Sep 2026· Turkish Journal of Mathematics· Vol 50, pp. 842-860· 0 citations
Abstract
Minimum distance bounds play a central role in the analysis of algebraic codes. For cyclic and constacyclic codes, several bounds based on their zero set have been developed. However, analogous results for quasi-twisted (QT) codes are comparatively limited. In this paper, we further investigate the spectral theory of QT codes and derive a general spectral bound on their minimum distance. Our bound unifies and generalizes previously known spectral bounds for quasi-cyclic (QC) and QT codes, and contains them as special cases. We present a new proof technique and show that the bound can be formulated with respect to an arbitrary subset of eigenvalues, thereby extending its applicability to the largest possible setting. Numerical examples and simulations demonstrate that the proposed bound often improves upon the Jensen bound and earlier spectral bounds.
We introduce lower bounds on the minimum distance of one-generator quasi-cyclic codes of arbitrary index, obtained by puncturing and shortening the code at the level of its blocks. For balanced codes, these bounds form a nondecreasing chain whose first term is the Jensen bound. We apply these bounds to conduct a comput...
K. Abdukhalikov, Duy Ho, Rasha M. Shat· 0 citations
We provide a positive answer to a conjecture proposed by Toh\v{a}neanu and Van Tuyl regarding the minimum distance of codes whose underlying set of points is a reduced complete intersection. Despite the technical nature of the conjecture, we show that it follows directly from a not-well-known refinement of the classica...
Function-correcting codes, recently introduced by Lenz et al.\ (2023), provide a framework for the receiver to provide a higher level of protection against channel errors than the level of protection for messages. In this paper, we derive a generalised Plotkin-type bound on the redundancy of FCCs under the Hamming metr...
This paper investigates the construction of linear codes via abelian codes over finite fields. By exploiting the algebraic structure of multivariate polynomial quotient rings, we derive lower bounds on the minimum distance using a generalized shift bound, which extends the classical van Lint-Wilson bound for cyclic cod...
Cong Yu, Hao Chen, Zhong-Hua Sun et al.· 0 citations
We study the asymptotic density of $\mathbb{F}_q$-point-free linear sections of geometrically irreducible projective varieties over finite fields. We then apply these results to rank-metric codes via determinantal varieties. Our approach recovers the known cases in which the density tends to $0$ or $1$ and determines t...
Shamil Asgarli, Lian Duan, Nathan Kaplan et al.· 0 citations
We prove an $\Omega(n^2)$ lower bound on the determinantal complexity of the power sum polynomial $\sum_{i=1}^n x_i^n$ over the field of complex numbers. A similar result was claimed in a recent paper of Sheshadri (arXiv:2606.13628), via an AI-assisted and AI-written proof. Assuming its correctness, this was the first...
Mrinal Kumar, Ben Lee Volk· 1 citation
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