounds when the noise level is relatively low. In fact, the majority of known optimal difference systems of sets are for exceptionally noisy channels, requiring a substantial amount of bits for synchronization. To address this problem, we present constructions for difference systems of sets that allow for higher information rates while sacrificing optimality to only a small extent. Our constructions utilize optimal difference systems of sets as ingredients and, when applied carefully, generate asymptotically optimal ones with higher information rates. We also give direct constructions for optimal difference systems of sets with high information rates and error tolerance that generate binary and ternary self-synchronizing codes.

As in classical coding theory, quantum analogs of low-density parity-check (LDPC) codes have offered good error correction performance and low decoding complexity by employing the Calderbank-Shor-Steane construction. However, special requirements in the quantum setting severely limit the structures such quantum codes can have. While the entanglement-assisted stabilizer formalism overcomes this limitation by exploiting maximally entangled states (ebits), excessive reliance on ebits is a substantial obstacle to implementation. This paper gives necessary and sufficient conditions for the existence of quantum LDPC codes which are obtainable from pairs of identical LDPC codes and consume only one ebit, and studies the spectrum of attainable code parameters.

Contrary to the assumption that most quantum error-correcting codes (QECC) make, it is expected that phase errors are much more likely than bit errors in physical devices. By employing the entanglement-assisted stabilizer formalism, we develop a new kind of error-correcting protocol which can flexibly trade error correction abilities between the two types of errors, such that high error correction performance is achieved both in symmetric and in asymmetric situations. The characteristics of the QECCs can be optimized in an adaptive manner during information transmission. The proposed entanglement-assisted QECCs require only one ebit regardless of the degree of asymmetry at a given moment and can be decoded in polynomial time.

Test response compaction for integrated circuits (ICs) with scan-based design-for-testability (DFT) support in the presence of unknown logic values (Xs) is investigated from a combinatorial viewpoint. The theoretical foundations of X-codes, employed in an X-tolerant compaction technique called X-compact, are examined. Through the formulation of a combinatorial model of X-compact, novel design techniques are developed for X-codes to detect a specified maximum number of errors in the presence of a specified maximum number of unknown logic values, while requiring only small fan-out. The special class of X-codes that results leads to an avoidance problem for configurations in combinatorial designs. General design methods and nonconstructive existence theorems to estimate the compaction ratio of an optimal X-compactor are also derived.

Frequency hopping spread spectrum and direct sequence spread spectrum are two main spread coding technologies in communication systems. Frequency hopping sequences are needed in frequency hopping code-division multiple-access (FH-CDMA) systems. In this paper, four algebraic and a combinatorial constructions of optimal sets of frequency hopping sequences with new parameters are presented, and a number of bounds on sets of frequency hopping sequences are described.