Upper Bounds On the ML Decoding Error Probability of General Codes over AWGN Channels

TL;DR

This paper introduces a generalized parameterized Gallager bounding technique to derive upper bounds on ML decoding error probabilities for general codes over AWGN channels, extending classical bounds to broader code classes.

Contribution

The paper generalizes three classical bounds to non-geometrically uniform codes and reveals equivalences among existing bounds, broadening their applicability.

Findings

Unified bounds for general codes over AWGN channels.

Revealed equivalence between Herzberg-Poltyrev and Kasami bounds.

Extended classical bounds to non-geometrically uniform codes.

Abstract

In this paper, parameterized Gallager's first bounding technique (GFBT) is presented by introducing nested Gallager regions, to derive upper bounds on the ML decoding error probability of general codes over AWGN channels. The three well-known bounds, namely, the sphere bound (SB) of Herzberg and Poltyrev, the tangential bound (TB) of Berlekamp, and the tangential-sphere bound (TSB) of Poltyrev, are generalized to general codes without the properties of geometrical uniformity and equal energy. When applied to the binary linear codes, the three generalized bounds are reduced to the conventional ones. The new derivation also reveals that the SB of Herzberg and Poltyrev is equivalent to the SB of Kasami et al., which was rarely cited in the literatures.