bibkey: angelinos2022narain authors: Nikolaos Angelinos; Debarghya Chakraborty; Anatoly Dymarsky year: 2022 title: “Optimal Narain CFTs from codes” doi: 10.48550/arXiv.2206.14825 url: https://arxiv.org/abs/2206.14825v1 claim: “For prime p, the full enumerator polynomial of the code with generating matrix (I | B^T), evaluated at x_ab = t_a t_b with t_a = t_-a and averaged over the p^(c(c-1)/2) antisymmetric matrices B with zero diagonal, is conjectured to equal t_0^(2c) + (sum_k (sum_(a,b) cos(2 pi k a b / p) t_a t_b)^c - p t_0^c (sum_a t_a)^c) / p^c (eq. barP).” strata_touched:

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Optimal Narain CFTs from codes

N. Angelinos, D. Chakraborty and A. Dymarsky, arXiv:2206.14825 (v1 2022-06-29, the only version); JHEP 11 (2022) 118. Subject: hep-th.

The paper builds Narain conformal field theories from codes over F_p × F_p. For prime p a code of length c has a generating matrix that “can always be brought to the form G = (I | B^T), where B is an integer valued antisymmetric c × c matrix defined mod p, B^T = −B mod p, and B_ii = 0”. Its codewords are (r, B^T r) with r ∈ Z_p^c, and its full enumerator polynomial is P_C({x_g}) = Σ_{(g_1, …, g_c) ∈ C} ∏_i x_{g_i} with g_i = (a_i, b_i). Averaging over the p^{c(c−1)/2} matrices B at x_ab = t_a t_b, t_a = t_{−a}, the authors write:

We conjecture the form of corresponding averaged enumerator polynomial based on invariance under MacWilliams identity and explicit checks for sufficiently small n and prime p

followed by eq. (barP): P̄({t_a t_b}) = t_0^{2c} + (Σ_{k=0}^{p−1} (Σ_{a,b} cos(2πkab/p) t_a t_b)^c − p t_0^c (Σ_a t_a)^c) / p^c (the printed upper limit p=1 is read as p − 1).

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2206.14825
• URL: https://arxiv.org/abs/2206.14825v1 (source of v1 retrieved 2026-09-30).
• Location: the section on codes over F_p × F_p for prime p, for the B-form of the generating matrix, the enumerator polynomial and eq. (barP).
• Published version: https://doi.org/10.1007/JHEP11(2022)118 (open-access PDF retrieved 2026-09-30); it states the same conjecture with the same formula as eq. (5.5), including the printed upper limit p=1.