Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2611.08945

The Schur products of some nonbinary narrow-sense antiprimitive BCH codes

Haojie Xu·Xia Wu·Wei Lu·Xiwang Cao
View original source ↗Read source document ↗Export metadata ↓

Abstract

In recent years, the Schur products of linear codes have been extensively studied due to their favorable applications in cryptography. In this paper, we aim to study the parameters of the Schur products of some nonbinary narrow-sense antiprimitive BCH codes. Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be two narrow-sense antiprimitive BCH codes with designed distances $δ_{1}$ and $δ_{2}$, respectively, where $2\le δ_{1},δ_{2}\le n$. Let $\mathcal{C}_1\star\mathcal{C}_2$ be the Schur product of $\mathcal{C}_1$ and $\mathcal{C}_2$. We present some necessary and sufficient conditions for $\mathcal{C}_1\star\mathcal{C}_2\ne\mathbb{F}_q^n$ by applying some constraints to $δ_{1}$ and $δ_{2}$. Furthermore, we determine the dimensions of these products and lower bounds on their minimum distance. Several linear codes with best known parameters are found.

Explore related subjects

cs.ITmath.IT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Haojie Xu, Xia Wu, Wei Lu, Xiwang Cao. 2026-11-06. The Schur products of some nonbinary narrow-sense antiprimitive BCH codes. https://arxiv.org/abs/2611.08945

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Graphical Analysis of Lifted Product Code Constructions

Lifted product codes are an important family of quantum low-density parity-check (QLDPC) codes, as they were the first QLDPC code family shown to be asymptotically good. Understanding the structure of their parity-check matrices $H_{\mathsf{X}}$ and $H_{\mathsf{Z}}$, as well as the associated Tanner graphs, is essential for analyzing their decoding behavior and error-floor performance. In this work, we show that the Tanner graphs of $H_{\mathsf{X}}$ and $H_{\mathsf{Z}}$ are indeed isomorphic, and investigate their graph-theoretical structure. We establish conditions ensuring the connectivity of these graphs and provide bounds on their minimal absorbing sets, providing new insight into the combinatorial structures influencing decoding performance.

cs.IT↗
arXiv2026

Sensing-Based Communication: A Non-Orthogonal Passive Reflection Framework for ISAC

While integrating uplink without disrupting the primary downlink remains a critical challenge for existing reflective communication systems, this paper proposes a non-orthogonal passive integrated sensing and communication (ISAC) mechanism termed Sensing-based Communication. Originating from and validated within an optical retro-reflective system, this mechanism possesses an underlying wave-agnostic architecture with the potential to be generalized across other physical modalities. It leverages incident downlink signals for parasitic modulation via passive tags. By relying solely on fundamental on-off keying (OOK) reflection, the tag unifies the reflective sensing channel and the uplink communication channel, and simultaneously achieves both sensing and duplex communication. Consequently, our approach introduces uplink capability without sacrificing the downlink data rate or consuming additional spectrum resources. At the receiver, we design a joint decoding architecture combining cross-correlation computation and successive interference cancelation (SIC) to fully exploit the degrees of freedom in broadband signals for multi-user interference suppression. We derive closed-form expressions for the interference and bit error rate (BER) assuming zero-mean Gaussian signals. Using an experimental testbed, we validate the pivotal theoretical finding, the scaling relationship of SINR with signal bandwidth and cross-correlation time--where linear regression yields a strong goodness-of-fit with $R^2 = 0.9688$. Furthermore, evaluations under noise-like orthogonal frequency division multiplexing (OFDM) waveforms exhibit high consistency between theoretical derivations and simulation data ($R^2 = 0.9372$, mean absolute percentage error, $\text{MAPE} = 14\%$), confirming the model's accuracy and waveform generalizability.

cs.IT↗
arXiv2026

Anchored Iterative Decoding of Reed-Solomon Product Codes for High-Rate Magnetic Tape Storage

We present a decoding algorithm for a Reed--Solomon product code targeted at the demanding reliability requirements of modern magnetic tape storage. For a code $RS(256, 224)\times RS(175, 173)$ in a BSC we achieve post-decoding bit error rate (BER) of $10^{-20}$ at $\approx 5.3\times 10^{-3}$ raw BER while maintaining the common requirement to detect and correct 4 out of 32 dead tracks. We resolve two common problems of miscorrections and stall patterns that arise under such a high raw BER via simplified versions of anchor decoding and binary image analysis. The highest efficiency is achieved by performing an unlimited iterative decoder, which performs decoding iterations until no changes are made. We give an empirical analysis of the average number of iterations: below the raw BER of $5\times 10^{-3}$, four half-iterations are performed on average while improving substantially over a decoder that always performs 4 half-iterations. We provide a CPU implementation achieving 100-200 megabytes/s single-core throughput on a regular laptop and a design of a potential hybrid FPGA-CPU implementation of our decoder.

cs.IT↗