On Parameterized Gallager's First Bounds for Binary Linear Codes over AWGN Channels

TL;DR

This paper introduces a parameterized framework for Gallager's first bounds on binary linear codes over AWGN channels, providing conditions for optimal parameters and revisiting existing bounds with new insights.

Contribution

It develops a unified parameterized approach to Gallager's bounds, establishes conditions for optimal parameters, and clarifies relationships among existing bounds.

Findings

Derived necessary and sufficient conditions for optimal parameters.

Identified conditions where optimal parameters are independent of SNR.

Revealed equivalence between Herzberg-Poltyrev and Kasami bounds.

Abstract

In this paper, nested Gallager regions with a single parameter is introduced to exploit Gallager's first bounding technique (GFBT). We present a necessary and sufficient condition on the optimal parameter. We also present a sufficient condition (with a simple geometrical explanation) under which the optimal parameter does not depend on the signal-to-noise ratio (SNR). With this general framework, three existing upper bounds are revisited, including the tangential bound (TB) of Berlekamp, the sphere bound (SB) of Herzberg and Poltyrev, and the tangential-sphere bound (TSB) of Poltyrev. This paper also reveals that the SB of Herzberg and Poltyrev is equivalent to the SB of Kasami et al., which was rarely cited in literature.