Asymptotically good codes from Siegel modular varieties

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.