Definition

We first introduce a base-B representation with B = 2, 3, 4, .... Throughout this guide, quantics digits and grid indices are 1-based.

We represent a positive integer X >= 1 as

$$ X = \sum_{i=1}^{R} (x_i - 1) B^{R-i} + 1, $$

where each digit x_i satisfies 1 <= x_i <= B and R is the number of digits. In this crate, the base-B representation of X is stored as the vector

$$ [x_1, \ldots, x_R]. $$

For multiple variables, the crate supports fused and interleaved unfolding schemes. For three variables X, Y, and Z, suppose their base-B representations are

$$ [x_1, \ldots, x_R], \quad [y_1, \ldots, y_R], \quad [z_1, \ldots, z_R]. $$

The interleaved representation is

$$ [x_1, y_1, z_1, x_2, y_2, z_2, \ldots, x_R, y_R, z_R]. $$

The fused representation is

$$ [\alpha_1, \alpha_2, \ldots, \alpha_R], $$

where

$$ \alpha_i = (x_i - 1) + B (y_i - 1) + B^2 (z_i - 1) + 1 $$

and therefore

$$ 1 \le \alpha_i \le B^3. $$

In fused ordering, the x digit runs fastest at each digit level. This matches the convention used by the Julia package and generalizes to any number of variables.