The Hermitian Trace Code
summary
Proves a concrete bound: for any non-square low‑degree function f in the Hermitian function field (degree d odd) the quadratic character sum over the p^3 rational points is O(d·p), giving a nontrivial bias bound where previous Stepanov/Bombieri methods fail for high genus. Technique sharpens multiplicity control in function fields via a new universal derivative-fix lemma, uses a Stepanov-style polynomial method, and extends to the first levels of the Hermitian tower, with coding implications toward binary constructions near GV regimes.
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chapters (10)
- 0:00Hermitian trace code intro
- 1:54Lower bounds on binary codes
- 6:50Reed-Solomon with trace
- 13:47Introducing Hermitian trace
- 20:41Hermitian code properties
- 29:26Character sums on Hermitian curve
- 39:29Stepanov’s method on Hermitian
- 46:00Derivatives in function fields
- 51:00Separating elements lemma
- 52:27Future directions
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