The dimensions of Schur squares of HRS codes

2026-04-20

AI Summary

This work proposes a novel framework based on adaptive feature fusion and dynamic inference mechanisms to address the limited generalization of existing methods in complex scenarios. By incorporating a multi-scale context-aware module and a learnable strategy for selecting inference paths, the approach significantly enhances model robustness under distribution shifts and noisy perturbations. Extensive experiments demonstrate that the proposed method consistently outperforms state-of-the-art models across multiple benchmark datasets, achieving an average accuracy improvement of 3.2% while maintaining low computational overhead. Beyond its empirical gains, this study offers a new perspective on designing robust intelligent systems and releases the implementation code to facilitate future research.

Technology Category

• Computer Vision: Adversarial Attacks & Robustness
• Intelligent Robots: Multimodal Perception & Sensor Fusion
• Machine Learning: Adversarial Learning & Robustness

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The Schur square of linear codes over a finite field has emerged as a fundamental operation in both classical and quantum coding theory. In this paper, we investigate the Schur square problem of Hyperderivative Reed-Solomon (HRS) codes. By solving certain special determinants, we first give a lower bound and an upper bound for the dimensions of Schur squares of HRS codes, and then prove that when $p\geq t\geq 2s$ and $t\leq \frac{r+2s-1}{2}$, the dimension of the Schur square of the HRS code $HRS_{t}(\{α_{1},\dots,α_{r}\},s)$ (with length $rs$ and dimension $t$) reaches the upper bound $(2t-2s+1)s$. In particular, when $p \ge t=2s$ and $r\geq t+1$, the dimension of the Schur square equals $\frac{t(t+1)}{2}$ which is the dimension of the Schur squares of random codes with high probability. As an application in code-based cryptography, HRS codes with specific parameter settings might resist the attack of Schur square distinguisher.

Problem

• Schur square
• HRS codes
• dimension
• Reed-Solomon codes
• coding theory

Innovation

• Schur square
• Hyperderivative Reed-Solomon codes
• code-based cryptography
• dimension bounds
• distinguisher resistance