TY - GEN

T1 - Constructing small-bias sets from algebraic-geometric codes

AU - Ben-Aroya, Avraham

AU - Ta-Shma, Amnon

PY - 2009

Y1 - 2009

N2 - We give an explicit construction of an ∈-biased set over k bits of size O (k/∈2log(1/∈))5/4. This improves upon previous explicit constructions when ∈ 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 ∈-biased sets with parameters nearly matching the lower bound, and in particular giving binary error correcting codes beating the Gilbert-Varshamov bound.

AB - We give an explicit construction of an ∈-biased set over k bits of size O (k/∈2log(1/∈))5/4. This improves upon previous explicit constructions when ∈ 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 ∈-biased sets with parameters nearly matching the lower bound, and in particular giving binary error correcting codes beating the Gilbert-Varshamov bound.

KW - Algebraic-geometric codes

KW - Small-bias sets

UR - http://www.scopus.com/inward/record.url?scp=77952341530&partnerID=8YFLogxK

U2 - 10.1109/FOCS.2009.44

DO - 10.1109/FOCS.2009.44

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AN - SCOPUS:77952341530

SN - 9780769538501

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 191

EP - 197

BT - Proceedings - 50th Annual Symposium on Foundations of Computer Science, FOCS 2009

T2 - 50th Annual Symposium on Foundations of Computer Science, FOCS 2009

Y2 - 25 October 2009 through 27 October 2009

ER -