Quantum weight enumerators are fundamental tools for analyzing quantum error-correcting codes and multipartite entanglement, offering insights into the existence of quantum error-correcting codes andk-uniform states. In this work, we establish a connection between quantum weight enumerators and then-qubit parallelized SWAP test. We demonstrate that each shadow enumerator corresponds to a probability derived from this test, providing a physical interpretation for the shadow enumerators. Leveraging the non-negativity of these probabilities, we present an elegant proof for the shadow inequalities. Additionally, we show that the Shor-Laflamme weight enumerators and the Rains unitary enumerators can be calculated using then-qubit parallelized SWAP test. For applications, we utilize this test to compute the distances of quantum error-correcting codes, determine thek-uniformity of pure states, and evaluate multipartite entanglement measures. Our results indicate that quantum weight enumerators can be efficiently estimated on quantum computers, opening a path to calculate and verify the distances of quantum error-correcting codes.

Abstract. We study the supersaturation problems of oddtown and eventown. Given a family [Formula: see text] of subsets of an [Formula: see text]-element set, let [Formula: see text] denote the number of distinct pairs [Formula: see text] for which [Formula: see text] is odd. We show that if [Formula: see text] consists of [Formula: see text] odd-sized subsets, then [Formula: see text], which is tight when [Formula: see text]. This disproves a conjecture by O’Neill on the supersaturation problem of oddtown. For the supersaturation problem of eventown, we show that for large enough [Formula: see text], if [Formula: see text] consists of [Formula: see text] even-sized subsets, then [Formula: see text] for any positive integer [Formula: see text]. This partially proves a conjecture by O’Neill on the supersaturation problem of eventown. Previously, the correctness of this conjecture was only verified for [Formula: see text] and 2. We further provide a lower bound which is a factor of two away from the conjectured bound for eventown, that is, [Formula: see text] for general [Formula: see text] and [Formula: see text] by using discrete Fourier analysis. Finally, some asymptotic results for the lower bounds of [Formula: see text] are given when [Formula: see text] is large for both problems.

Do N-partite k-uniform states always exist when$k\leq \left \lfloor {{\frac {N}{2}}}\right \rfloor -1$? In this work, we provide new upper bounds on the parameter k for the existence of k-uniform states in$(\mathbb {C}^{d})^{\otimes N}$when$d=3,4,5$, which extend Rains’ bound in 1999 and improve Scott’s bound in 2004. Since a k-uniform state in$(\mathbb {C}^{d})^{\otimes N}$corresponds to a pure$((N,1,k+1))_{d}$quantum error-correcting code, we also give new upper bounds on the minimum distance$k+1$of pure$((N,1,k+1))_{d}$quantum error-correcting codes. Furthermore, we generalize Scott’s bound to heterogeneous systems, and show some non-existence results of absolutely maximally entangled states in$\mathbb {C}^{d_{1}}\otimes (\mathbb {C}^{d_{2}})^{\otimes 2n}$.

Function-correcting codes (FCCs), introduced by Lenz, Bitar, Wachter-Zeh, and Yaakobi, protect specific function values of a message rather than the entire message. A central challenge is determining the optimal redundancy—the minimum additional information required to recover function values amid errors. This redundancy depends on both the number of correctable errorstand the structure of message vectors yielding identical function values. While prior works established bounds, key questions remain, such as the optimal redundancy for functions like Hamming weight and Hamming weight distribution, along with efficient code constructions. In this paper, we make the following contributions: 1) For the Hamming weight function, we improve the lower bound on optimal redundancy from 10(t-1)/3 to 4t− 4/3√6t+ 2 + 2. On the other hand, we provide a systematical approach to constructing explicit FCCs via a novel connection with Gray codes, which also improve the previous upper bound from 4t-2/1−2√ln(2t)/(2t) to 4t− [logt]. Consequently, we almost determine the optimal redundancy for Hamming weight function. 2) The Hamming weight distribution function is defined by the value of Hamming weight divided by a given integerT∈ N. Previous work established that the optimal redundancy is 2twhenT> 2t, while the caseT≤ 2tremained unclear. We show that the optimal redundancy remains 2twhenT≥t+ 1. However, in the surprising regime whereT=o(t), we achieve near-optimal redundancy of 4t−o(t). Our results reveal a significant distinction in behavior of redundancy for distinct choices of T.

