Pattern Recognition, 4th Edition

Discrete-Valued Vectors

We will now consider vectors x whose coordinates belong to the finite set F = {0, 1,…, k − 1}, where k is a positive integer. It is clear that there are exactly k^l vectors x ∈ F^l. One can imagine these vectors as vertices in an l-dimensional grid as depicted in Figure 11.4. When k = 2, the grid collapses to the H^l (unit) hypercube.

FIGURE 11.4. (a) The l = 2 dimensional grid for k = 4. (b) The H² hypercube (square).

Consider x, y ∈ F^l and let(11.25)be a k × k matrix, where the element a^ij is the number of places where ...