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Second moments of pattern counts in dual Melas codewords

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theorycs.IT — Information Theory

Contributed by David Niedbala Giraudin ↗

Version 1 / Nov 10, 2026 / CC BY 4.0

Abstract

Let α\alpha be a primitive element of F2k\mathbb{F}_{2^k} and, for c∈F2k∗c\in\mathbb{F}_{2^k}^*, let Sc=(Tr(αi+cα−i))0≤i≤2k−2S_c=(\mathrm{Tr}(\alpha^i+c\alpha^{-i}))_{0\le i\le 2^k-2} be the corresponding codeword of the dual Melas code. Let Xk,c(w)X_{k,c}(w) be the number of occurrences of a uniformly random pattern w∈F2kw\in\mathbb{F}_2^k in ScS_c. Sac Himelfarb and Schwartz conjectured that Xk,ckX_{k,c_k} converges to the Poisson law P(1)\mathcal P(1) for every choice of ck∈F2k∗c_k\in\mathbb{F}_{2^k}^*; they proved that the second factorial moment satisfies lim sup⁡M2≤2\limsup M_2\le 2, and M2→1M_2\to1 for almost every sequence (ck)(c_k). We prove that ∣M2(k,c)−1∣≤2−(1/22+o(1))k|M_2^{(k,c)}-1|\le 2^{-(1/22+o(1))k} uniformly in c∈F2k∗c\in\mathbb{F}_{2^k}^* and in the primitive element α\alpha. Thus the second factorial moment converges to 11 for every sequence (ck)(c_k). The proof expresses M2M_2 through a norm form of the quadratic extension F2(x)/F2(x+x−1)\mathbb{F}_2(x)/\mathbb{F}_2(x+x^{-1}), which turns it into a bilinear Kloosterman sum of full length, and bounds the latter in characteristic two. We also show that the counts Nw(Sc)N_w(S_c), as functions of (w,c)(w,c), are the fibre sizes of a family of quadratic maps, and deduce that for almost every sequence of patterns wkw_k the first two factorial moments of c↦Nwk(Sc)c\mapsto N_{w_k}(S_c) converge to 11, as they would under the Poisson law conjectured by Sac Himelfarb and Schwartz for fixed patterns.

Author comments: 7 pages. Proves, uniformly in c, the second-moment case of Conjecture 35 of Sac Himelfarb-Schwartz (arXiv:2610.38017), and the first two moments of their fixed-pattern conjecture for almost every pattern. Ancillary verifier VERIF_MELAS_M2.py (pure Python).

Provenance statement

The research and the writing were carried out by the author with Claude (Anthropic). The proofs were checked by an independent AI review and every identity used was verified by exact computation. No professional mathematician has reviewed the note yet.

Tools used

Anthropic
Claude Opus

References

  1. C. Bagshaw, Bilinear forms with Kloosterman and Gauss sums in function fields, Finite Fields Appl. 94 (2024), 102356; arXiv:2304.05014.arXiv
  2. C. Bagshaw, Bilinear Kloosterman sums in function fields and the distribution of irreducible polynomials, Canad. J. Math. (2024); arXiv:2401.10399.arXiv
  3. C. Bagshaw and I. E. Shparlinski, Energy bounds, bilinear forms and their applications in function fields, Finite Fields Appl. 82 (2022), 102048.
  4. J. Bourgain and M. Z. Garaev, Kloosterman sums in residue rings, Acta Arith. 164 (2014), 43--64.
  5. G. Sac Himelfarb and M. Schwartz, On the bb-symbol weight distribution of dual Melas codes and a Poisson randomness postulate, arXiv:2610.38017.arXiv
  6. A. Weil, On some exponential sums, Proc. Nat. Acad. Sci. USA 34 (1948), 204--207.

Version history

  1. v1 · 7 pagesSubmitted by David Niedbala Giraudin7 pages. Proves, uniformly in c, the second-moment case of Conjecture 35 of Sac Himelfarb-Schwartz (arXiv:2610.38017), and the first two moments of their fixed-pattern conjecture for almost every pattern. Ancillary verifier VERIF_MELAS_M2.py (pure Python).CurrentNov 09, 2026