The averaged enumerator of B-form codes

Abstract

For every prime p, every c and every function t on Z/p with t(-a) = t(a), the full enumerator polynomial of the code with generating matrix (I | B^T), evaluated at x_ab = t_a t_b and averaged over the p^(c(c-1)/2) antisymmetric matrices B with zero diagonal, equals the cosine formula (barP) conjectured by N. Angelinos, D. Chakraborty and A. Dymarsky (arXiv:2206.14825).

Definition 1.1 (B-form matrices).

$$\mathrm{IsBForm}\left(B\right)\iff ((B^T=-B)\wedge (\forall i\in \mathrm{Fin}\left(c\right),B_{ii}=0))$$

Formalization. D5/S3/Quantum/Information/BFormCodeAveragedEnumerator.IsBForm (✓ std3).

Citation. Nikolaos Angelinos; Debarghya Chakraborty; Anatoly Dymarsky (2022). Optimal Narain CFTs from codes. DOI: 10.48550/arXiv.2206.14825. URL: https://arxiv.org/abs/2206.14825v1.

Commentary.

A c x c matrix B over Z/p is of B-form when it is antisymmetric and has zero diagonal; the code it generates has generating matrix (I | B^T).

Definition 1.2 (The full enumerator at x_ab = t_a t_b).

$$\mathrm{enumerator}\left(t,B\right)=\sum _{r\in (\mathbb{Z}/p{)}^c}\prod _{i}t\left(r_i\right)\cdot t\left((B^{T}r{)}_i\right)$$

Formalization. D5/S3/Quantum/Information/BFormCodeAveragedEnumerator.enumerator (✓ std3).

Citation. Nikolaos Angelinos; Debarghya Chakraborty; Anatoly Dymarsky (2022). Optimal Narain CFTs from codes. DOI: 10.48550/arXiv.2206.14825. URL: https://arxiv.org/abs/2206.14825v1.

Commentary.

The codewords of the code of B are the pairs (r, B^T r) with r in (Z/p)^c. Its full enumerator is the sum over the codewords of the product over i of x_(r_i, (B^T r)_i); the paper evaluates it at x_ab = t_a t_b.

Definition 1.3 (The conjectured average).

$$claim\iff (\forall p\in \mathbb{N},(\mathrm{Prime}\left(p\right))\Rightarrow \left(\forall c\in \mathbb{N},\forall t\in \mathbb{Z}/p\to \mathbb{C},(\forall a\in \mathbb{Z}/p,t\left(-a\right)=t\left(a\right))\Rightarrow \frac{{\sum }_{B\in \mathrm{Mat}\left(c,\mathbb{Z}/p\right),\mathrm{IsBForm}\left(B\right)}\mathrm{enumerator}\left(t,B\right)}{p^{\mathrm{natDiv}\left(c\cdot (c-1),2\right)}}=t{\left(0\right)}^{2\cdot c}+\frac{{\sum }_{k\in \mathbb{Z}/p}({\sum }_{a\in \mathbb{Z}/p}{\sum }_{b\in \mathbb{Z}/p}\cos \left(\frac{2\pi \cdot \mathrm{val}\left(k\right)\cdot \mathrm{val}\left(a\right)\cdot \mathrm{val}\left(b\right)}{p}\right)\cdot t\left(a\right)\cdot t\left(b\right){)}^c-p\cdot t{\left(0\right)}^c\cdot ({\sum }_{a\in \mathbb{Z}/p}t\left(a\right){)}^c}{p^c}\right))$$

Formalization. D5/S3/Quantum/Information/BFormCodeAveragedEnumerator.claim (✓ std3).

Citation. Nikolaos Angelinos; Debarghya Chakraborty; Anatoly Dymarsky (2022). Optimal Narain CFTs from codes. DOI: 10.48550/arXiv.2206.14825. URL: https://arxiv.org/abs/2206.14825v1.

Commentary.

Eq. (barP) of the paper: the average of the enumerator over all B-form matrices, for every prime p, every c and every t with t(-a) = t(a). In the cosine, k, a and b are read as their representatives 0, …, p - 1. In the exponent c(c - 1) is a natural number, natDiv is division of natural numbers rounded down and c - 1 is subtraction of natural numbers (0 at c = 0); c(c - 1) is even, so natDiv(c(c - 1), 2) = c(c - 1)/2 is the number of entries above the diagonal, and there are p^(c(c-1)/2) B-form matrices.

Theorem 1.4 (Proof of the averaged formula).

$$claim$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Information/BFormCodeAveragedEnumerator.result (✓ std3). ∎

Resolves. Problems/angelinos-2022-bform-averaged-enumerator (proved) by D5/S3/Quantum/Information/BFormCodeAveragedEnumerator.result.

Source. Repository-derived.

Acknowledgement. Nikolaos Angelinos; Debarghya Chakraborty; Anatoly Dymarsky (2022). Optimal Narain CFTs from codes. DOI: 10.48550/arXiv.2206.14825. URL: https://arxiv.org/abs/2206.14825v1.

Commentary.

Exchange the sums over B and r. The term r = 0 gives t_0^(2c) for every B. For r nonzero, antisymmetry with zero diagonal gives r . B^T r = 0, and the linear map B -> B^T r is onto the hyperplane orthogonal to r: if r_j is nonzero, a matrix supported on row and column j reaches any s orthogonal to r. So each such s has the same number of preimages, and the sum over B equals p^(c(c-1)/2) / p^(c-1) times the sum over s orthogonal to r. The constraint r . s = 0 is written as p^(-1) times the sum over k of the standard additive character psi(k r . s); the sum over r and s then factors into the c-th power of the sum over a and b of psi(kab) t_a t_b, and the row r = 0 contributes p t_0^c (sum of t_a)^c. Since t is even, replacing (a, b) by (-a, b) turns psi(kab) into its complex conjugate, so the character sum equals the cosine sum. Counting the B-form matrices as p^(c(c-1)/2) gives the formula.

References

• Truth anchor: D5/S3/Quantum/Information/BFormCodeAveragedEnumerator.IsBForm
• Truth anchor: D5/S3/Quantum/Information/BFormCodeAveragedEnumerator.claim
• Truth anchor: D5/S3/Quantum/Information/BFormCodeAveragedEnumerator.enumerator
• Truth anchor: D5/S3/Quantum/Information/BFormCodeAveragedEnumerator.result