math.NT — Number Theory

Second moments of pattern counts in dual Melas codewords

Contributed by David Niedbala Giraudin

Let $\alpha$ be a primitive element of $\mathbb{F}_{2^k}$ and, for $c\in\mathbb{F}_{2^k}^*$, let $S_c=(\mathrm{Tr}(\alpha^i+c\alpha^{-i}))_{0\le i\le 2^k-2}$ be the corresponding codeword of the dual Melas code. Let $X_{k,c}(w)$ be the number of occurrences of a uniformly random pattern $w\in\mathbb{F}_2^k$ in $S_c$. Sac Himelfarb and Schwartz conjectured that $X_{k,c_k}$ converges to the Poisson law $\mathcal P(1)$ for every choice of $c_k\in\mathbb{F}_{2^k}^*$; they proved that the second factorial moment satisfies $\limsup M_2\le 2$, and $M_2\to1$ for almost every sequence $(c_k)$. We prove that $|M_2^{(k,c)}-1|\le 2^{-(1/22+o(1))k}$ uniformly in $c\in\mathbb{F}_{2^k}^*$ and in the primitive element $\alpha$. Thus the second factorial moment converges to $1$ for every sequence $(c_k)$. The proof expresses $M_2$ through a norm form of the quadratic extension $\mathbb{F}_2(x)/\mathbb{F}_2(x+x^{-1})$, which turns it into a bilinear Kloosterman sum of full length, and bounds the latter in characteristic two. We also show that the counts $N_w(S_c)$, as functions of $(w,c)$, are the fibre sizes of a family of quadratic maps, and deduce that for almost every sequence of patterns $w_k$ the first two factorial moments of $c\mapsto N_{w_k}(S_c)$ converge to $1$, 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).