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 classical B\'ezout bound for overdetermined polynomial systems. For completeness, this paper presents a self-contained proof of this refined bound. Furthermore, we show that using the same approach, it is possible to obtain a bound on the generalized Hamming weights of such a code and, more generally, to control the minimum distance of the codes obtained by evaluating forms of degree $d$ on the points of a zero-dimensional complete intersection.
We study generalized Hamming weights along the one-point code flag of an AJ-Gorenstein curve. We organize these weights in a graded array, the zero diagram, whose entries are generalized coweights: the largest numbers of evaluation points on which subcodes of prescribed dimensions vanish simultaneously. Twisted and Wei...
E. Sarmiento-Rosales, J. A. Guzmán-Vega, Juan Carlos Jiménez-Cervantes· 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...
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
Jurij Vol\v{c}i\v{c} conjectured that a noncommutative polynomial $g$ belongs to the two-sided ideal generated by $f_1,\ldots,f_l$ if and only if, for matrices of every size, the rank of the evaluation of $g$ is bounded by a constant times the maximum rank of the evaluations of $f_1,\ldots,f_l$. In this paper, we show...
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 Q...
Buket Özkaya· Turkish Journal of Mathemati...· 0 citations
The generalized packing--covering conjecture asks whether, at every order, the packing radius of a linear code is at most its covering radius. We prove the conjecture for every linear code of redundancy at most fourteen over every finite field, extending the previously established range of redundancies at most seven. W...
Isaac Barouch Essayag, Aryeh Lev Zabokritskiy· 2 citations· ⚡1
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