Current metabolomics approaches predominantly rely on group-level differential abundance screening. While effective for identifying biomarkers, this paradigm often overlooks upstream regulatory hubs and fails to resolve the inter-individual heterogeneity inherent in complex diseases. To bridge this gap, we present ssNetShift, a single-sample network framework that identifies personalized topological driver metabolites by quantifying topological rewiring rather than static concentration deviations. By integrating linear interpolation-based network estimation with an extended neighbor-shift metric, ssNetShift systematically characterizes how specific metabolites alter their connectivity and centrality within individual patient networks. We applied ssNetShift to a multicohort gastric cancer dataset comprising 389 patients and 313 controls. Benchmarking analyses demonstrated that ssNetShift consistently outperformed conventional approaches: unlike group-level methods (e.g. NetShift), it recovered survival-associated driver metabolites masked by population averaging; unlike single-sample abundance methods (e.g. personalized perturbation profiles), it prioritized silent drivers, metabolites with stable abundance but drastic topological reorganization, thereby capturing system-level dysregulation. Crucially, ssNetShift revealed hidden prognostic subtypes within the same clinical stage, separating patients with identical tumor-node-metastasis (TNM) staging into distinct risk classes characterized by specific metabolic wiring patterns (e.g. nucleotide and tryptophan hubs) and significantly divergent survival outcomes. Collectively, ssNetShift provides a risk stratification dimension orthogonal to traditional staging, offering a robust tool for uncovering mechanistic drivers and refining prognostic resolution in heterogeneous malignancies.

In a recent work, quantum locally recoverable codes (qLRCs) have been introduced for their potential application in large-scale quantum data storage and implication for quantum LDPC codes. This work focuses on the bounds and constructions of qLRCs derived from the Hermitian construction, which solves an open problem proposed by Luo $et~al.$ (IEEE Trans. Inf. Theory, 71 (3): 1794-1802, 2025). We present four bounds for qLRCs and give comparisons in terms of their asymptotic formulas. We construct several new infinite families of NMDS codes, with general and flexible dimensions, that support t-designs for $t\in \{2,3\}$, and apply them to obtain Hermitian dual-containing classical LRCs (cLRCs). As a result, we derive three explicit families of optimal qLRCs. Compared to the known qLRCs obtained by the CSS construction, our optimal qLRCs offer new and more flexible parameters. It is also worth noting that the constructed cLRCs themselves are interesting as they are optimal with respect to four distinct bounds for cLRCs.

The existence of optimal binary self-orthogonal codes has been well characterized. In this paper, we develop general methods involving residual codes and the MacWilliams identities to prove the nonexistence of several infinite families of binary self-orthogonal codes, despite the existence of binary linear codes with the same parameters. In particular, we focus on the largest minimum distances of optimal binary self-orthogonal codes with dimension eight.

The error coefficient of a linear code, defined as the number of minimum-weight codewords, plays a central role in evaluating the performance of the code. In this paper, we establish two recursive bounds on the minimum possible error coefficient among optimal linear codes with prescribed parameters. We prove that these bounds are tight in infinitely many cases by constructing two explicit infinite families of optimal linear codes that attain them with equality, and we further show that MDS codes also meet one of the proposed bounds with equality. Beyond the recursive-bound framework, we determine the minimum possible error coefficient for three explicit families of optimal linear codes: two families arising from simplex codes and one family associated with MacDonald codes. Moreover, employing tools from combinatorial design theory, we solve a problem proposed by Guanet al.[12] on the eventual constancy of the minimum error coefficient of optimal codes.

