Bias

Let ${\displaystyle X}$ be a probability distribution over ${\displaystyle \{0,1\}^{n}}$. The bias of ${\displaystyle X}$ with respect to a set of indices ${\displaystyle I\subseteq \{1,\dots ,n\}}$ is defined as^[1]

${\displaystyle {\text{bias}}_{I}(X)=\left|\Pr _{x\sim X}\left(\sum _{i\in I}x_{i}=0\right)-\Pr _{x\sim X}\left(\sum _{i\in I}x_{i}=1\right)\right|=\left|2\cdot \Pr _{x\sim X}\left(\sum _{i\in I}x_{i}=0\right)-1\right|\,,}$

where the sum is taken over ${\displaystyle \mathbb {F} _{2}}$, the finite field with two elements. In other words, the sum ${\displaystyle \sum _{i\in I}x_{i}}$ equals ${\displaystyle 0}$ if the number of ones in the sample ${\displaystyle x\in \{0,1\}^{n}}$ at the positions defined by ${\displaystyle I}$ is even, and otherwise, the sum equals ${\displaystyle 1}$. For ${\displaystyle I=\emptyset }$, the empty sum is defined to be zero, and hence ${\displaystyle {\text{bias}}_{\emptyset }(X)=1}$.

ϵ-biased sample space

A probability distribution ${\displaystyle X}$ over ${\displaystyle \{0,1\}^{n}}$ is called an ${\displaystyle \epsilon }$-biased sample space if ${\displaystyle {\text{bias}}_{I}(X)\leq \epsilon }$ holds for all non-empty subsets ${\displaystyle I\subseteq \{1,2,\ldots ,n\}}$.

ϵ-biased set

An ${\displaystyle \epsilon }$-biased sample space ${\displaystyle X}$ that is generated by picking a uniform element from a multiset ${\displaystyle X\subseteq \{0,1\}^{n}}$ is called ${\displaystyle \epsilon }$-biased set. The size ${\displaystyle s}$ of an ${\displaystyle \epsilon }$-biased set ${\displaystyle X}$ is the size of the multiset that generates the sample space.

ϵ-biased generator

An ${\displaystyle \epsilon }$-biased generator ${\displaystyle G:\{0,1\}^{\ell }\to \{0,1\}^{n}}$ is a function that maps strings of length ${\displaystyle \ell }$ to strings of length ${\displaystyle n}$ such that the multiset ${\displaystyle X_{G}=\{G(y)\;\vert \;y\in \{0,1\}^{\ell }\}}$ is an ${\displaystyle \epsilon }$-biased set. The seed length of the generator is the number ${\displaystyle \ell }$ and is related to the size of the ${\displaystyle \epsilon }$-biased set ${\displaystyle X_{G}}$ via the equation ${\displaystyle s=2^{\ell }}$.

Theoretical bounds

The probabilistic method gives a non-explicit construction that achieves size ${\displaystyle s=O(n/\epsilon ^{2})}$.^[2] The construction is non-explicit in the sense that finding the ${\displaystyle \epsilon }$-biased set requires a lot of true randomness, which does not help towards the goal of reducing the overall randomness. However, this non-explicit construction is useful because it shows that these efficient codes exist. On the other hand, the best known lower bound for the size of ${\displaystyle \epsilon }$-biased sets is ${\displaystyle s=\Omega (n/(\epsilon ^{2}\log(1/\epsilon ))}$, that is, in order for a set to be ${\displaystyle \epsilon }$-biased, it must be at least that big.^[2]

Explicit constructions

There are many explicit, i.e., deterministic constructions of ${\displaystyle \epsilon }$-biased sets with various parameter settings:

• Naor & Naor (1990) achieve ${\displaystyle \displaystyle s={\frac {n}{{\text{poly}}(\epsilon )}}}$. The construction makes use of Justesen codes (which is a concatenation of Reed–Solomon codes with the Wozencraft ensemble) as well as expander walk sampling.
• Alon et al. (1992) achieve ${\displaystyle \displaystyle s=O\left({\frac {n}{\epsilon \log(n/\epsilon )}}\right)^{2}}$. One of their constructions is the concatenation of Reed–Solomon codes with the Hadamard code; this concatenation turns out to be an ${\displaystyle \epsilon }$-balanced code, which gives rise to an ${\displaystyle \epsilon }$-biased sample space via the connection mentioned above.
• Concatenating Algebraic geometric codes with the Hadamard code gives an ${\displaystyle \epsilon }$-balanced code with ${\displaystyle \displaystyle s=O\left({\frac {n}{\epsilon ^{3}\log(1/\epsilon )}}\right)}$.^[2]
• Ben-Aroya & Ta-Shma (2009) achieves ${\displaystyle \displaystyle s=O\left({\frac {n}{\epsilon ^{2}\log(1/\epsilon )}}\right)^{5/4}}$.
• Ta-Shma (2017) achieves ${\displaystyle \displaystyle s=O\left({\frac {n}{\epsilon ^{2+o(1)}}}\right)}$ which is almost optimal because of the lower bound.

These bounds are mutually incomparable. In particular, none of these constructions yields the smallest ${\displaystyle \epsilon }$-biased sets for all settings of ${\displaystyle \epsilon }$ and ${\displaystyle n}$.

k-wise independent spaces

A random variable ${\displaystyle Y}$ over ${\displaystyle \{0,1\}^{n}}$ is a k-wise independent space if, for all index sets ${\displaystyle I\subseteq \{1,\dots ,n\}}$ of size ${\displaystyle k}$, the marginal distribution ${\displaystyle Y|_{I}}$ is exactly equal to the uniform distribution over ${\displaystyle \{0,1\}^{k}}$. That is, for all such ${\displaystyle I}$ and all strings ${\displaystyle z\in \{0,1\}^{k}}$, the distribution ${\displaystyle Y}$ satisfies ${\displaystyle \Pr _{Y}(Y|_{I}=z)=2^{-k}}$.

Almost k-wise independent spaces

A random variable ${\displaystyle Y}$ over ${\displaystyle \{0,1\}^{n}}$ is a ${\displaystyle \delta }$-almost k-wise independent space if, for all index sets ${\displaystyle I\subseteq \{1,\dots ,n\}}$ of size ${\displaystyle k}$, the restricted distribution ${\displaystyle Y|_{I}}$ and the uniform distribution ${\displaystyle U_{k}}$ on ${\displaystyle \{0,1\}^{k}}$ are ${\displaystyle \delta }$-close in 1-norm, i.e., ${\displaystyle {\Big \|}Y|_{I}-U_{k}{\Big \|}_{1}\leq \delta }$.