Minjia Shi

Minjia Shi, Jing Wang, Patrick Solé

The Lloyd Theorem of (Solé, 1989) is combined with the Schwartz-Zippel Lemma of theoretical computer science to derive non-existence results for perfect codes in the Lee metric, NRT metric, mixed Hamming metric, and for the sum-rank distance. The proofs are based on asymptotic enumeration of integer partitions. The framework is the new concept of {\em polynomial} weakly metric association schemes. A connection between this notion and the recent theory of multivariate P-polynomial schemes of ( Bannai et al. 2025) and of $m$-distance regular graphs ( Bernard et al 2025) is pointed out.

Yuhao Zhou, Minjia Shi, Yuxin Tian et al.

Federated learning (FL) is identified as a crucial enabler for large-scale distributed machine learning (ML) without the need for local raw dataset sharing, substantially reducing privacy concerns and alleviating the isolated data problem. In reality, the prosperity of FL is largely due to a centralized framework called FedAvg, in which workers are in charge of model training and servers are in control of model aggregation. However, FedAvg's centralized worker-server architecture has raised new concerns, be it the low scalability of the cluster, the risk of data leakage, and the failure or even defection of the central server. To overcome these problems, we propose Decentralized Federated Trusted Averaging (DeFTA), a decentralized FL framework that serves as a plug-and-play replacement for FedAvg, instantly bringing better security, scalability, and fault-tolerance to the federated learning process after installation. In principle, it fundamentally resolves the above-mentioned issues from an architectural perspective without compromises or tradeoffs, primarily consisting of a new model aggregating formula with theoretical performance analysis, and a decentralized trust system (DTS) to greatly improve system robustness. Note that since DeFTA is an alternative to FedAvg at the framework level, \textit{prevalent algorithms published for FedAvg can be also utilized in DeFTA with ease}. Extensive experiments on six datasets and six basic models suggest that DeFTA not only has comparable performance with FedAvg in a more realistic setting, but also achieves great resilience even when 66% of workers are malicious. Furthermore, we also present an asynchronous variant of DeFTA to endow it with more powerful usability.

Minjia Shi, Xuan Wang, Bouazzaoui Zakariae et al.

Wall-Sun-Sun primes (shortly WSS primes) are defined as those primes $p$ such that the period of the Fibonacci recurrence is the same modulo $p$ and modulo $p^2.$ This concept has been generalized recently to certain second order recurrences whose characteristic polynomials admit as a zero the principal unit of $\mathbb{Q}(\sqrt{d}),$ for some integer $d>0.$ Primes of the latter type we call $WSS(d).$ They correspond to the case when $\mathbb{Q}(\sqrt{d})$ is not $p$-rational. For such a prime $p$ we study the weight distributions of the cyclic codes over $\mathbb{F}_p$ and $\mathbb{Z}_{p^2}$ whose check polynomial is the reciprocal of the said characteristic polynomial. Some of these codes are MDS (reducible case) or NMDS (irreducible case).

Minjia Shi, Xuan Wang, Junmin An et al.

We study linear codes over Gaussian integers equipped with the Mannheim distance. We develop Mannheim-metric analogues of several classical bounds. We derive an explicit formula for the volume of Mannheim balls, which yields a sphere packing bound and constraints on the parameters of two-error-correcting perfect codes. We prove several other useful bounds, and exhibit families of codes meeting these bounds for some parameters, thereby showing that these bounds are tight. We also discuss self-dual codes over Gaussian integers and obtain upper bounds on their minimum Mannheim distance for certain parameter regions using a Mannheim version of the Macwilliams-type identity. Finally, we present decoding algorithms for codes over Gaussian integer residue rings. We give examples showing that certain errors which are not correctable under the Hamming metric become correctable under the Mannheim metric.

Yuhao Zhou, Minjia Shi, Yuxin Tian et al.

Federated Learning (FL) presents an innovative approach to privacy-preserving distributed machine learning and enables efficient crowd intelligence on a large scale. However, a significant challenge arises when coordinating FL with crowd intelligence which diverse client groups possess disparate objectives due to data heterogeneity or distinct tasks. To address this challenge, we propose the Federated cINN Clustering Algorithm (FCCA) to robustly cluster clients into different groups, avoiding mutual interference between clients with data heterogeneity, and thereby enhancing the performance of the global model. Specifically, FCCA utilizes a global encoder to transform each client's private data into multivariate Gaussian distributions. It then employs a generative model to learn encoded latent features through maximum likelihood estimation, which eases optimization and avoids mode collapse. Finally, the central server collects converged local models to approximate similarities between clients and thus partition them into distinct clusters. Extensive experimental results demonstrate FCCA's superiority over other state-of-the-art clustered federated learning algorithms, evaluated on various models and datasets. These results suggest that our approach has substantial potential to enhance the efficiency and accuracy of real-world federated learning tasks.

Minjia Shi, Hongwei Zhu, Tor Helleseth

The $r$-th generalized Hamming metric and the $b$-symbol metric are two different generalizations of Hamming metric. The former is used on the wire-tap channel of Type II, and the latter is motivated by the limitations of the reading process in high-density data storage systems and applied to a read channel that outputs overlapping symbols. In this paper, we study the connections among the three metrics (that is, Hamming metric, $b$-symbol metric, and $r$-th generalized Hamming metric) mentioned above and give a conjecture about the $b$-symbol Griesmer Bound for cyclic codes. %Furthermore, we explore the combinatorial function of the size of the $b$-symbol weight set of a code $C$.

Minjia Shi, Tor Helleseth, Patrick Sole

Let $p$ be a prime number. Irreducible cyclic codes of length $p^2-1$ and dimension $2$ over the integers modulo $p^h$ are shown to have exactly two nonzero Hamming weights. The construction uses the Galois ring of characteristic $p^h$ and order $p^{2h}.$ When the check polynomial is primitive, the code meets the Griesmer bound of (Shiromoto, Storme) (2012). By puncturing some projective codes are constructed. Those in length $p+1$ meet a Singleton-like bound of (Shiromoto , 2000). An infinite family of strongly regular graphs is constructed as coset graphs of the duals of these projective codes. A common cover of all these graphs, for fixed $p$, is provided by considering the Hensel lifting of these cyclic codes over the $p$-adic numbers.

Minjia Shi, Li Xu, Patrick Sole

Double polycirculant codes are introduced here as a generalization of double circulant codes. When the matrix of the polyshift is a companion matrix of a trinomial, we show that such a code is isodual, hence formally self-dual. Numerical examples show that the codes constructed have optimal or quasi-optimal parameters amongst formally self-dual codes. Self-duality, the trivial case of isoduality, can only occur over $ \F_2$ in the double circulant case. Building on an explicit infinite sequence of irreducible trinomials over $\F_2,$ we show that binary double polycirculant codes are asymptotically good.