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Resolution of an Open Problem on Quasi-Cyclic Codes over $\mathbb{Z}_4$ and New Quaternary Linear Codes — arXiv:2609.21601

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.21601
License (as archived): CC0-1.0
Archived: 2026-10-10T02:20:08.060Z
SHA-256 of the archived text: d0cf02748d2f7dcbb494210b91d30411a7ae7821e7f0f57d987a2237eeb48f2c
Stable URL of this page: https://metraipolis.com/documento/dc_2ec2dceb8705

How to cite

Resolution of an Open Problem on Quasi-Cyclic Codes over $\mathbb{Z}_4$ and New Quaternary Linear Codes — arXiv:2609.21601 — arXiv (en), https://arxiv.org/abs/2609.21601 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_2ec2dceb8705 (Metraipolis archive dc_2ec2dceb8705, sha256 d0cf02748d2f7dcbb494210b91d30411a7ae7821e7f0f57d987a2237eeb48f2c)

BibTeX

@misc{dc2ec2dceb8705,
  title = {Resolution of an Open Problem on Quasi-Cyclic Codes over $mathbbZ_4$ and New Quaternary Linear Codes — arXiv:2609.21601},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.21601},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_2ec2dceb8705; sha256 d0cf02748d2f7dcbb494210b91d30411a7ae7821e7f0f57d987a2237eeb48f2c},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.21601. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Given a cyclic code $C_g = \langle g(x) \rangle$ of odd length $m$ over $\mathbb{Z}_4$, one common way to build a quasi-cyclic (QC) code is to pick $f_1, \dots, f_\ell \in \mathbb{Z}_4[x]$ and let $C = \langle (f_1 g, \dots, f_\ell g) \rangle$. Because $\mathbb{Z}_4$ is not a field, the type of $C_g$ ($4^{k_1}2^{k_2}$) is not necessarily inherited by $C$. Determining conditions under which the type of $C_g$ is inherited by $C$ was posed as an open problem recently in \cite{AydinLuOnta2023}. In this paper, we settle this problem. We first give two sufficient conditions for the type to be preserved: one requires a single $f_i$ to be coprime to $x^m-1$ over $\mathbb{F}_2$, the other only requires the $f_i$ to be jointly coprime to it. Neither condition is necessary in general. Using the fact that $x^m-1$ is squarefree for odd $m$, we decompose $\mathbb{Z}_4[x]/\langle x^m-1\rangle$ into a product of finite chain rings using Chinese remainder theorem (CRT) and derive a condition on $f_1, \dots, f_\ell$ that is both necessary and sufficient for $C$ to match the type of $C_g$. This condition depends only on the irreducible factors of $x^m-1$ where $g$ does not already vanish. This also yields a simple test for when $C$ is a free $\mathbb{Z}_4$-module. Finally, we report many new QC codes over $\mathbb{Z}_4$, found by computer searches using Magma software~\cite{Magma1997} guided by this criterion, with Lee distances greater than previously known codes of the same type.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-21 in math.IT).
Categorie arXiv: Information Theory (cs.IT).
Pagina del lavoro: https://arxiv.org/abs/2609.21601 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-10T02:04:28.746Z dalla pagina https://arxiv.org/catchup/math.IT/2026-09-21?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).