A low-power error-correcting cooling (LPECC) code was introduced as a coding scheme for communication over a bus by Chee et al. to control the peak temperature, the average power consumption of on-chip buses, and error-correction for the transmitted information, simultaneously. Specifically, an (n, t,w, e)-LPECC code is a coding scheme overnwires that avoids state transitions on the t hottest wires and allows at most w state transitions in each transmission, and can correct up toetransmission errors. In this paper, we study the maximum possible size of an (n, t,w, e)-LPECC code, denoted byC(n, t,w, e). Whenw = e+ 2 is large, we establish a general upper boundC(n, t,w,w− 2) ≤ ⌊ (n+1 2) / (w+t2) ⌋; whenw = e+ 2 = 3, we proveC(n, t, 3, 1) ≤ ⌊n(n+1) 6(t+1) ⌋. Both bounds are tight for largensatisfying some divisibility conditions. Previously, tight bounds were known only forw = e+ 2 = 3, 4 andt≤ 2. In general, whenw = e + dis large for a constantd, we determine the asymptotic value ofC(n, t,w,w−d) ∼ (n d) / (w+t d) asngoes to infinity, which can be extended toq-ary codes.

Facial landmark tracking and 3D reconstruction have gained considerable attention due to their numerous applications such as human-computer interactions, facial expression analysis, and emotion recognition, etc. Traditional approaches require users to be confined to a particular location and face a camera under constrained recording conditions, which prevents them from being deployed in many application scenarios involving human motions. In this paper, we propose the first single-earpiece lightweight biosensing system,BioFace-3D, that can unobtrusively, continuously, and reliably sense the entire facial movements, track 2D facial landmarks, and further render 3D facial animations. Our single-earpiece biosensing system takes advantage of the cross-modal transfer learning model to transfer the knowledge embodied in ahigh-gradevisual facial landmark detection model to thelow-gradebiosignal domain. After training, ourBioFace-3Dcan directly perform continuous 3D facial reconstruction from the biosignals, without any visual input. Additionally, by utilizing biosensors, we also showcase the potential for capturing both behavioral aspects, such as facial gestures, and distinctive individual physiological traits, establishing a comprehensive two-factor authentication/identification framework. Extensive experiments involving 16 participants demonstrate thatBioFace-3Dcan accurately track 53 major facial landmarks with only 1.85 mm average error and 3.38% normalized mean error, which is comparable with most state-of-the-art camera-based solutions. Experiments also show that the system can authenticate users with high accuracy (e.g., over 99.8% within two trials for three gestures in series), low false positive rate (e.g., less 0.24%), and is robust to various types of attacks.

To prevent service time bottlenecks in distributed storage systems, the access balancing problem has been studied by designing almost supermagic edge labelings of certain graphs to balance the access requests to different servers. In this paper, we introduce the concept ofrobustnessof edge labelings under limited-magnitude swaps, which is important for studying thedynamicalaccess balancing problem with respect to changes in data popularity. We provide upper and lower bounds on the robustness ratio for complete graphs withnvertices, and constructO(n)-almost supermagic labelings that are asymptotically optimal in terms of the robustness ratio.

Frameproof codes have been extensively studied for many years due to their application in copyright protection and their connection to extremal set theory. This paper investigates the upper bounds of the cardinalities of wide-sense t-frameproof codes. For$t=2$, we apply results from Sperner theory to give a better upper bound, which significantly improves a recent bound by Zhou and Zhou. For$t\geq 3$, we provide a general upper bound by establishing a relation between wide-sense frameproof codes and cover-free families. Finally, when the code length n is at most$\frac {15+\sqrt {33}}{24}(t-1)^{2}$, we show that a wide-sense t-frameproof code has at most n codewords, and the unique optimal code consists of all weight-one codewords. As byproducts, our results improve or imply several best-known results on binary t-frameproof codes.

Reconstruction codes are generalizations of error-correcting codes that can correct errors by a given number of noisy reads. The study of such codes was initiated by Levenshtein in 2001 and developed recently due to applications in modern storage devices such as racetrack memories and DNA storage. The central problem on this topic is to design codes with redundancy as small as possible for a given number$N$of noisy reads. In this paper, the minimum redundancy of such codes for binary channels with exactly two insertions is determined asymptotically for all values of$N\ge 5$. Previously, such codes were studied only for channels with single edit errors or two-deletion errors.

Maximum Distance Separable (MDS) codes with a sparse and balanced generator matrix are appealing in distributed storage systems for balancing and minimizing the computational load. Such codes have been constructed via Reed-Solomon codes over large fields. In this paper, we focus on small fields. We prove that there exists an$[n,k]_{q}$MDS code that has a sparse and balanced generator matrix for any$q\geq n-1$provided that$n\leq 2k$, by designing several algorithms with complexity running in polynomial time in$k$and$n$. For the case$n>2k$, we give some constructions for$q=n=p^{s}$and$k=p^{e}m$based on sumsets, when$e\leq s-2$and$m\leq p-1$, or$e=s-1$and$m < \frac {p}{2}$.

