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Covering 1024 syndromes with 50 columns — arXiv:2608.27494

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.27494
License (as archived): CC0-1.0
Archived: 2026-10-17T11:23:11.340Z
SHA-256 of the archived text: 784c8669102d5da17226e338c475a464c831d2cb59063fee8415e7ad570b1121
Stable URL of this page: https://metraipolis.com/documento/dc_12ed4bb486af

How to cite

Covering 1024 syndromes with 50 columns — arXiv:2608.27494 — arXiv (en), https://arxiv.org/abs/2608.27494 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_12ed4bb486af (Metraipolis archive dc_12ed4bb486af, sha256 784c8669102d5da17226e338c475a464c831d2cb59063fee8415e7ad570b1121)

BibTeX

@misc{dc12ed4bb486af,
  title = {Covering 1024 syndromes with 50 columns — arXiv:2608.27494},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.27494},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_12ed4bb486af; sha256 784c8669102d5da17226e338c475a464c831d2cb59063fee8415e7ad570b1121},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.27494. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We exhibit a binary linear $[50,40]_2$ code of covering radius $2$, so $\ell_2(10,2)\le 50$, one column below the Kaikkonen--Rosendahl length $51$ that has stood since 2003 and that still seeds the $R=2$ family of Davydov--Marcugini--Pambianco ( arXiv:2511.02542 ). The new matrix admits a $(2,0)$-partition into ten blocks, so Construction $\mathrm{QM}_2^2$ propagates it to exhaustively verified codes of lengths $815$ and $1631$ at $r=18$ and $r=20$, and to the family $n=51\cdot 2^{r/2-5}-1$ of asymptotic density $2601/2048$. The matrices, verifiers, and source are at this https URL , pin problems/covering/share/2026-08-24/ at commit 736a38f.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-08-31 in math.CO).
Categorie arXiv: Information Theory (cs.IT) ; Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2608.27494 — 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 2026-10-17T11:06:51.775Z dalla pagina https://arxiv.org/catchup/math.CO/2026-08-31?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).