{"paper":{"title":"Intermediate Constacyclic Codes and Scalar-Residue Reed--Muller Layers","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"The minimum distance of intermediate constacyclic codes equals an explicit case formula in the field size q and the parameters a and b of the degree ℓ.","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Yaoran Yang, Yutong Zhang","submitted_at":"2026-05-16T14:48:31Z","abstract_excerpt":"A 2024 paper of Sun, Ding and Wang introduced a second class of constacyclic codes over finite fields, denoted $C(q,m,r,\\ell)$, with length $(q^m-1)/r$, where $r\\mid(q-1)$ and the defining monomials have total $q$-ary degree congruent to $r-1$ modulo $r$. In the non-projective intermediate range $2<r<q-1$ the paper gave a sharp-looking upper bound and a BCH-type lower bound, and left the minimum distance open. We prove that the upper bound is the exact minimum distance for every admissible intermediate parameter. More precisely, if $\\ell=(q-1)a+b<(q-1)m-1$, $0\\le b\\le q-2$, and $b\\equiv r-1\\pm"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"If ℓ=(q-1)a+b<(q-1)m-1, 0≤b≤q-2, and b≡r-1 (mod r), then for every prime power q, every divisor r of q-1 with 2<r<q-1, and every m≥2, d(C(q,m,r,ℓ)) equals (q-1)/r *(q-b+1)q^{m-a-2} when 0≤a≤m-2 and (q-b+r-2)/r when a=m-1.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The proof relies on the hidden scalar homogeneity of the evaluation model for these codes, which is invoked to enable the orbit-counting obstruction and the homogeneous pencil construction that attain the claimed distances and supports.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Proves that the minimum distance of intermediate constacyclic codes C(q,m,r,ℓ) equals a specific piecewise formula and determines the minimum affine support for non-terminal scalar-residue layers of generalized Reed-Muller codes.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The minimum distance of intermediate constacyclic codes equals an explicit case formula in the field size q and the parameters a and b of the degree ℓ.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"135a079d6792a87e64f193bb555938b5aee11abe12367dd23dcea659372327c9"},"source":{"id":"2605.17022","kind":"arxiv","version":1},"verdict":{"id":"4e34c233-e40b-495f-99ff-d7774903e907","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T18:33:54.304762Z","strongest_claim":"If ℓ=(q-1)a+b<(q-1)m-1, 0≤b≤q-2, and b≡r-1 (mod r), then for every prime power q, every divisor r of q-1 with 2<r<q-1, and every m≥2, d(C(q,m,r,ℓ)) equals (q-1)/r *(q-b+1)q^{m-a-2} when 0≤a≤m-2 and (q-b+r-2)/r when a=m-1.","one_line_summary":"Proves that the minimum distance of intermediate constacyclic codes C(q,m,r,ℓ) equals a specific piecewise formula and determines the minimum affine support for non-terminal scalar-residue layers of generalized Reed-Muller codes.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The proof relies on the hidden scalar homogeneity of the evaluation model for these codes, which is invoked to enable the orbit-counting obstruction and the homogeneous pencil construction that attain the claimed distances and supports.","pith_extraction_headline":"The minimum distance of intermediate constacyclic codes equals an explicit case formula in the field size q and the parameters a and b of the degree 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