We study$k$-uniform states in heterogeneous systems whose local dimensions are not all the same. Based on the connections between mixed orthogonal arrays with certain minimum Hamming distance, irredundant mixed orthogonal arrays and$k$-uniform states, we present two constructions of 2-uniform states in heterogeneous systems. We also construct two families of 3-uniform states in heterogeneous systems, which solves a question posed in [D. Goyenecheet al., Phys. Rev. A94, 012346 (2016)]. We show two methods of generating$(k-1)$-uniform states from$k$-uniform states. Some results on the nonexistence of absolutely maximally entangled states are provided. For the applications, we present an orthogonal basis consisting of$k$-uniform states with the minimum support, and we show that some$k$-uniform bases are locally irreducible. Moreover, we connect$k$-uniform states with quantum information masking.

Motivated by the duplication-correcting problem for data storage in live DNA, we study the construction of constant-weight codes inl1-metric. By using packings and group divisible designs in combinatorial design theory, we give constructions of optimal codes over non-negative integers and optimal ternary codes withl1-weight w ≤ 4 for all possible distances. In general, we derive the size of the largest ternary code with constant weight w and distance 2w-2 for sufficiently large length n satisfying n ≡ 1, w,- w+2,-2w+3 mod w(w-1).

This paper focuses on constructions of MDS self-dual codes from (extended) generalized Reed-Solomon (GRS) codes. Let$q = r^{2}$be an odd prime power. We show that, there exists a$q$-ary self-dual (extended) GRS code for each even length in the range$[{2r,3r-3}]$, and for each singly even length in the range$[3r-1,4r]$. This extends the only known consecutive range$[2,2r]$to$[{2,3r-3}]$for this case. Furthermore, our general constructions provide many MDS self-dual codes with new parameters which, to the best of our knowledge, were not reported before.

The study of constant-weight codes in$\ell _{1}$-metric was motivated by the duplication-correcting problem for data storage in live DNA. It is interesting to determine the maximum size of a code given the length${n}$, weight${w}$, minimum distance${d}$and the alphabet size${q}$. In this paper, based on graph decompositions, we determine the maximum size of ternary codes with constant weight w and distance$2{w}-2$for all sufficiently large length${n}$. Previously, this was known only for a very sparse family${n}$of density$4/{w}({w}-1)$.

To solve the access-balancing problem in distributed storage systems, we introduce a new combinatorial model, called MinVar model for fractional repetition (FR) codes. Since FR codes are based on graphs or set systems, our MinVar model is characterized by the property that the variance among the sums of block-labels incident to a fixed vertex is minimized. This characterization is different from Dau and Milenkovic's MaxMinSum model, while the minimum sum of labels is maximized. We show that our MinVar model is meaningful by distinguishing labelings with different variances but with the same MaxMin value for some FR codes. By reformulating the MinVar model to an equivalent vertex-labeling problem of graphs, we find several families of optimal FR codes with balanced access frequency, and provide fundamental results for both problems. It is interesting that MinVar model is closely related to the concept of magic-labeling in graph theory.

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

Splitter sets have been widely studied due to their applications in flash memories, and their close relations with lattice tilings and conflict avoiding codes. In this paper, we give necessary and sufficient conditions for the existence of nonsingular perfect splitter sets, B[-k1, k2](p) sets, where 0 ≤ k1≤ k2= 4. Meanwhile, constructions of nonsingular perfect splitter sets are given. When perfect splitter sets do not exist, we present four new constructions of quasi-perfect splitter sets. Finally, we give a connection between nonsingular splitter sets and Cayley graphs, and as a byproduct, a general lower bound on the maximum size of nonsingular splitter sets is given.

We demonstrate that certain Johnson-type bounds are asymptotically exact for a variety of classes of codes, namely, constant-composition codes, nonbinary constant-weight codes, group divisible codes, and multiply constant-weight codes. We achieve this via an application of the theory of decomposition of edge-colored digraphs.

The Johnson-type upper bound on the maximum size of a code of length n, distance d = 2w - 1, and constant n composition w̅ is [n/w1] the largest component of w̅. Recently, Chee et al. proved that this upper bound can be achieved for all constant-composition codes of sufficiently large lengths. Let Nccc(w) be the smallest such length. The determination of Nccc(w̅) is trivial for binary codes. This paper provides a lower bound on Nccc(w̅), which is shown to be tight for all ternary and quaternary codes by giving new combinatorial constructions. Consequently, by the refining method, we determine the values of Nccc(w̅), for all q-ary constant-composition codes, provided that 3w1≥ w with finite possible exceptions.

