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The dual first-order generalized Reed–Muller code 3-design conjecture

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Let qq be a prime power, let mm be a positive integer, and let C=RMq(1,m)C=RM_q(1,m) be the first-order generalized Reed–Muller code. For ue0\boldsymbol{u} e 0, write Cℓ⊥C^{\perp}_{\ell} for the shell of the dual code consisting of codewords of Hamming weight ℓ\ell.

Dual Reed–Muller 3-design conjecture. If q≥3q\geq 3, then for every ℓ∈N\ell\in\mathbb{N}, (C⊥)ℓ(C^{\perp})_{\ell} is not a combinatorial 33-design.

The preceding calculation gives evidence for this claim by expressing the difference of the relevant Jacobi polynomials for two triples as a nonzero multiple of (wy−xz)3(wy-xz)^3. The conjecture concerns all weight shells of the dual code and remains unresolved in the supplied source.

References

Primary source

Ryosuke Yamaguchi, “Jacobi polynomials for the first-order generalized Reed–Muller codes”, arXiv:2309.01119 (2024).

coding-theorydesign-theorygeneralized-reed-muller-codes
Posted

Progress summary

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Claimed solved

A published 2024 paper proves that every nonempty dual shell fails to be a three-design for all prime powers at least three.

The conjecture asserts that, for C=RMq(1,m)C=RM_q(1,m) with q≥3q\geq 3, no nonempty shell (C⊥)ℓ(C^\perp)_\ell is a combinatorial 33-design. A 2023 source presented this as an unresolved conjecture.

Known results

  • Computational checks covered parameters with q2m<109q^{2m}<10^9 (2023).
  • The primal shells CℓC_\ell were proved not to be combinatorial 33-designs for q≥3q\geq 3 and m≥2m\geq 2, while relevant shells are 22-designs (2023).

2024 published proof

A Springer paper proves the dual assertion for every nonempty shell. Its Jacobi-polynomial difference factors as a nonzero multiple of (wy−xz)3(wy-xz)^3, establishing failure of the 33-design condition; no contrary result or standing objection was found.

Current status (as of August 2026): The conjecture is settled affirmatively by the published 2024 proof for all q≥3q\geq 3 and all nonempty dual shells.

Sources

Solutions 0

No solutions have been posted yet.