The error coefficient of a linear code is defined as the number of minimum-weight codewords. In an additive white Gaussian noise channel, optimal linear codes with the smallest error coefficients achieve the best possible asymptotic frame error rate (AFER) among all optimal linear codes under maximum likelihood decoding. Such codes are referred to as AFER-optimal linear codes. The Griesmer bound is essential for determining the optimality of linear codes. However, establishing tight lower bounds on the error coefficients of Griesmer optimal linear codes is challenging, and the linear programming bound often performs inadequately. In this paper, we propose several iterative lower bounds for the error coefficients of Griesmer optimal linear codes. Specifically, for binary linear codes, our bounds are tight in most cases when the dimension does not exceed 5. To evaluate the performance of our bounds when they are not tight, we also determine the parameters of the remaining 5-dimensional AFER-optimal linear codes. Our final comparison demonstrates that even when our bounds are not tight, they remain very close to the actual values, with a gap of less than or equal to 2.

A binary linear code is calledasymptotic frame error rate (AFER)-optimalif it achieves the maximum possible value of the minimum distance while having the smallest value of the corresponding error coefficient. Over the additive white Gaussian noise channel and under maximum-likelihood decoding, AFER-optimal codes attain the best possible asymptotic frame error rate at high signal-to-noise ratio. In this paper, we present several bounds on the smallest error coefficients of binary linear codes and give several constructions of AFER-optimal binary linear codes. Many examples confirm that our bounds are sharp on numerous occasions. In addition, we give two families of AFER-optimal codes that respectively attain the proposed bounds with equality.

Carlet et al. showed that for$q\gt 2$, there exists a q-ary linear complementary pair (LCP) of codes whose security parameter is as good as the minimum distance of the best linear code with the same length and dimension. In this paper, we study the best security parameters of binary LCPs of codes. As a result, we solve an open problem proposed by Carlet et al. (IEEE Trans. Inf. Theory 65(3): 1694-1704, 2019) and a conjecture proposed by Choi et al. (Cryptogr. Commun. 15(2): 469-486, 2023).

The hull of a linear code over a finite field is the intersection of the code and its dual, which was introduced by Assmus and Key to classify finite projective planes. The main objective of this paper is to obtain a closed mass formula for linear codes with prescribed hull dimension. We simplify the mass formula obtained by Sendrier and provide an alternative proof for the mass formula for self-orthogonal codes obtained by Pless. Finally, we obtain a classification of (optimal) ternary linear codes with small parameters.

For an integer$s\ge 1$, let${\mathcal {C}}_{s}(q_{0})$be the generalized Zetterberg code of length$q_{0}^{s}+1$over the finite field${\mathbb {F}}_{q_{0}}$of odd characteristic. Recently, Shi et al. determined the covering radius of${\mathcal {C}}_{s}(q_{0})$for$q_{0}^{s} \cancel {\equiv }7 \pmod {8}$, and left the remaining case as an open problem. In this paper, we develop a general technique involving arithmetic of finite fields and algebraic curves over finite fields to determine the covering radius of all generalized Zetterberg codes for$q_{0}^{s} \equiv 7 \pmod {8}$, which therefore solves this open problem. We also introduce the concept of twisted half generalized Zetterberg codes of length$\frac {q_{0}^{s}+1}{2}$, and show the same results hold for them. As a result, we obtain some quasi-perfect codes.

In future mobile networks, the demand for real-time, accurate localization of multiple signal sources is paramount, but the facilities are often resource-constrained and the deploying environments are complex. In this context, we present a lightweight deep neural network in this work, which is tailored for multi-source angle-of-arrival (AoA) estimation under low signal-to-noise-ratio (SNR) conditions. The network employs mobile inverted bottleneck convolution (MBConv), known for its enhanced feature extraction capabilities and resilience to noise. By leveraging a scale attention mechanism, we effectively integrate the outputs of each layer without the need for neural architecture search. Trained on multi-channel data under low SNR, the network formulates angle estimation as a multi-label classification task. Experimental results confirm that, the proposed network demonstrates superior accuracy in extreme noise conditions and with limited snapshots, outperforming existing methodologies in multi-source scenarios.