Codes that can correct up to t symmetric errors and detect all unidirectional errors, known as t-EC-AUED codes, are studied in this paper. Given positive integers q, a, and t, let nq(a, t + 1) denote the length of the shortest q-ary t-EC-AUED code of size a. We introduce combinatorial constructions for q-ary t-EC-AUED codes via one-factorizations of complete graphs, and concatenation of MDS codes and codes from resolvable set systems. Consequently, we determine the exact values of nq(a, t + 1) for several new infinite families of q, a, and t.

Multiply constant-weight codes (MCWCs) have been recently studied to improve the reliability of certain physically unclonable function response. In this paper, we give combinatorial constructions for the MCWCs, which yield several new infinite families of optimal MCWCs. Furthermore, we demonstrate that the Johnson-type upper bounds of the MCWCs are asymptotically tight for fixed Hamming weights and distances. Finally, we provide bounds and constructions of the 2-D MCWCs.

Splitter sets are closely related to lattice tilings, and have applications in flash memories and conflict-avoiding codes. The study of k-radius sequences was motivated by some problems occurring in large data transfer. It is observed that the existence of splitter sets yields k-radius sequences of short length. In this paper, we obtain several new results contributing to splitter sets and k-radius sequences. We give some new constructions of perfect splitter sets, as well as some nonexistence results on them. As a byproduct, we obtain some new results on optimal conflict-avoiding codes. Furthermore, we provide several explicit constructions of short k-radius sequences for certain values of n, by establishing the existence of k-additive sequences. In particular, we show that for any fixed k, there exist infinitely many values of n such that fk(n) = 2k/n2+ O(n), where fk(n) denotes the shortest length of an n-ary k-radius sequence. This result partially affirms a conjecture posed by Bondy, Lonc, and Rza̧żewski.

We construct codes for the Chebyshev permutation channels whose study was initiated by Langberg et al. (2015). We establish several recursive code constructions and present efficient decoding algorithms for our codes. In particular, our constructions yield a family of binary codes of rate 0.643 when r = 1. The upper bound on the rate in this case is 2/3 and the previous highest rate is 0.609.

Perfect deletion-correcting codes of the same length over the same alphabet can have different sizes. The interesting problem of determining the possible sizes of perfect deletion-correcting codes has previously been studied. In this paper, we study the corresponding problem for burst deletion-correcting codes. We completely determine the spectrum of sizes for perfect burst deletion-correcting codes for certain classes of parameters and also construct new classes of perfect deletion-correcting codes.

We construct the first class of permutation codes that are capable of correcting a burst of up to s unstable deletions, for general s. Efficient decoding algorithms are presented to show the correctness of our constructions.

The study of Armstrong codes is motivated by the problem of understanding complexities of dependences in relational database systems, where attributes have bounded domains. A (q, k, n)-Armstrong code is a q-ary code of length n with minimum Hamming distance n - k + 1, and for any set of k - 1 coordinates, there exist two codewords that agree exactly there. Let f (q, k) be the maximum n for which such a code exists. In this paper, f (q, 3) = 3q -1 is determined for all q ≥ 5 with three possible exceptions. This disproves a conjecture of Sali. Furthermore, we introduce generalized Armstrong codes for branching, or (s, t)-dependences, construct several classes of optimal Armstrong codes, and establish lower bounds for the maximum length n in this more general setting.

Guruswami et al. showed that the list-decodability of random linear codes is as good as that of general random codes. In this paper, we further strengthen the result by showing that the list-decodability of random Euclidean self-orthogonal codes is as good as that of general random codes as well, i.e., achieves the classical Gilbert-Varshamov bound. In particular, we show that, for any fixed finite field Fq, error fraction δ ∈ (0,1 - 1/q) satisfying 1 - Hq(δ) ≤ 1/2, and small ε > 0, with high probability a random Euclidean self-orthogonal code over Fqof rate 1 - Hq(δ) - ε is (δ, O(1/ε))-list-decodable. This generalizes the result of linear codes to Euclidean self-orthogonal codes. In addition, we extend the result to list decoding symplectic dual-containing codes by showing that the list-decodability of random symplectic dual-containing codes achieves the quantum Gilbert-Varshamov bound as well. This implies that list-decodability of quantum stabilizer codes can achieve the quantum Gilbert-Varshamov bound. The counting argument on self-orthogonal codes is an important ingredient to prove our result.

