Determining the number of clusters in a data set

In theoretical computer science and coding theory, the long code is an error-correcting code that is locally decodable. Long codes have an extremely poor rate, but play a fundamental role in the theory of hardness of approximation.

Definition

Let $f_1,\dots ,f_{2^n}:\{0,1{\}}^k\to \{0,1\}$ for $k=\log n$ be the list of all functions from $\{0,1{\}}^k\to \{0,1\}$. Then the long code encoding of a message $x\in \{0,1{\}}^k$ is the string $f_1(x)\circ f_2(x)\circ \dots \circ f_{2^n}(x)$ where $\circ$ denotes concatenation of strings. This string has length $2^n=2^{2^k}$.

The Walsh-Hadamard code is a subcode of the long code, and can be obtained by only using functions $f_i$ that are linear functions when interpreted as functions ${\mathbb{𝔽}}_2^k\to {\mathbb{𝔽}}_2$ on the finite field with two elements. Since there are only $2^k$ such functions, the block length of the Walsh-Hadamard code is $2^k$.

An equivalent definition of the long code is as follows: The Long code encoding of $j\in [n]$ is defined to be the truth table of the Boolean dictatorship function on the $j$th coordinate, i.e., the truth table of $f:\{0,1{\}}^n\to \{0,1\}$ with $f(x_1,\dots ,x_n)=x_i$. Thus, the Long code encodes a $(\log n)$-bit string as a $2^n$-bit string.

Properties

The long code does not contain repetitions, in the sense that the function $f_i$ computing the $i$th bit of the output is different from any function $f_j$ computing the $j$th bit of the output for $j\ne i$. Among all codes that do not contain repetitions, the long code has the longest possible output. Moreover, it contains all non-repeating codes as a subcode.

References

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