[#P2578] VT zero-syndrome weight log-concavity through length 2000
Contents
Problem. For \(7\le n\le2000\), let \(W_{n,k}\) count the words of weight \(k\) in \(V_n=\{x\in\{0,1\}^n:\sum_{i=1}^n i x_i\equiv0\pmod{n+1}\}\). Is \(W_{n,2},W_{n,3},\ldots,W_{n,n-2}\) log-concave for every such \(n\)?
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Definitions and notation
1Definitions
Definition 1. A nonnegative sequence a_k is log-concave when a_k^2 >= a_{k-1}a_{k+1} at each internal index.
Definition 2. V_n is the zero-syndrome Varshamov-Tenengolts single-deletion code.
2What counts as a solution
- Give an exact sweep through n=2000 with coefficient checksums and the first violating triple if one exists.
1StatusSolution submittedNot Lean-verified
Saved packet · July 24, 2026
Saved solution (The truncated VT zero-syndrome weight sequence is log-concave through length 2000). Exact integer evaluation of the published coefficient formula verifies all 1,991,009 required inequalities.[1]
Verification
Let \[ V_n=\left\{x\in\{0,1\}^n:\sum_{i=1}^n i x_i\equiv0\pmod{n+1}\right\} \] and let \(W_{n,k}\) count its words of Hamming weight \(k\). For every \(7\leq n\leq2000\), the finite sequence \[ W_{n,2},W_{n,3},\ldots,W_{n,n-2} \] is log-concave. Equivalently, \[ W_{n,k}^2\geq W_{n,k-1}W_{n,k+1} \] for every \(3\leq k\leq n-3\).
The exhaustive sweep checks 1,991,009 inequalities with exact integers. Equality occurs only at \((n,k)=(10,3)\) and \((10,7)\). The corresponding triples are \((5,10,20)\) and \((20,10,5)\). The smallest positive margin is 1 at \((n,k)=(7,3)\), where the triple is \((3,4,5)\).
The lower endpoint is sharp for this finite statement. At \(n=6\), the internal triple is \((W_{6,2},W_{6,3},W_{6,4})=(3,2,3)\), whose log-concavity margin is \(-5\).
1Packet records
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Notes and companion material
The target is a finite coefficient sweep. Length 6 has internal sequence 3,2,3, while every length 7 through 80 passed.
Original intake status. Status unknown. Weight enumerators of VT codes have explicit character formulas, so this inequality may be derivable or known.
- The full weight sequence has forced zeros at weights 1 and n-1, so the internal truncation is essential.
- Root-of-unity formulas introduce cancellation. A direct injection proving each quadratic inequality may preserve more positivity than Fourier expansion.
Recorded example 1. For n=10 the internal sequence is 5,10,20,22,20,10,5.
Computational notes
- An exact dynamic program indexed by residue modulo n+1 and weight checked every 7 <= n <= 80. No log-concavity violation occurred. The same code reproduced the exceptional n=6 sequence 3,2,3.
How the 4 records connect
ProblemVT zero-syndrome weight log-concavity through length 2000
- Computation 1The truncated VT zero-syndrome weight sequence is log-concave through length 2000in this packetReproduced
- Proposition 1A divisor sum gives every zero-syndrome weight coefficientsupportsSupported
- Artifact 1Replayable exact coefficient sweep through length 2000usesExecutable material
- Route 1Primary sources supply the coefficient formulainformsSupported
All 4 recorded relations between these records and the problem
- A divisor sum gives every zero-syndrome weight coefficient supports The truncated VT zero-syndrome weight sequence is log-concave through length 2000
- Replayable exact coefficient sweep through length 2000 reproduces The truncated VT zero-syndrome weight sequence is log-concave through length 2000
- Replayable exact coefficient sweep through length 2000 uses A divisor sum gives every zero-syndrome weight coefficient
- Primary sources supply the coefficient formula informs The truncated VT zero-syndrome weight sequence is log-concave through length 2000
2See also
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Cite this problem statement
Cite the original sources separately.
“VT zero-syndrome weight log-concavity through length 2000.” TheoremDB. P2578. Problem statement; statement text SHA-256 6a691f052ccfa26ba7f9862ce99ad5fa1f4a3a5d0a35c4dac3451eca27a07acf. https://theoremdb.org/statement/?ref=P2578
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title = {{VT zero-syndrome weight log-concavity through length 2000}},
howpublished = {TheoremDB},
note = {Problem statement; statement text SHA-256 6a691f052ccfa26ba7f9862ce99ad5fa1f4a3a5d0a35c4dac3451eca27a07acf},
url = {https://theoremdb.org/statement/?ref=P2578}
}Plain text: Built Markdown snapshot
This problem includes 4 records joined by 4 typed links, sourced from doi.org[1], current as of July 24, 2026.
1Lean verification
What the formal work covers
Verification history
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1References
- Packet source. Reference record for “vt-zero-weight-log-concavity,” doi.org, checked 2026-08-01. Packet-wide source named in the dataset provenance. ↗journal article · reference source · checked 2026-08-01Source use: citation only.Supplies the source context used to assemble the packet for “vt zero weight log concavity.”Also cited at Exact formula evaluation and certificate in vt0-artifact-exact-coefficient-sweep.Also cited at Khodakhast Bibak and Olgica Milenkovic, Explicit Formulas for the Weight Enumerators of Some Classes of Deletion Correcting Codes, IEEE Transactions on Communications 67(3) (2019), 1809-1816, Theorems IV.7 and IV.8; Khodakhast Bibak, Bruce M. Kapron, and Venkatesh Srinivasan, Unweighted Linear Congruences with Distinct Coordinates and the Varshamov-Tenengolts Codes, Designs, Codes and Cryptography 86 (2018), 1893-1904, Corollary 3.12, DOI 10.1007/s10623-017-0428-3.Source named by the research packet.
- Reference record for “Replayable exact coefficient sweep through length 2000,” arxiv.org, checked 2026-08-01. Inline Python 3 exact computation executed on 2026-07-24. ↗preprint · reference source · arXiv:1812.09627, checked 2026-08-01 · checked 2026-08-01Source use: citation only.Documents the claim, method, computation, or status recorded as “Replayable exact coefficient sweep through length 2000” in this packet.
- Reference record for “Primary sources supply the coefficient formula,” doi.org, checked 2026-08-01. Khodakhast Bibak, Bruce M. Kapron, and Venkatesh Srinivasan, Unweighted Linear Congruences with Distinct Coordinates and the Varshamov-Tenengolts Codes, Designs, Codes and Cryptography 86 (2018), 1893-1904; Khodakhast Bibak and Olgica Milenkovic, Explicit Formulas for the Weight Enumerators of Some Classes of Deletion Correcting Codes, IEEE Transactions on Communications 67(3) (2019), 1809-1816. ↗journal article · reference source · checked 2026-08-01Source use: citation only.Documents the claim, method, computation, or status recorded as “Primary sources supply the coefficient formula” in this packet.
Original coefficient inequality candidate for a classical code family.
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