We demonstrate that certain Johnson-type bounds are asymptotically exact for a variety of classes of codes, namely, constant-composition codes, nonbinary constant-weight codes and multiply constant-weight codes. This was achieved via an interesting application of the theory of decomposition of edge-colored digraphs.

We introduce the class of multiply constant-weight codes to improve the reliability of certain physically unclonable function response, and extend classical coding methods to construct multiply constant-weight codes from known \(q\) -ary and constant-weight codes. We derive analogs of Johnson bounds and give constructions showing these bounds to be asymptotically tight up to a constant factor under certain conditions. We also examine the rates of multiply constant-weight codes and demonstrate that these rates are the same as those of constant-weight codes of corresponding parameters.

The study of Armstrong codes is motivated by the problem of understanding complexities of dependencies in relational database systems, where attributes have bounded domains. A (q, k, n)-Armstrong code is a q-ary code of length n with minimum Hamming distance n - k + 1, and for any set of k - 1 coordinates there exist two codewords that agree exactly there. Let f(q, k) be the maximum n for which such a code exists. In this paper, f(q, 3) = 3q - 1 is determined for all q ≥ 5 with three possible exceptions. This disproves a conjecture of Sali. Further, we introduce generalized Armstrong codes for branching, or (s, t)-dependencies and construct several classes of optimal Armstrong codes in this more general setting.

Coding in a new metric space, the Enomoto-Katona space, is considered recently in connection to the study of implication structures of functional dependencies and their generalizations in relational databases. The central problem here is the determination of C(n, k, d), the size of an optimal code of length n, weight k, and distance d in the Enomoto-Katona space. The value of C(n, k, d) is known only for some congruence classes of n when (k, d) ∈ {(2, 3), (3, 5)}. In this paper, we obtain new infinite families of optimal codes in the Enomoto-Katona space. In particular, C(n, k, 2k-1) is determined for all sufficiently large n satisfying either n ≡ 1 mod k and n(n-1) ≡ 0 mod 2k2, or n ≡ 0 mod k.

Frameproof codes are used to preserve the security in the context of coalition when fingerprinting digital data. Let Mc,l(q) be the largest cardinality of a q-ary c-frameproof code of length l and Rc,l=limq→∞Mc,l(q)/q[ l/c]. It has been determined by Blackburn that Rc,l=1 when l≡1(mod c), Rc,l=2 when c=2 and l is even, and R3,5=5/3. In this paper, we give a recursive construction for c-frameproof codes of length l with respect to the alphabet size q . As applications of this construction, we establish the existence results for q-ary c-frameproof codes of length c+2 and size c+2/c(q-1)2+1 for all odd q when c=2 and for all q≡4 when c=3 . Furthermore, we show that Rc,c+2=(c+2)/c meeting the upper bound given by Blackburn, for all integers c such that c+1 is a prime power.

Constant-weight codes (CWCs) play an important role in coding theory. The problem of determining the sizes for optimal ternary CWCs with length$n$, weight 4, and minimum Hamming distance 5 ($(n,5,4)_{3}$code) has been settled for all positive integers$n\leq 10$or$n > 10$and$n\equiv 1\pmod {3}$with$n\in \{13,52,58\}$undetermined. In this paper, we investigate the problem of constructing optimal$(n,5,4)_{3}$codes for all lengths$n$with the tool of group divisible codes. We determine the size of an optimal$(n,5,4)_{3}$code for each integer$n\geq 4$leaving the lengths$n\in \{12,13,21,27,33,39,45,52\}$unsolved.

α-acyclicity is an important notion in database theory. The α-arboricity of a hypergraph [Formula: see text] is the minimum number of α-acyclic hypergraphs that partition the edge set of [Formula: see text]. The α-arboricity of the complete 3-uniform hypergraph is determined completely.

Motivated by applications in combinatorial group testing for consecutive positives, we investigate a block sequence of a maximum packing $MP (t,k,v)$ which contains the blocks exactly once such that the collection of all blocks together with all unions of two consecutive blocks of this sequence forms an error correcting code with minimum distance d. Such a sequence is usually called a block sequence with consecutive unions having minimum distance d, and denoted by $BSCU (t,k,v|d)$. In this paper, we show that the necessary conditions for the existence of $BSCU (3,4,v|4)$s of Steiner quadruple systems, namely, $v\equiv2,4$ (mod 6) and $v\geq4$, are also sufficient, excepting $v=8,10$.