Hierarchies of Local-Optimality Characterizations in Decoding of Tanner Codes

TL;DR

This paper introduces hierarchical local-optimality conditions in Tanner code decoding, linking local code properties and iteration counts to improve understanding of decoding certificates and their inclusion properties.

Contribution

It defines hierarchies of local-optimality based on local code distance and iteration count, revealing inclusion relations and implications for decoding certificates.

Findings

Hierarchies satisfy inclusion properties as parameters increase.

Decoding certificates after h iterations imply certificates after k·h iterations.

Provides a structured understanding of local-optimality in Tanner code decoding.

Abstract

Recent developments in decoding of Tanner codes with maximum-likelihood certificates are based on a sufficient condition called local-optimality. We define hierarchies of locally-optimal codewords with respect to two parameters. One parameter is related to the minimum distance of the local codes in Tanner codes. The second parameter is related to the finite number of iterations used in iterative decoding. We show that these hierarchies satisfy inclusion properties as these parameters are increased. In particular, this implies that a codeword that is decoded with a certificate using an iterative decoder after $h$ iterations is decoded with a certificate after $k\cdot h$ iterations, for every integer $k$.