A new method for constructing small-bias spaces from hermitian codes
WAIFI'12 Proceedings of the 4th international conference on Arithmetic of Finite Fields
A new family of locally correctable codes based on degree-lifted algebraic geometry codes
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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We give an explicit construction of an $\eps$-biased set over $k$ bits of size $O(\frac{k}{\eps^2 \log(1/\eps)})^{5/4}$. This improves upon previous explicit constructions when $\eps$ is roughly (ignoring logarithmic factors) in the range $[k^{-1.5}, k^{-0.5}]$. The construction builds on an algebraic-geometric code. However, unlike previous constructions we use low-degree divisors whose degree is significantly smaller than the genus. Studying the limits of our technique, we arrive at a hypothesis that if true implies the existence of $\eps$-biased sets with parameters nearly matching the lower bound, and in particular giving binary error correcting codes beating the Gilbert-Varshamov bound.