Abstract

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.

Highlights

  • Constructing combinatorial objects with certain properties is an intriguing challenge in computer science

  • It is easy to verify that a random object satisfies the required property with high probability, while it is difficult to pin down such an explicit object

  • Perhaps the most remarkable example of this type is that of algebraic-geometric codes (AG codes)

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Summary

Introduction

Constructing combinatorial objects with certain properties (such as expander graphs, extractors, error correcting codes and others) is an intriguing challenge in computer science. Finding an explicit construction that attains this bound is an open problem Another closely related question is that of finding an [n, k, (1/2 − ε)n]2 binary code, in which the relative weight of every non-zero codeword is in the range [1/2 − ε, 1/2 + ε]. When the “degree” of G is larger than the genus g of the function field F (again, defined in Section 3) the Riemann-Roch Theorem [8, Thm I.5.17] tells us exactly what the dimension dim(G) is, and it turns out that dim(G) = deg(G) − g + 1 This almost matches the Singleton bound, dim(G) ≤ deg(G) + 1, except for a loss of g.

A self-contained elementary description of the construction
Algebraic geometry
Geometric Goppa codes
Concatenating AG codes with Hadamard
The construction
Limits of the approach
The bound
An open problem
Full Text
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