Uniformity Properties of Construction C

TL;DR

This paper investigates the geometric uniformity of Construction C, a multi-level coding scheme, revealing that two-level constructions are geometrically uniform while higher levels are generally not.

Contribution

It proves that two-level Construction C is geometrically uniform and provides counterexamples showing higher levels lack this property.

Findings

Two-level Construction C is geometrically uniform.

Higher levels ($L extgreater 2$) are not geometrically uniform.

Distance spectrum varies in multi-level constructions with $L extgreater 2$.

Abstract

Construction C (also known as Forney's multi-level code formula) forms a Euclidean code for the additive white Gaussian noise (AWGN) channel from $L$ binary code components. If the component codes are linear, then the minimum distance is the same for all the points, although the kissing number may vary. In fact, while in the single level ($L=1$) case it reduces to lattice Construction A, a multi-level Construction C is in general not a lattice. We show that the two-level ($L=2$) case is special: a two-level Construction C satisfies Forney's definition for a geometrically uniform constellation. Specifically, every point sees the same configuration of neighbors, up to a reflection of the coordinates in which the lower level code is equal to 1. In contrast, for three levels and up ($L\geq 3$), we construct examples where the distance spectrum varies between the points, hence the constellation is not geometrically uniform.