{"paper":{"title":"Discrepancy for Random Linear Codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","cs.CR","math.CO","math.IT"],"primary_cat":"cs.IT","authors_text":"Dean Doron, Henrique Navas, Jo\\~ao Ribeiro, Jonathan Mosheiff, Nicolas Resch, Tal Leonov","submitted_at":"2026-06-23T12:01:54Z","abstract_excerpt":"We prove that random linear codes have nearly optimal discrepancy properties in a broad range of regimes. Our main results are two general theorems: one controlling all translates of a fixed test, and another controlling large families of Fourier-pseudorandom tests. Two motivating applications follow.\n  First, random linear codes match unstructured random codes for list-decoding from errors above capacity. If $C\\subseteq\\mathbb F_q^n$ is a random linear code of rate $1-\\frac1n\\log_q |B_\\rho|+\\epsilon$, where $B_\\rho$ is a radius-$\\rho$ Hamming ball, then with high probability $$ |C\\cap B|=(1\\p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.24471","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.24471/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}