Asymptotically good codes from Siegel modular varieties
Year: 2026
Type: preprint
Abstract: We construct asymptotically good codes by evaluating Siegel modular forms at superspecial points on congruence towers of Siegel modular varieties. For every genus $g\ge2$ and every fixed weight $w\ge g+1$, the construction gives such a family in sufficiently large characteristic, over a finite field which is independent of the level of the tower. The towers can be chosen so that this field is $\mathbb{F}_{p^2}$. We give an explicit lower bound for the asymptotic rate. The lower bound for the relative minimum distance follows from the Pieri formula on the space of flags in the Hodge bundle and an intersection argument along its strata. We also prove that, for $p\ge5$ and $0\le w\le p+1$, the kernel of evaluation in every genus consists precisely of the multiples of the ordinary Hasse invariant. For $g=2,3$, the geometry of the supersingular locus gives explicit bounds for the relative minimum distance and recovery sets whose sizes are bounded independently of the level.
Source: arXiv (Cornell University)
Author: Maeda, Yota
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225832754
Cites:
Cited by:
Related to: