{"paper":{"title":"On the existence of asymptotically good linear codes in minor-closed classes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Peter Nelson, Stefan H.M. van Zwam","submitted_at":"2014-04-30T15:59:02Z","abstract_excerpt":"Let $\\mathcal{C} = (C_1, C_2, \\ldots)$ be a sequence of codes such that each $C_i$ is a linear $[n_i,k_i,d_i]$-code over some fixed finite field $\\mathbb{F}$, where $n_i$ is the length of the codewords, $k_i$ is the dimension, and $d_i$ is the minimum distance. We say that $\\mathcal{C}$ is asymptotically good if, for some $\\varepsilon > 0$ and for all $i$, $n_i \\geq i$, $k_i/n_i \\geq \\varepsilon$, and $d_i/n_i \\geq \\varepsilon$. Sequences of asymptotically good codes exist. We prove that if $\\mathcal{C}$ is a class of GF$(p^n)$-linear codes (where $p$ is prime and $n \\geq 1$), closed under pun"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.7771","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}