The hull of a linear code over finite fileds is the intersection of the code and its dual, which was introduced by Assmus and Key to classify finite projective planes. The main purpose of this paper is to obtain the closed mass formula for binary linear codes with various hull dimensions, which simplifies the mass formula obtained by Sendrier in (SIAM J. Discrete Math., 10(2): 282-293, 1997). We show that almost all binary linear codes with ℓ-dimensional hull are odd-like codes with odd-like duals for fixed ℓ. We also study the largest minimum distance of a binary linear [n, k] code with ℓ-dimensional hull. Most importantly, we give a complete classification of binary linear codes with various hull dimensions for n ≤ 12 using a building-up construction, which is confirmed by double-checking with our mass formula. We also give the classification of optimal binary linear [n, k] codes with various hull dimensions for n ≤ 13. Combining with known results, we obtain the classification of (optimal) binary linear codes with small parameters.

Determining the weight distribution of a code is an old and fundamental topic in coding theory that has been thoroughly studied. In 1977, Helleseth, Kløve, and Mykkeltveit presented a weight enumerator polynomial of the lifted code over${\mathbb {F}}_{q^{\ell } }$of a q-ary linear code with significant combinatorial properties, which can determine the support weight distribution of this linear code. The Solomon-Stiffler codes are a family of famous Griesmer codes, which were proposed by Solomon and Stiffler in 1965. In this paper, we determine the weight enumerator polynomials of the lifted codes of the projective Solomon-Stiffler codes using some combinatorial properties of subspaces. As a result, we determine the support weight distributions of the projective Solomon-Stiffler codes. In particular, we determine the weight hierarchies of the projective Solomon-Stiffler codes.

The purpose of this paper is to solve the two conjectures on the largest minimum distance$d_{so}(n,5)$of a binary self-orthogonal$[n, 5]$code proposed by Kim and Choi (2022). The determination of$d_{so}(n,k)$has been a fundamental and difficult problem in coding theory because there are too many binary self-orthogonal codes as the dimension$k$increases. Recently, Kim et al. (2021) considered the shortest self-orthogonal embedding of a binary linear code, and many binary optimal self-orthogonal$[n,k]$codes were constructed for$k=4,5$. Kim and Choi (2022) improved some results of Kim et al. (2021) and made two conjectures on$d_{so}(n,5)$. In this paper, we develop a general method to determine the exact value of$d_{so}(n,k)$for$k=5,6$and show that the two conjectures made by Kim and Choi (2022) are true.

We study the codes of the title by the CRT method, that decomposes such codes into constituent codes, which are shorter codes over larger alphabets. Criteria on these constituent codes for self-duality and linear complementary duality of the decomposed codes are derived. The special class of the one-generator codes is given a polynomial representation and exactly enumerated. In particular, we present some illustrative examples of binary optimal linear codes with respect to the Griesmer bound derived from the$\mathbb {Z}_{2} \mathbb {Z}_{4}$-additive quasi-cyclic codes.

Fingerprinting localization systems are outstanding for its convenient deployment, where a major challenge is the high cost for collecting a huge number of received signal strength (RSS) fingerprints. Mobile crowdsensing (MCS) paradigm is cost-effective for large-scale data collection; however, a quality-aware data pricing mechanism dedicated to MCSed fingerprints accommodating practical application situations including budget constraints and online data submission is still unavailable. In this paper, we present a data pricing scheme dedicated to MCSed fingerprints by enhancing the online learning technique. We first reveal the principle of fingerprints quality assessment for accurate localization. Based on the principle, we design corresponding loss and regret function, reflecting the values of the fingerprints with respect to localization accuracy. We then present an online pricing scheme for MCSed data, which results in that the worker's payoff is a random variable following an optimal probability density function (PDF) leading to the minimum expected regret. Further, we extend our scheme to application scenarios with different budget settings, where the pricing strategies for the scenarios of regret minimization with fixed budget and budget minimization for certain fingerprints quality level are investigated. Experimental results are presented to verify our theoretical analysis.